Calculus is a complex branch in mathematics that deals with the study of change. Knowledge of how functions change over certain periods or conditions can only be answered with methods derived from calculus. There are 3 core topics which comprise calculus: Limits, Derivation and Integration. Limits describe how functions approach values. Derivation describes the rate of change at certain values of a function. Integration describes the area under the curve of a function. The beauty in calculus lies in how these 3 core topics relate to one another.
\"},\"riemann-sums\":{\"title\":\"Riemann Sums\",\"sections\":[{\"text\":\"...\"}]},\"proof-of-the-fundamental-theorem-of-calculus-part-i\":{\"title\":\"Proof of the Fundamental Theorem of Calculus Part I\",\"sections\":[\"
Proof of the Fundamental Theorem of Calculus Part I
Alexander De Carlo
Proving the fundamental theorem of calculus consists of two parts. Interestingly, the 'first part' was proved secondly, and the 'second part' was proved first. However, it makes much more sense for any calculus student to learn the first part's proof, then the second one, despite the order that they were proven. So let's jump right into the first part!
The first part of the fundamental theorem of calculus connects the idea of a derivative and an integral. It states that the derivative of an integral returns the original function. In mathematical notation: \\\\[\\\\frac{d}{dx}\\\\left[\\\\int f(x) \\\\, dx\\\\right] = f(x)\\\\]This can also be used to say that the integral of f(x) (area under the curve) is just the antiderivative of f(x), or in simpler terms, a function g(x) where it's derivative of f(x). As we can see, proving this theorem will give us a gateway into actually solving integrals, one of the hardest tasks in calculus.
But why would we even consider a relationship in the first place? Well, let's consider a very general function, f(x).
Observe f(x) and A(x), where A(x) is the area function. Basically, we are assuming that the area under any curve can be considered some function of x. As we see with the blue line to the right, changing that also changes the overall area of the function. If f(x) is decreasing, then the added area to A(x) will become less and less large. Here, we can intuitively see a connection to the derivative of f(x) and A(x). It just so happens that the area function IS the antiderivative of f(x), which is less intuitive to see.
Lets begin with the proof:
Consider A(x) and f(x) where A(x) is the area function to f(x). We are to prove that A'(x) = f(x).
Let's begin with the definition of a derivative, just to see if we can work with that somehow: \\\\[ A'(x) = \\\\lim_{h \\\\to 0} \\\\left[ \\\\frac{A(x+h) - A(x)}{h} \\\\right] \\\\]
Notice A(x + h) and A(x). These appear to be the difference of two areas. Let's represent this graphically.
As we see, A(x + h) - A(x) must be some definite area, but what? Here, we will use interesting logic in our proof. The first thing we notice is that the value of f(x) at point x shown is at a minimum, and f(x+h) is at a maximum for the interval x to x + h. That seems unique, why isn't x + h moved a bit more to the right, where it wouldn't be a maximum of the interval? Well, the key is that the expression A(x + h) - A(x) is a limit, so h is meant to be a slight nudge to x. It seems a lot more like a nudge on the graph to show the idea of the area under a curve. Since h approaches 0, it is meant to be a \u201csufficiently small\u201d value. This means that either f(x) or f(x + h) is a max or a min, depending if the function is increasing or not, because the distance between x and x + h will approach 0. Although it doesn't really matter as we will see, the function is increasing so f(x + h) will be the max. and f(x) will be the min.
Now with that settled, why does that even matter? Well, we know the area between x and x + h is an area, and that f(x) is the lowest point on the graph, and f(x + h) is the highest. Now, we know that the area in the interval is some area value, A. Let's simplify the area for A greatly. Let's say that the area for A is just the area of some rectangle with the width of the interval h. Then, A = h * f(c), where f(c) is some height between f(x) and f(x + h). Ok\u2026 but how does that help us? Well, we know that f(x) is a minimum, so f(x) * h MUST be smaller than the actual area for the interval, because all other points on the graph are greater than f(x). So, if you view the interval as the sum of a bunch of rectangles, we just took the smallest rectangle and spread it across the interval, meaning it must be smaller. With similar logic, since f(x + h) is a maximum, it MUST be larger than the actual area. With that, we know that f(c) must be somewhere between f(x) and f(x + h). Since our function is continuous, we know that every value must exist between f(x) and f(x + h), which means every value must exist between f(x) * h and f(x + h) * h. This means that area A must exist in the form f(c) * h, where c \u2208 [x, x + h]. \\\\[ A'(x) = \\\\lim_{h \\\\to 0} \\\\left[ \\\\frac{A(x+h) - A(x)}{h} \\\\right] \\\\]
Back to the actual math, we've essentially just logically proven that A(x + h) - A(x) = f(c) * h. Let's substitute that in! \\\\[ A'(x) = \\\\lim_{h \\\\to 0} \\\\left[ \\\\frac{f(c) * h}{h} \\\\right] \\\\] \\\\[ A'(x) = \\\\lim_{h \\\\to 0} \\\\left[ f(c) \\\\right] \\\\]
Now, we know that c \u2208 [x, x + h] then, \\\\[ x \\\\leq c \\\\leq x + h \\\\]
But, h\u21920, so: \\\\[ x \\\\leq c \\\\leq x \\\\]
Which means c = x, meaning f(c) = f(x), solving the limit.
A'(x) = f(x), which is what we intended to prove.
\"]},\"proof-of-the-fundamental-theorem-of-calculus-part-ii\":{\"title\":\"Proof of the Fundamental Theorem of Calculus Part II\",\"sections\":[\"
Proof of the Fundamental Theorem of Calculus Part II
Alexander De Carlo
There is a problem with the first part of our theorem. But\u2026 didn't we just prove it true? We did, but not to where we can do anything meaningful. What do we mean by this?
Note that we just proved A'(x) = f(x), meaning the derivative of the area function of a graph is f(x). But what precisely is the area function? Where does it begin, and where does it conclude? How can it be effectively expressed? The graph used in proving the first part of our fundamental theorem was convenient because it had a definite starting point. However, most graphs do not. Fortunately, we can choose the starting point of a graph without a clear beginning to be anywhere. But what implications does this have for our area function?
To answer this, let's consider the derivative of x2:
\\\\[ \\\\frac{d}{dx}\\\\left[ x^2 + C \\\\right] = 2x \\\\]
As we see, many things can share a derivative. Since the derivative of a constant is always 0, the antiderivative of a function always has \\\"+ C\\\" after it, indicating that the antiderivative of a function is actually a family of functions that all differ from a constant value.
So, if A(x) is just the antiderivative of f(x), which we had just proven, then A(x) = F(x) + C. F(x) is just denoting the antiderivative function for A(x). It actually makes sense why a C needs to be there for such a general area function.
Whether we start our area function at D or E, all the areas to the left of both D and E are the same. This means the area functions for both SHOULD be the same, just differing by some constant, C. This is called the indefinite integral.
\\\\[ \\\\int f(x) \\\\, dx = F(x) + C \\\\]
The explanation for the notation will be shown in a different article. Here, we are essentially finding an area function for f(x), but we have no bounds on that function. The changing of bounds will just differ by a constant C, shown above.
We want to know how to evaluate integrals WITH bounds, however. The good thing is, we already know it should really include the area function. Let's say we start our integral at defined point a, and it goes to some variable point x.
\\\\[ A(x) = \\\\int_{a}^{x} f(t) \\\\, dt \\\\]
The variable t here is a dummy variable, because we are trying to show that x could be any point on the graph f(t), which originally would be called f(x). The \\\"x\\\"-axis would be called the \\\"t\\\"-axis to make things clearer. Essentially it's just changing the name of variables so as to not get the x in f(x) confused with the x on the integral.
We know that A(x) is the antiderivative of f(x), so A(x) = F(x) + C. Thus:
\\\\[ F(x) + C = \\\\int_{a}^{x} f(t) \\\\, dt \\\\]
The important thing to note here is that we defined one boundary for the integral, at A. this means that we should be able to somehow solve for C, as C can no longer be just any constant. This integral with one bound at A should no longer be a family of antiderivatives, but just one specific antiderivative. But how can we solve it? Well, we can do a sneaky trick. Let's make x some defined point B, essentially finding the area of some interval.
Then:
\\\\[ C = \\\\int_{a}^{b} f(t) \\\\, dt - F(b) \\\\]
That didn't really help us. But since we start at A, C must be the same, for any other arbitrary endpoint we pick. Then, how can we get rid of the nasty integral? Let's just make b = a! It's sneaky, but x can technically just be a. Now, an integral from a to a is just clearly 0, because there is no area. Thus, C = -F(a). Now, we've essentially discovered the second part of the fundamental theorem of calculus.
\\\\[ \\\\begin{array}{l} F(x) + C = \\\\int_{a}^{x} f(t) \\\\, dt \\\\\\\\ C = -F(a) \\\\end{array} \\\\]
We are pretty much done here, but the proof is shown for an interval between defined point A to defined point B, so we can just let x = a. This also allows us to not have to use the dummy variable t anymore and just make that x.
Simple, yet impactful. Basically, it doesn't matter at all what happens between a and b, the value for the integral on an interval is just F(b) - F(a).
\"]}}")}}]);
\ No newline at end of file
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deleted file mode 100644
index 43c7d40..0000000
--- a/static/js/156.b1189339.chunk.js
+++ /dev/null
@@ -1 +0,0 @@
-"use strict";(self.webpackChunkmathinfo=self.webpackChunkmathinfo||[]).push([[156],{156:function(t){t.exports=JSON.parse('{"Summary":{"text":"
Calculus
Calculus is a complex branch in mathematics that deals with the study of change. Knowledge of how functions change over certain periods or conditions can only be answered with methods derived from calculus. There are 3 core topics which comprise calculus: Limits, Derivation and Integration. Limits describe how functions approach values. Derivation describes the rate of change at certain values of a function. Integration describes the area under the curve of a function. The beauty in calculus lies in how these 3 core topics relate to one another.
"},"riemann-sums":{"title":"Riemann Sums","sections":[{"text":"..."}]},"proof-of-the-fundamental-theorem-of-calculus-part-i":{"title":"Proof of the Fundamental Theorem of Calculus Part I","sections":[{"text":"
Proof of the Fundamental Theorem of Calculus Part I
Alexander De Carlo
Proving the fundamental theorem of calculus consists of two parts. Interestingly, the \'first part\' was proved secondly, and the \'second part\' was proved first. However, it makes much more sense for any calculus student to learn the first part\'s proof, then the second one, despite the order that they were proven. So let\'s jump right into the first part!
The first part of the fundamental theorem of calculus connects the idea of a derivative and an integral. It states that the derivative of an integral returns the original function. In mathematical notation: \\\\[\\\\frac{d}{dx}\\\\left[\\\\int f(x) \\\\, dx\\\\right] = f(x)\\\\]This can also be used to say that the integral of f(x) (area under the curve) is just the antiderivative of f(x), or in simpler terms, a function g(x) where it\'s derivative of f(x). As we can see, proving this theorem will give us a gateway into actually solving integrals, one of the hardest tasks in calculus.
But why would we even consider a relationship in the first place? Well, let\'s consider a very general function, f(x).
Observe f(x) and A(x), where A(x) is the area function. Basically, we are assuming that the area under any curve can be considered some function of x. As we see with the blue line to the right, changing that also changes the overall area of the function. If f(x) is decreasing, then the added area to A(x) will become less and less large. Here, we can intuitively see a connection to the derivative of f(x) and A(x). It just so happens that the area function IS the antiderivative of f(x), which is less intuitive to see.
Lets begin with the proof:
Consider A(x) and f(x) where A(x) is the area function to f(x). We are to prove that A\'(x) = f(x).
Let\'s begin with the definition of a derivative, just to see if we can work with that somehow: \\\\[ A\'(x) = \\\\lim_{h \\\\to 0} \\\\left[ \\\\frac{A(x+h) - A(x)}{h} \\\\right] \\\\]
Notice A(x + h) and A(x). These appear to be the difference of two areas. Let\'s represent this graphically.
As we see, A(x + h) - A(x) must be some definite area, but what? Here, we will use interesting logic in our proof. The first thing we notice is that the value of f(x) at point x shown is at a minimum, and f(x+h) is at a maximum for the interval x to x + h. That seems unique, why isn\'t x + h moved a bit more to the right, where it wouldn\'t be a maximum of the interval? Well, the key is that the expression A(x + h) - A(x) is a limit, so h is meant to be a slight nudge to x. It seems a lot more like a nudge on the graph to show the idea of the area under a curve. Since h approaches 0, it is meant to be a \u201csufficiently small\u201d value. This means that either f(x) or f(x + h) is a max or a min, depending if the function is increasing or not, because the distance between x and x + h will approach 0. Although it doesn\'t really matter as we will see, the function is increasing so f(x + h) will be the max. and f(x) will be the min.
Now with that settled, why does that even matter? Well, we know the area between x and x + h is an area, and that f(x) is the lowest point on the graph, and f(x + h) is the highest. Now, we know that the area in the interval is some area value, A. Let\'s simplify the area for A greatly. Let\'s say that the area for A is just the area of some rectangle with the width of the interval h. Then, A = h * f(c), where f(c) is some height between f(x) and f(x + h). Ok\u2026 but how does that help us? Well, we know that f(x) is a minimum, so f(x) * h MUST be smaller than the actual area for the interval, because all other points on the graph are greater than f(x). So, if you view the interval as the sum of a bunch of rectangles, we just took the smallest rectangle and spread it across the interval, meaning it must be smaller. With similar logic, since f(x + h) is a maximum, it MUST be larger than the actual area. With that, we know that f(c) must be somewhere between f(x) and f(x + h). Since our function is continuous, we know that every value must exist between f(x) and f(x + h), which means every value must exist between f(x) * h and f(x + h) * h. This means that area A must exist in the form f(c) * h, where c \u2208 [x, x + h]. \\\\[ A\'(x) = \\\\lim_{h \\\\to 0} \\\\left[ \\\\frac{A(x+h) - A(x)}{h} \\\\right] \\\\]
Back to the actual math, we\'ve essentially just logically proven that A(x + h) - A(x) = f(c) * h. Let\'s substitute that in! \\\\[ A\'(x) = \\\\lim_{h \\\\to 0} \\\\left[ \\\\frac{f(c) * h}{h} \\\\right] \\\\] \\\\[ A\'(x) = \\\\lim_{h \\\\to 0} \\\\left[ f(c) \\\\right] \\\\]
Now, we know that c \u2208 [x, x + h] then, \\\\[ x \\\\leq c \\\\leq x + h \\\\]
But, h\u21920, so: \\\\[ x \\\\leq c \\\\leq x \\\\]
Which means c = x, meaning f(c) = f(x), solving the limit.
A\'(x) = f(x), which is what we intended to prove.
"}]},"proof-of-the-fundamental-theorem-of-calculus-part-ii":{"title":"Proof of the Fundamental Theorem of Calculus Part II","sections":[{"text":"
Proof of the Fundamental Theorem of Calculus Part II
Alexander De Carlo
There is a problem with the first part of our theorem. But\u2026 didn\'t we just prove it true? We did, but not to where we can do anything meaningful. What do we mean by this?
Note that we just proved A\'(x) = f(x), meaning the derivative of the area function of a graph is f(x). But what precisely is the area function? Where does it begin, and where does it conclude? How can it be effectively expressed? The graph used in proving the first part of our fundamental theorem was convenient because it had a definite starting point. However, most graphs do not. Fortunately, we can choose the starting point of a graph without a clear beginning to be anywhere. But what implications does this have for our area function?
To answer this, let\'s consider the derivative of x2:
\\\\[ \\\\frac{d}{dx}\\\\left[ x^2 + C \\\\right] = 2x \\\\]
As we see, many things can share a derivative. Since the derivative of a constant is always 0, the antiderivative of a function always has \\"+ C\\" after it, indicating that the antiderivative of a function is actually a family of functions that all differ from a constant value.
So, if A(x) is just the antiderivative of f(x), which we had just proven, then A(x) = F(x) + C. F(x) is just denoting the antiderivative function for A(x). It actually makes sense why a C needs to be there for such a general area function.
Whether we start our area function at D or E, all the areas to the left of both D and E are the same. This means the area functions for both SHOULD be the same, just differing by some constant, C. This is called the indefinite integral.
\\\\[ \\\\int f(x) \\\\, dx = F(x) + C \\\\]
The explanation for the notation will be shown in a different article. Here, we are essentially finding an area function for f(x), but we have no bounds on that function. The changing of bounds will just differ by a constant C, shown above.
We want to know how to evaluate integrals WITH bounds, however. The good thing is, we already know it should really include the area function. Let\'s say we start our integral at defined point a, and it goes to some variable point x.
\\\\[ A(x) = \\\\int_{a}^{x} f(t) \\\\, dt \\\\]
The variable t here is a dummy variable, because we are trying to show that x could be any point on the graph f(t), which originally would be called f(x). The \\"x\\"-axis would be called the \\"t\\"-axis to make things clearer. Essentially it\'s just changing the name of variables so as to not get the x in f(x) confused with the x on the integral.
We know that A(x) is the antiderivative of f(x), so A(x) = F(x) + C. Thus:
\\\\[ F(x) + C = \\\\int_{a}^{x} f(t) \\\\, dt \\\\]
The important thing to note here is that we defined one boundary for the integral, at A. this means that we should be able to somehow solve for C, as C can no longer be just any constant. This integral with one bound at A should no longer be a family of antiderivatives, but just one specific antiderivative. But how can we solve it? Well, we can do a sneaky trick. Let\'s make x some defined point B, essentially finding the area of some interval.
Then:
\\\\[ C = \\\\int_{a}^{b} f(t) \\\\, dt - F(b) \\\\]
That didn\'t really help us. But since we start at A, C must be the same, for any other arbitrary endpoint we pick. Then, how can we get rid of the nasty integral? Let\'s just make b = a! It\'s sneaky, but x can technically just be a. Now, an integral from a to a is just clearly 0, because there is no area. Thus, C = -F(a). Now, we\'ve essentially discovered the second part of the fundamental theorem of calculus.
\\\\[ \\\\begin{array}{l} F(x) + C = \\\\int_{a}^{x} f(t) \\\\, dt \\\\\\\\ C = -F(a) \\\\end{array} \\\\]
We are pretty much done here, but the proof is shown for an interval between defined point A to defined point B, so we can just let x = a. This also allows us to not have to use the dummy variable t anymore and just make that x.
Simple, yet impactful. Basically, it doesn\'t matter at all what happens between a and b, the value for the integral on an interval is just F(b) - F(a).
"}]}}')}}]);
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+++ /dev/null
@@ -1 +0,0 @@
-"use strict";(self.webpackChunkmathinfo=self.webpackChunkmathinfo||[]).push([[404],{404:function(s){s.exports=JSON.parse('{"Summary":{"text":"
Trigonometry
Trigonometry deals with the study of angles. The etymology of the word also provides insight into what branch of math it encompasses, as \\"trigon\\" refers to triangles, or a shape consisting of 3 angles. Although trigonometry is a sub-branch of geometry, it deserves its own in-depth section as trigonometry appears and is necessary to understand for many other aspects of mathematics. Trigonometry will introduce many new ideas with 2 core functions that incorporate angles called sine and cosine that all relate back to the triangle. The inner-workings of trigonometry include right angle trig., general angle trig, trigonometric function analysis, and even complex trigonometry.
Trigonometry, as in the name, deals with the study of triangles, angles, ratios, trigonometric functions, graphs and a lot more. It is one of the most important topics in Mathematics and is the gateway for numerous other advanced concepts. While math revolves around Trigonometry, Trigonometry revolves around right triangles. A major reason for this is that only right triangles satisfy the Pythagorean Theorem, which states a2 + b2 = c2, where sides a and b are the legs, while side c is the hypotenuse. One of the key aspects of right triangles is their strict vertical and horizontal side due to the 90\xb0 angle. This also means side c is essentially the slope (rise / run) of the triangle. Let\'s take a look at this right triangle:
","image":"
Diagram of right triangle.
"},{"text":"
With just this diagram, we can identify many different ratios between the sides. There are a total of 6 ratios, each a trigonometric function. These operations are designated as functions because they completely rely on the angle between both sides.
Let\'s focus on angle \u03b1. Starting with the sine function, we know sin = opposite / hypotenuse. Applying that to angle \u03b1, we find that sin(\u03b1) = a / c, because side a is opposite to angle \u03b1, and side c is the hypotenuse of the triangle. Sine works the exact same way with any other angle. For example, sin(\u03b2) = b / c, as side b is opposite to angle \u03b2, and side c is again the hypotenuse of the triangle. Now that we know all about sine, let\'s look at the other core trigonometric functions:
\u03b8 for any given angle: Sine -> sin(\u03b8) = opposite / hypotenuse Cosine -> cos(\u03b8) = adjacent / hypotenuse Tangent -> tan(\u03b8) = opposite / adjacent
Swapping the numerator and denominator of these core functions gives us their reciprocals:
These 6 functions are all extremely important, with relationships to be further elucidated on as we go through trigonometry. For now though, it\'s important to simply understand where these functions come from and how they\'re used in relation to right triangles.
","image":"
All 6 trigonometric functions of an angle on a triangle of side lengths 5, 4, and 3.
"},{"text":"
Trigonometry will help us solve triangles like these. Our goal is to find side length c, but we aren\'t given both sides a and b. So what is side c? Which trigonometric function will encompass both side length11 and side c to return the angle? Looking at the 42\xb0 angle, we can see the side with length 11 is adjacent to it, and side c is the hypotenuse of the triangle. Going back to the trigonometric functions, cosine conveniently uses both the adjacent and hypotenuse sides of a right triangle.
cos(42\xb0) = 11 / c c * cos(42\xb0) = 11 c = 11 / cos(42\xb0)
Here cos(42\xb0) is just a constant, as cos(42\xb0) will be the same for all triangles with a 42\xb0 angle. Using a calculator equipped with the cosine function, we arrive at c \u2248 14.8.
","image":"
Can you find the length of side b? (b \u2248 6.3)
"}]},"right-angle-trigonometry-test-problem":{"title":"Right Angle Trigonometry Test Problem","sections":[{"text":"
Right Angle Trigonometry Test Problem
Alexander De Carlo
Solution:Let y = the base of the entire triangle. tan(30\xb0) = 50 / y y * tan(30\xb0) = 50 y = 50 / tan(30\xb0)
Let z = the base of the inner right triangle, z = y - x. tan(60\xb0) = 50 / z z * tan(60\xb0) = 50 z * tan(60\xb0) = 50
x = y - z x = 50 / tan(30\xb0) - 50 / tan(60\xb0) x \u2248 57.7
","image":"
All 6 trigonometric functions of an angle on a triangle of side lengths 5, 4, and 3.
How can you extend the use of trigonometry to non-right angle triangles with trigonometric functions? The key is in breaking up the \u201cGeneral Triangle\u201d into right triangles and seeing what can be done from there. Let\'s look at the general triangle, where every angle, not just two, are unknown.
By dropping an altitude at the height to the base of the general triangle, we have created two right triangles. Looking more closely, we can observe a relationship between the two triangles.
sin(B) = h / a sin(A) = h / b
Then, we can solve for the height.
a * sin(B) = h b * sin(A) = h
Taking advantage of their equality, we can merge the equations into the following:
a * sin(B) = b * sin(A)
And finally, dividing both sides by ab yields us our final result.
sin(B) / b = sin(A) / a
Realizing that we placed the altitude at an arbitrary angle, we could have rotated the triangle and placed it anywhere else for the same relationship. Therefore, sin(C) / c must also be equivalent.
\u2234 sin(A) / a = sin(B) / b = sin(C) / c
This very simple proof establishes the theorem known as The Law of Sines.
So what kinds of triangles can we solve with this?
"},{"text":"
AAS Triangles
Solving triangles refers to finding every angle and side length associated with that triangle. AAS - Angle Angle Side, means you know two angles and a side, with the order of appearance being angle-angle-side. Because the direction of rotation is arbitrary to its properties, SAA is functionally the same thing. ASA means you know two angles and a side in between. Looping back to AAS/SAA triangles, they are the easiest to solve.
To find missing angle C in the above description, simply subtract the known angles by 180\xb0 as all triangle degrees add to 180\xb0. C = 110. Finding the other sides is simply a matter of using The Law of Sines.
sin(A) / a = sin(B) / b = sin(C) / c, meaning the angle maintains a ratio with its opposite side, enabling us to find a missing angle/side assuming we have its counterpart and a full angle/side pair.
sin(30\xb0) / a = sin(40\xb0) / 11 a * sin (40\xb0) = 11sin(30\xb0) a = 11sin(30\xb0) / sin(40\xb0), a \u2248 8.56
Solving for angle C would follow the same process.
"},{"text":"
SSA Triangles
SSA Triangles are more tricky to solve, as they lead to ambiguity which could result in two possible ways to create the triangle.
Here, we must use the inverse sine function. Basically, if sin(\u1340) gives us a side length, then sin-1(side length) gives us our angle. The only thing is, two possible angles satisfy the sine inverse in all cases. When you plug sin-1(9sin(70\xb0) / 10) into the calculator, you will get one value, the PRINCIPLE value. An explanation for all of this will be in later articles regarding trigonometric functions. sin-1(9sin(70\xb0) / 10) \u2248 57.7. Our other possible angle is simply 180 - PRINCIPLE ANGLE, in this case, 180 - 57.7 = 122.3. (For those who understand the unit circle or are revisiting from a later article, we are essentially reaching the equivalent sin(\u1340) value from QII or QIII.)
To see if both of these triangles are possible, let\'s try to get the 3rd value and see if it all makes sense.
180 - 57.7 - 70 = 52.3 180 - 122.3 - 70 = -12.3
We cannot have negative angles, so we can reject the 122.3\xb0 triangle. If it yielded a positive third angle, we would have to solve for the rest of our values for BOTH the triangles, and simply have two possibilities.
The Law of Sines is an important relationship in the general triangle, but there\'s another core relationship we can implement when dealing with triangles, the Pythagorean Theorem. Consider the general triangle with an altitude dropped, creating two right triangles composing the general triangle. From side c, let\'s call x the base length of the smaller triangle and c - x the base length of the larger inner triangle.
Instead of relying on sine, let\'s use cosine on angle B to try to do something with that x variable.
cos(B) = x / a a * cos(B) = x
Now we have x in terms of our side lengths and angles of our general triangle. As we can see, we have two right triangles, so we can create some relationships using the Pythagorean Theorem. Note that we want to find an end relationship that doesn\'t contain variables such as h or x, as these aren\'t the side lengths or angles of our triangle.
h2 + x2 = a2 h2 = a2 - x2
h2 + (c - x)2 = b2 h2 = b2 - (c - x)2
To remove h, we can set these two equations equal to one another.
Remember, x = a * cos(B) so to remove x, substitute this in.
a2 + c2 - 2cb * cos(B) = b2
Since this is a general triangle, variable names can be changed and swapped as they are arbitrary, the only thing that has to stay is the relationship between the cosine angle and the side.
Therefore, to make it look better, we can change this to: a2 + b2 - 2ab * cos(\u03b3) = c2. The cosine angle stays opposite to side c, with the previous case opposite to side b.
This is called The Law of Cosines, for its use of the cos function instead of the sin function.
If you notice, The Law of Cosines is actually just an extension of the Pythagorean Theorem! When cos is 90\xb0, cos(90\xb0) = 0. This removes the -2ab term and simply gives us a2 + b2 = c2, which is pretty much a special case of The Law of Cosines.
"},{"text":"
SAS triangles
In this case, The Law of Sines is unusable, leaving us to rely on The Law of Cosines.
We can find the missing side with The Law of Cosines.
We know all three sides, so to find an angle we can simply use The Law of Sines. We do NOT have to worry about the possibility of there being two triangles because if you know the lengths of all three sides, there can be only one possibility for what the angles can be.
"},{"text":"
SSS triangles
To solve SSS triangles, we can use The Law of Cosines. To eliminate ambiguity when solving, solve for the angle of the largest side first. This removes the possibility of a second triangle because solving from the longest side gives us the largest angle.
Proceed with The Law of Sines to get the other angles.
"}]},"the-unit-circle-i":{"title":"The Unit Circle I","sections":[{"text":"
The Unit Circle I
Alexander De Carlo
To truly understand trigonometry, one must comprehend how angles work, understanding the unit circle, its relation to angles, and a novel angle measurement called radians.
Angles reach a maximum of 360\xb0, completing a full rotation in circular motion. This range from 0 - 360\xb0 exists because 360 is a conveniently divisible number (by 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, etc.). While practical for everyday use, it lacks the mathematical precision we desire.
Consider a circle with a radius of 1, making the circumference C = 2\u03c0. This equates a full 360-degree rotation to 2\u03c0 radians, a unit used in advanced trigonometry. To convert between radians and degrees, multiply by 180/\u03c0 or \u03c0/180, respectively.
Now, observe the unit circle on the coordinate plane with an angle traversing it:
Two angles are measured, originating from the right x-axis and extending outward. The blue and red lines, termed terminal rays, gauge the angle size. Notably, the red terminal ray measures externally, while the blue one measures internally, following the counterclockwise motion. For example, angle a is around 135\xb0 or 3\u03c0/4 radians, while angle b is greater. For angles exceeding 360\xb0, they continue counterclockwise, such as 500\xb0 being equivalent to 140\xb0 after a full rotation. Negative angles rotate clockwise; -1\xb0 equals 359\xb0. This insight allows us to divide the unit circle into four quadrants.
Every \u03c0/4 (90\xb0) defines a new quadrant, with x and y lines acting as quadrantal separators.
Now that we understand this improved angle system and its application around the unit circle, let\'s explore its significance and connection to right triangles. Unlike general right triangles, special triangles like the 45-45-90 and 30-60-90 offer exact angle and side measurements.
These special triangles, created using the Pythagorean theorem and specific base splits, provide exact values without trigonometry. Placing these triangles within the unit circle, with a hypotenuse of length 1, allows us to compress them by dividing by 2 and 3, respectively. This results in three triangles per quadrant: 45-45-90, 30-60-90, and 60-30-90. This process enables the mapping of coordinates for various angles on the unit circle, forming a helpful diagram for memorization.
How does this relate to trigonometric functions? The crucial point lies in the fact that the hypotenuse of these triangles is always 1. Thus, sin(\u03b8) = opp, where \\"opp\\" is the y-coordinate, and cos(\u03b8) = x-coordinate. This insight leads to the realization that coordinates on the circle are simply (cos(\u03b8), sin(\u03b8)). For instance, with our special triangle values, sin(\u03c0/4) = sqrt(2)/2.
Examining the unit circle reveals essential insights into trigonometric functions:
Sine and cosine values range between -1 and 1.
Sin and cosine values are equal at two values: \u03c0/4, 5\u03c0/4
In QI and QII, sin is positive; in QIV and QI, cos is positive.
As sin increases, cos decreases, and vice versa. Two different angles return the same value, either in the top or bottom half for sin and left or right for cos (sin(\u03c0 - \u1340) = sin\u1340, cos(2\u03c0 - \u1340) = cos\u1340).
Finally, recognizing that cos(\u1340) = x and sin(\u1340) = y, tan(\u1340), the slope, is found as y/x, making tan(\u1340) = sin(\u1340)/cos(\u1340). Discovering these relationships is termed finding identities, a concept explored further in our trigonometry studies.
"}]},"the-unit-circle-ii":{"title":"The Unit Circle II","sections":[{"text":"
The Unit Circle II
Alexander De Carlo
Let\'s explore some intriguing relationships between trigonometric functions using the unit circle while maintaining clarity and coherence. First, let\'s examine the coordinates of points on circles with radii greater or less than 1.
sin(x) = y * r r * sin(x) = y r * cos(x) = x
Simply multiply by r, the radius. The ratios are still the same.
(r * sin(x), r * cos(x))
Consider the unit circle.
Here, four angles are presented with colored terminal rays. The red and blue rays are reflected across the y-axis, and the blue and yellow across the x-axis. If we examine cos(blue) and cos(yellow), they are the same. Yellow is simply the negative angle of blue; thus, cos(-x) = cos(x). With sine, sin(-x) = -sin(x). Examining the relationship between blue and red, or yellow and purple, we observe that sin(\u03c0 - x) = sin(x), and cos(\u03c0 - x) = -cos(x).
We also observe that cos(2\u03c0 - x) = cos(x), and sin(2\u03c0 - x) = -sin(x). Knowing that tan(x) is the slope, when cos or sin is negative, tan will be negative. Therefore, tan(x + \u03c0) = tan(x), tan(x - \u03c0) = -tan(x), and tan(-x) = -tan(x). These relationships are intuitive and can be quickly deduced from the unit circle.
Now, let\'s explore the relationship between sine and cosine.
When rotating 90\xb0, cos(x), which is x, would be rotated completely vertically upward, becoming sine, and sine would become cosine. Starting from the triangle in the first quadrant, upon rotation, the cosine and sine values swap places, and the sine becomes negative. This process is repeated in the second triangle, yielding the following relationships:
If we examine the tangent values, the cosine and sines are swapped, and one value becomes negative. Adding a right angle, \u03c0/2, yields the negative reciprocal of the original. Therefore:
tan(x + \u03c0/2) = tan(x - \u03c0/2)
More succinctly:
tan(x + \u03c0/2) = -cot(x)
These relationships showcase the beauty of trigonometric functions.
"}]},"the-unit-circle-iii":{"title":"The Unit Circle III","sections":[{"text":"
The Unit Circle III
Alexander De Carlo
While we are acquainted with the representations of cos and sin on the unit circle, it\'s essential to explore how all trigonometric functions manifest and derive identities from these visual connections.
Acknowledging that cos and sin correspond to our x and y values, the question arises: how do we express tan as a length measure? Leveraging the unit circle\'s triangle with a hypotenuse of 1, we can establish a right angle from the (cos, sin) point.
Upon closer inspection, we observe that tan represents that line, given that tan is opp/adj, and adj is 1. This elucidates the name \\"tan,\\" signifying its tangent relationship to the circle. Additionally, it visually explains why tan(\u03c0/2) and tan(\u03c0/4) are undefined, extending infinitely without intersecting the x-axis. To determine sec, representing adj/opp, we leverage the triangle formed with tan, realizing that sec must denote the hypotenuse of that triangle.
Applying a similar process, we can ascertain cotangent.
Similarly, the derivation of cosecant follows a comparable procedure.
Exploring the relationships between trigonometric functions through the Pythagorean theorem, we unveil crucial connections:
These relationships are commonly known as the Pythagorean Identities. Among them, the most significant is sin2(x) + cos2(x) = 1, a fundamental identity that merits in-depth study.
"}]},"using-the-pythagorean-identity-sine-and-cosine":{"title":"Using the Pythagorean Identity: Sine and Cosine","sections":[{"text":"
Then we can get the adjacent side from how we calculate cosine:
cos(\u03b8) = adj/hyp -sqrt(5) / 3 = x/3 x = -sqrt(5)"}]},"trigonometric-functions-graphs-and-problems-i":{"title":"Trigonometric Functions Graphs and Problems I","sections":[{"text":"
Trigonometric Functions Graphs and Problems I
Alexander De Carlo
While we are familiar with the six trigonometric functions through the unit circle, let\'s explore what these functions look like when graphed. Our x value will represent the angle value in radians, and the y value will be the corresponding return value.
Let\'s examine the special values in the sin function that are rational: 0 - 0, \u03c0/2 - 1, \u03c0 - 0, 3\u03c0/2 - -1, and 2\u03c0 - 0. We observe this cycling motion every 2\u03c0 radians, known as the period. Let\'s visualize this graphically.
Upon graphing and examination, a beautiful cycling pattern emerges.
Since sin(x) = cos(x - \u03c0/2), we anticipate a \u03c0/2 shift to the right for the cosine graph.
"},{"text":"
Now, let\'s apply standard function transformations to see how these functions change with applications such as f(x) = A sin[B(x - h)] + k. This applies to both sin and cos.
Let\'s examine each transformation individually:
When looking at sin and cos, which cycle between 1 and -1, we notice a change by a value from 1 in opposite directions from the average line, 0. Therefore, |A| is the amplitude, representing the distance the maximum and minimum values deviate from the midline point.
Note: A negative sign on A could flip everything across the x-axis. A = (max - min) / 2, as this finds the average distance from the midline.
B represents the period. In a normal wave, it repeats every 2\u03c0. When B increases, the entire thing compresses, resulting in a period of 2\u03c0 / B.
h denotes the horizontal shift, and k is the vertical shift. To find k, you can find the actual average for the midline: (max + min) / 2.
k is 1, making the midline 1, and A is 3, so it will extend up and down by 3. The 2x means this will happen during the period \u03c0.
For g(x):
A is 2. No reflection is needed. B is \u03c0, so the period is 2\u03c0 / \u03c0, which is just 2. x is shifted left by 1, and the midline is -3.
What problems can we model with our knowledge of the graphs of sines and cosines?
Consider the word problem: A Ferris wheel with a diameter of 520 feet has a low point of 30 feet. Every 30 minutes, it completes 1 rotation (starting at the bottom). Let f(x) model the height of a point on the Ferris wheel.
We begin at the lowest point, and since a cos graph always begins at 1, a negative cos must start at the bottom.
If the diameter is 520 feet, then we go 260 feet up and down from the midline, meaning a = 260.
If a full rotation is 30 minutes, then 30 = 2\u03c0 / B. B = \u03c0 / 15.
To start at 30, instead of the current -260, we need to add 290.
So, f(x) = -260cos(\u03c0 / 15 * x) + 290.
"}]},"trigonometric-functions-graphs-and-problems-ii":{"title":"Trigonometric Functions Graphs and Problems II","sections":[{"text":"
Trigonometric Functions Graphs and Problems II
Alexander De Carlo
We graphed cos and sin, but how would we graph their reciprocal functions, csc and sec? Remember that csc(x) = 1/sin(x) and sec(x) = 1/cos(x). This is very important. Realizing that sin and cos are in the denominator, we can also see that whenever they return zero, the fraction would be 1/0, which is undefined. These occur every \u03c0 period and are known as asymptotes.
Here for the csc graph, these are all the points we know are undefined. What happens when we approach those points? They must either approach infinity or negative infinity, as when the denominator gets smaller, it returns a larger and larger value. When the sin graph is in the positive half of the coordinate plane, it must blow up to infinity, and when the sin graph is negative, it will expand to negative infinity. Let\'s use a sine graph to visualize this:
As we see, some parts stay between 0 and 1 and others go from 0 to -1. Our csc graph will touch the sin graph when it peaks at 1 or -1, as 1/1 = 1, but will diverge from there. In the negative parts, to negative infinity. With this information, we can create our cosecant graph in orange.
The secant graph is done the exact same way. This process helps you graph any sec or csc graph, as you simply graph the sin or cos graph and then apply the asymptote concept.
tan(x) and cot(x) graphs:
tan(x) is undefined at \u03c0/2 + 2\u03c0k and 3\u03c0/2 + 2\u03c0k. Graphing tan is quite simple for this reason. We know when tan(x) goes negative and goes positive, so we can make a graph easily.
With cot(x), we notice that the 0\'s will become the asymptotes, BUT the asymptotes in tan(x) will become the 0\'s for cot(x). These ideas are common to see with reciprocal functions.
Comparing the two graphs, they almost look reflected. This comes from the identity that we previously derived that states tan(x + \u03c0/2) = -cot(x). Also, take note that for these graphs, the period is not 2\u03c0, just \u03c0.
y = 3cot(3\u03c0/2(x)).
Here we would have a stretch of 3, and a period of 3/2. Given that cot(x) has asymptotes at 0, \u03c0, 2\u03c0, etc.., we can solve for the asymptotes by taking both of these base asymptotes and then applying the proper transformations. :
3\u03c0/2(x) = 2\u03c0 x = 4/3
3\u03c0/2(x) = \u03c0 x = 2/3
We can see here that the steps in the asymptotes are in increments of 2/3, something we could also have calculated by knowing that the function repeats every \u03c0. Now, you could also calculate the 0\'s by directly applying transformations, but a much simpler way is to realize that the asymptotes of any trigonometric function are equidistant from the x intercept on the base function. This way, we can see that the 0\'s would be 1/3, 1, etc\u2026 Since this is a cot function, the function will continuously decrease. Given all of these baby steps calculated, try graphing this for yourself, since the challenging parts are out of the way."}]},"the-inverse-trigonometric-functions":{"title":"The Inverse Trigonometric Functions","sections":[{"text":"
The Inverse Trigonometric Functions
Alexander De Carlo
The inverse trig functions, instead of returning ratios, give the angle that provides that ratio. The naming of these is quite easy, and simply just adds an \'arc\' to the functions. They can also be named with the inverse sign, what we are familiar with from the right and general angle trigonometry.
arccos(x) OR cos-1(x) arcsin(x) OR sin-1(x) arctan(x) OR tan-1(x) arccsc(x) OR csc-1(x) arcsec(x) OR sec-1(x) arccot(x) OR cot-1(x)
For example, arcsin(1/2) = \u03c0/6
Why not also 5\u03c0/6? To make it a function, we have to restrict its domain to the right half of the unit circle, [-\u03c0/2, \u03c0/2]. arctan(x) is also only between [-\u03c0/2, \u03c0/2]. cos however must be defined as the top half of the unit circle,[0, \u03c0]. Similar to the square root, the domain restriction only exists when these functions are used alone, however in trigonometric equations, every angle will be allowed.
"},{"text":"
Composition
Consider sin(arccos(x)). How would we solve a composite function like this?
sin(arccos(x)) = \u221a(1 - x2)
We can resemble x/1 as a triangle since these are the lengths of the hypotenuse and adjacent.
sin is opp / hyp. So, what is opp? x2 + opp2 = 1. opp = \u221a(1 - x2)
Therefore, sin(arccos(x)) = \u221a(1 - x2)
We can also notice the same process would work for: cos(arcsin(x)).
The arccos and cos would cancel out, leaving \u03c0/2 - x.
\u2234 arccos(sin(x)) = \u03c0/2 - x arcsin(cos(x)) = \u03c0/2 - x
"}]},"introduction-to-trigonometric-equations":{"title":"Introduction to Trigonometric Equations","sections":[{"text":"
Introduction to Trigonometric Equations
Alexander De Carlo
With trigonometric equations, domain restrictions no longer apply; we consider every possible angle unless specified otherwise. For instance, 3\u03c0 / 4 + 2\u03c0k indicates solutions at 3\u03c0 / 4 and all angles obtained by adding or subtracting multiples of 2\u03c0, where k belongs to the set of integers (k \u220a \u2124).
"},{"text":"
2cos(\u1340) = sqrt(3) cos(\u1340) = sqrt(3) / 2 \u1340 = \u03c0 / 6, but can also be 11\u03c0 / 6 because of the unit circle. You can also go fully 2\u03c0 around the circle, allowing the following (in all these cases k \u220a \u2124). \u1340 = \u03c0 / 6 + 2\u03c0k, 11\u03c0 / 6 + 2\u03c0k
This derivation yields the formula: cos(\u03b1 - \u03b2) = cos(\u03b1)cos(\u03b2) + sin(\u03b1)sin(\u03b2), a valuable identity in sum and difference identities.
Now, let\'s explore other identities by setting \u03b2 = -\u03b3:
The product to sum identities are particularly useful anytime we want to convert multiplication of trigonometric functions into addition and subtraction. This will become useful when we eventually put these new identities to use. Now that we know product -> sum identities, how would we go upon getting sum -> product identities? The key is in rearranging \u03b1 and \u03b2 in our product to sum identities.
let u = \u03b1 + \u03b2, v = \u03b1 - \u03b2. We must get u and v in terms of \u03b1 and \u03b2 so we can change the entire equation to be in terms of u and v.
"}]},"using-the-sum-and-difference-identity":{"title":"Using the Sum and Difference Identity","sections":[{"text":"
Using the Sum and Difference Identity
Alexander De Carlo
The sum and difference identities, the most fundamental of the more complex trigonometric identities, can be used in many ways, including finding exact values to previously unknown angles.
We can find unknown angle values by adding or subtracting known ones. For example, let\'s consider \u03c0/6 and \u03c0/4. Summing these together, we get 5\u03c0/12, an angle that we currently have no clue what the exact value is.
What about a base fraction, like \u03c0/12? Knowing the exact value of that could help us find the exact value of a lot more angles. Here, we can use the difference formula.
What subtracts to equal \u03c0/12? \u03c0/3 and \u03c0/4 do!
Certainly isn\'t as nice looking as something like sin(\u03c0/3), which is why these values usually do not need to be memorized. This all goes to show that any trigonometric function will yield an exact value if the angle is in terms of \u03c0, due to the transcendentality of the trigonometric functions and the transcendentality of \u03c0.
sin(5 radians) will never be in exact value (it would be transcendental), but sin(3\u03c0/19) does have an exact value.
"},{"text":"
Simplification of the Sum and Difference Identities
Oftentimes, we find that the sum and difference identities are a bit too general for what we need to deal with. These can simplify to what are known as the double angle identities.
Notice that there may be a relationship between the overall degree of the relationship and the inner coefficient of sin(3\u03b1). We will dive deeper into this in later articles.
"}]},"using-identities-single-sine-expression":{"title":"Using Identities: Single Sine Expression","sections":[{"text":"
Using Identities: Single Sine Expression
Alexander De Carlo
As we journey through mathematics, we may find it very useful to rewrite trigonometric expressions as sine functions. This becomes helpful when solving trigonometric equations in some cases, but also with calculus and graphing. Let\'s see what we can do with a general sine expression.
The general form is A * sin(Bx + C) where A, B and C are constants. Let\'s try to separate this out. We want to work backwards from the answer because we are trying to convert complex expressions to this general form.
Using the sine addition identity, we can expand to A * [sin(Bx) * cos(C) + cos(Bx) * sin(C)]. Notice that sin(C) and cos(C) are constants. Carrying on, let\'s distribute out constant A and move the constants in front.
A * cos(C)sin(Bx) + A * sin(C)cos(Bx)
Since A * cos(C) and A * sin(B) are constant values, let\'s rewrite them as M and N respectively.
M * sin(Bx) + N * cos(Bx) M = A * cos(C) N = A * sin(C)
Hold on a second. We can use one of our identities to find a relationship between M, N, and A to remove those nasty sine and cosine functions.
Now that we have derived some identities and found some tricks within trigonometry, let\'s solve some equations that we could not previously solve before. Let\'s start with a problem involving sine and cosine.
2sin(4x) - 8cos(4x) = sin(2x) * cos(2x) + 3
This looks challenging, but it\'s quite simple with our knowledge of identities. First, we can use the single sine trick, then the product to sum identity. We could also multiply both sides of this equation by 2, and use the double angle identity. Both work here, but generally try to find the fastest solution for when things get harder.
Let\'s solve another, simpler equation for some exact values.
sin(x) * sin(2x) + cos(x) * cos(2x) = sqrt(3) / 2
Here, use the difference formula for cosine. When solving for equations, we should condense without losing a possible root.
cos(x - 2x) = sqrt(3) / 2 cos(-x) = sqrt(3) / 2
Remember that cosine isn\'t affected by that negative value, you can memorize this for efficiency, but it never hurts to sketch the unit circle to see this simple relationship.
cos(x) = sqrt(3) / 2
Here x = \u03c0/6, 11\u03c0/6.
Let\'s solve one more.
sin(x) + sin(3x) = cos(x) [0, 2\u03c0]
We can\'t use previous methods in this article, but we luck out because there isn\'t a constant term. This means if we can get a product form and factor, we can solve it. Good thing there\'s an identity for that!
x = \u03c0/12, 5\u03c0/12 (add \u03c0 to navigate through the interval) x = 13\u03c0/12, 17\u03c0/12
In the end, x = \u03c0/2, 3\u03c0/2, \u03c0/12, 5\u03c0/12, 13\u03c0/12, 17\u03c0/12.
"}]},"power-reduction-and-half-angle-identities":{"title":"Power Reduction and Half Angle Identities","sections":[{"text":"
Power Reduction and Half Angle Identities
Alexander De Carlo
We\'ve moved beyond fundamental trigonometric identities, which are essential for most trigonometry topics, to more advanced identities useful in calculus. The \\"Power Reduction\\" identity simplifies trigonometric functions raised to high powers, making them easier to solve or graph, using the Pythagorean identity:
sin2(\u03b8) + cos2(\u03b8) = 1
For higher-degree functions, we use the double angle identity for cosine:
cos(2\u03b8) = cos2(\u03b8) - sin2(\u03b8)
From the Pythagorean identity, we know:
cos2(\u03b8) = 1 - sin2(\u03b8)
Substituting, we get:
cos(2\u03b8) = 1 - 2sin2(\u03b8)
Solving for sin2(\u03b8):
sin2(\u03b8) = (1 - cos(2\u03b8)) / 2
Similarly, solving for cos2(\u03b8):
cos2(\u03b8) = (1 + cos(2\u03b8)) / 2
Using these, we derive the half-angle identities. Let \ud835\udf03 = \u03b1 / 2 to substitute and simplify:
sin2(\u03b8) = (1 - cos(2\u03b8)) / 2 cos2(\u03b8) = (1 + cos(2\u03b8)) / 2 cos(\u03b1 / 2) = sqrt((1 + cos(\u03b1)) / 2) sin(\u03b1 / 2) = sqrt((1 - cos(\u03b1)) / 2)"}]}}')}}]);
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Trigonometry
Trigonometry deals with the study of angles. The etymology of the word also provides insight into what branch of math it encompasses, as \\"trigon\\" refers to triangles, or a shape consisting of 3 angles. Although trigonometry is a sub-branch of geometry, it deserves its own in-depth section as trigonometry appears and is necessary to understand for many other aspects of mathematics. Trigonometry will introduce many new ideas with 2 core functions that incorporate angles called sine and cosine that all relate back to the triangle. The inner-workings of trigonometry include right angle trig., general angle trig, trigonometric function analysis, and even complex trigonometry.
Trigonometry, as in the name, deals with the study of triangles, angles, ratios, trigonometric functions, graphs and a lot more. It is one of the most important topics in Mathematics and is the gateway for numerous other advanced concepts. While math revolves around Trigonometry, Trigonometry revolves around right triangles. A major reason for this is that only right triangles satisfy the Pythagorean Theorem, which states a2 + b2 = c2, where sides a and b are the legs, while side c is the hypotenuse. One of the key aspects of right triangles is their strict vertical and horizontal side due to the 90\xb0 angle. This also means side c is essentially the slope (rise / run) of the triangle. Let\'s take a look at this right triangle:
","
With just this diagram, we can identify many different ratios between the sides. There are a total of 6 ratios, each a trigonometric function. These operations are designated as functions because they completely rely on the angle between both sides.
Let\'s focus on angle \u03b1. Starting with the sine function, we know sin = opposite / hypotenuse. Applying that to angle \u03b1, we find that sin(\u03b1) = a / c, because side a is opposite to angle \u03b1, and side c is the hypotenuse of the triangle. Sine works the exact same way with any other angle. For example, sin(\u03b2) = b / c, as side b is opposite to angle \u03b2, and side c is again the hypotenuse of the triangle. Now that we know all about sine, let\'s look at the other core trigonometric functions:
\u03b8 for any given angle: Sine -> sin(\u03b8) = opposite / hypotenuse Cosine -> cos(\u03b8) = adjacent / hypotenuse Tangent -> tan(\u03b8) = opposite / adjacent
Swapping the numerator and denominator of these core functions gives us their reciprocals:
These 6 functions are all extremely important, with relationships to be further elucidated on as we go through trigonometry. For now though, it\'s important to simply understand where these functions come from and how they\'re used in relation to right triangles.
","
Trigonometry will help us solve triangles like these. Our goal is to find side length c, but we aren\'t given both sides a and b. So what is side c? Which trigonometric function will encompass both side length11 and side c to return the angle? Looking at the 42\xb0 angle, we can see the side with length 11 is adjacent to it, and side c is the hypotenuse of the triangle. Going back to the trigonometric functions, cosine conveniently uses both the adjacent and hypotenuse sides of a right triangle.
cos(42\xb0) = 11 / c c * cos(42\xb0) = 11 c = 11 / cos(42\xb0)
Here cos(42\xb0) is just a constant, as cos(42\xb0) will be the same for all triangles with a 42\xb0 angle. Using a calculator equipped with the cosine function, we arrive at c \u2248 14.8.
"]},"right-angle-trigonometry-test-problem":{"title":"Right Angle Trigonometry Test Problem","sections":["
Right Angle Trigonometry Test Problem
Alexander De Carlo
Solution:Let y = the base of the entire triangle. tan(30\xb0) = 50 / y y * tan(30\xb0) = 50 y = 50 / tan(30\xb0)
Let z = the base of the inner right triangle, z = y - x. tan(60\xb0) = 50 / z z * tan(60\xb0) = 50 z * tan(60\xb0) = 50
x = y - z x = 50 / tan(30\xb0) - 50 / tan(60\xb0) x \u2248 57.7
How can you extend the use of trigonometry to non-right angle triangles with trigonometric functions? The key is in breaking up the \u201cGeneral Triangle\u201d into right triangles and seeing what can be done from there. Let\'s look at the general triangle, where every angle, not just two, are unknown.
By dropping an altitude at the height to the base of the general triangle, we have created two right triangles. Looking more closely, we can observe a relationship between the two triangles.
sin(B) = h / a sin(A) = h / b
Then, we can solve for the height.
a * sin(B) = h b * sin(A) = h
Taking advantage of their equality, we can merge the equations into the following:
a * sin(B) = b * sin(A)
And finally, dividing both sides by ab yields us our final result.
sin(B) / b = sin(A) / a
Realizing that we placed the altitude at an arbitrary angle, we could have rotated the triangle and placed it anywhere else for the same relationship. Therefore, sin(C) / c must also be equivalent.
\u2234 sin(A) / a = sin(B) / b = sin(C) / c
This very simple proof establishes the theorem known as The Law of Sines.
So what kinds of triangles can we solve with this?
","
AAS Triangles
Solving triangles refers to finding every angle and side length associated with that triangle. AAS - Angle Angle Side, means you know two angles and a side, with the order of appearance being angle-angle-side. Because the direction of rotation is arbitrary to its properties, SAA is functionally the same thing. ASA means you know two angles and a side in between. Looping back to AAS/SAA triangles, they are the easiest to solve.
To find missing angle C in the above description, simply subtract the known angles by 180\xb0 as all triangle degrees add to 180\xb0. C = 110. Finding the other sides is simply a matter of using The Law of Sines.
sin(A) / a = sin(B) / b = sin(C) / c, meaning the angle maintains a ratio with its opposite side, enabling us to find a missing angle/side assuming we have its counterpart and a full angle/side pair.
sin(30\xb0) / a = sin(40\xb0) / 11 a * sin (40\xb0) = 11sin(30\xb0) a = 11sin(30\xb0) / sin(40\xb0), a \u2248 8.56
Solving for angle C would follow the same process.
","
SSA Triangles
SSA Triangles are more tricky to solve, as they lead to ambiguity which could result in two possible ways to create the triangle.
Here, we must use the inverse sine function. Basically, if sin(\u1340) gives us a side length, then sin-1(side length) gives us our angle. The only thing is, two possible angles satisfy the sine inverse in all cases. When you plug sin-1(9sin(70\xb0) / 10) into the calculator, you will get one value, the PRINCIPLE value. An explanation for all of this will be in later articles regarding trigonometric functions. sin-1(9sin(70\xb0) / 10) \u2248 57.7. Our other possible angle is simply 180 - PRINCIPLE ANGLE, in this case, 180 - 57.7 = 122.3. (For those who understand the unit circle or are revisiting from a later article, we are essentially reaching the equivalent sin(\u1340) value from QII or QIII.)
To see if both of these triangles are possible, let\'s try to get the 3rd value and see if it all makes sense.
180 - 57.7 - 70 = 52.3 180 - 122.3 - 70 = -12.3
We cannot have negative angles, so we can reject the 122.3\xb0 triangle. If it yielded a positive third angle, we would have to solve for the rest of our values for BOTH the triangles, and simply have two possibilities.
The Law of Sines is an important relationship in the general triangle, but there\'s another core relationship we can implement when dealing with triangles, the Pythagorean Theorem. Consider the general triangle with an altitude dropped, creating two right triangles composing the general triangle. From side c, let\'s call x the base length of the smaller triangle and c - x the base length of the larger inner triangle.
Instead of relying on sine, let\'s use cosine on angle B to try to do something with that x variable.
cos(B) = x / a a * cos(B) = x
Now we have x in terms of our side lengths and angles of our general triangle. As we can see, we have two right triangles, so we can create some relationships using the Pythagorean Theorem. Note that we want to find an end relationship that doesn\'t contain variables such as h or x, as these aren\'t the side lengths or angles of our triangle.
h2 + x2 = a2 h2 = a2 - x2
h2 + (c - x)2 = b2 h2 = b2 - (c - x)2
To remove h, we can set these two equations equal to one another.
Remember, x = a * cos(B) so to remove x, substitute this in.
a2 + c2 - 2cb * cos(B) = b2
Since this is a general triangle, variable names can be changed and swapped as they are arbitrary, the only thing that has to stay is the relationship between the cosine angle and the side.
Therefore, to make it look better, we can change this to: a2 + b2 - 2ab * cos(\u03b3) = c2. The cosine angle stays opposite to side c, with the previous case opposite to side b.
This is called The Law of Cosines, for its use of the cos function instead of the sin function.
If you notice, The Law of Cosines is actually just an extension of the Pythagorean Theorem! When cos is 90\xb0, cos(90\xb0) = 0. This removes the -2ab term and simply gives us a2 + b2 = c2, which is pretty much a special case of The Law of Cosines.
","
SAS triangles
In this case, The Law of Sines is unusable, leaving us to rely on The Law of Cosines.
We can find the missing side with The Law of Cosines.
We know all three sides, so to find an angle we can simply use The Law of Sines. We do NOT have to worry about the possibility of there being two triangles because if you know the lengths of all three sides, there can be only one possibility for what the angles can be.
","
SSS triangles
To solve SSS triangles, we can use The Law of Cosines. To eliminate ambiguity when solving, solve for the angle of the largest side first. This removes the possibility of a second triangle because solving from the longest side gives us the largest angle.
Proceed with The Law of Sines to get the other angles.
"]},"the-unit-circle-i":{"title":"The Unit Circle I","sections":["
The Unit Circle I
Alexander De Carlo
To truly understand trigonometry, one must comprehend how angles work, understanding the unit circle, its relation to angles, and a novel angle measurement called radians.
Angles reach a maximum of 360\xb0, completing a full rotation in circular motion. This range from 0 - 360\xb0 exists because 360 is a conveniently divisible number (by 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, etc.). While practical for everyday use, it lacks the mathematical precision we desire.
Consider a circle with a radius of 1, making the circumference C = 2\u03c0. This equates a full 360-degree rotation to 2\u03c0 radians, a unit used in advanced trigonometry. To convert between radians and degrees, multiply by 180/\u03c0 or \u03c0/180, respectively.
Now, observe the unit circle on the coordinate plane with an angle traversing it:
Two angles are measured, originating from the right x-axis and extending outward. The blue and red lines, termed terminal rays, gauge the angle size. Notably, the red terminal ray measures externally, while the blue one measures internally, following the counterclockwise motion. For example, angle a is around 135\xb0 or 3\u03c0/4 radians, while angle b is greater. For angles exceeding 360\xb0, they continue counterclockwise, such as 500\xb0 being equivalent to 140\xb0 after a full rotation. Negative angles rotate clockwise; -1\xb0 equals 359\xb0. This insight allows us to divide the unit circle into four quadrants.
Every \u03c0/4 (90\xb0) defines a new quadrant, with x and y lines acting as quadrantal separators.
Now that we understand this improved angle system and its application around the unit circle, let\'s explore its significance and connection to right triangles. Unlike general right triangles, special triangles like the 45-45-90 and 30-60-90 offer exact angle and side measurements.
These special triangles, created using the Pythagorean theorem and specific base splits, provide exact values without trigonometry. Placing these triangles within the unit circle, with a hypotenuse of length 1, allows us to compress them by dividing by 2 and 3, respectively. This results in three triangles per quadrant: 45-45-90, 30-60-90, and 60-30-90. This process enables the mapping of coordinates for various angles on the unit circle, forming a helpful diagram for memorization.
How does this relate to trigonometric functions? The crucial point lies in the fact that the hypotenuse of these triangles is always 1. Thus, sin(\u03b8) = opp, where \\"opp\\" is the y-coordinate, and cos(\u03b8) = x-coordinate. This insight leads to the realization that coordinates on the circle are simply (cos(\u03b8), sin(\u03b8)). For instance, with our special triangle values, sin(\u03c0/4) = sqrt(2)/2.
Examining the unit circle reveals essential insights into trigonometric functions:
Sine and cosine values range between -1 and 1.
Sin and cosine values are equal at two values: \u03c0/4, 5\u03c0/4
In QI and QII, sin is positive; in QIV and QI, cos is positive.
As sin increases, cos decreases, and vice versa. Two different angles return the same value, either in the top or bottom half for sin and left or right for cos (sin(\u03c0 - \u1340) = sin\u1340, cos(2\u03c0 - \u1340) = cos\u1340).
Finally, recognizing that cos(\u1340) = x and sin(\u1340) = y, tan(\u1340), the slope, is found as y/x, making tan(\u1340) = sin(\u1340)/cos(\u1340). Discovering these relationships is termed finding identities, a concept explored further in our trigonometry studies.
"]},"the-unit-circle-ii":{"title":"The Unit Circle II","sections":["
The Unit Circle II
Alexander De Carlo
Let\'s explore some intriguing relationships between trigonometric functions using the unit circle while maintaining clarity and coherence. First, let\'s examine the coordinates of points on circles with radii greater or less than 1.
sin(x) = y * r r * sin(x) = y r * cos(x) = x
Simply multiply by r, the radius. The ratios are still the same.
(r * sin(x), r * cos(x))
Consider the unit circle.
Here, four angles are presented with colored terminal rays. The red and blue rays are reflected across the y-axis, and the blue and yellow across the x-axis. If we examine cos(blue) and cos(yellow), they are the same. Yellow is simply the negative angle of blue; thus, cos(-x) = cos(x). With sine, sin(-x) = -sin(x). Examining the relationship between blue and red, or yellow and purple, we observe that sin(\u03c0 - x) = sin(x), and cos(\u03c0 - x) = -cos(x).
We also observe that cos(2\u03c0 - x) = cos(x), and sin(2\u03c0 - x) = -sin(x). Knowing that tan(x) is the slope, when cos or sin is negative, tan will be negative. Therefore, tan(x + \u03c0) = tan(x), tan(x - \u03c0) = -tan(x), and tan(-x) = -tan(x). These relationships are intuitive and can be quickly deduced from the unit circle.
Now, let\'s explore the relationship between sine and cosine.
When rotating 90\xb0, cos(x), which is x, would be rotated completely vertically upward, becoming sine, and sine would become cosine. Starting from the triangle in the first quadrant, upon rotation, the cosine and sine values swap places, and the sine becomes negative. This process is repeated in the second triangle, yielding the following relationships:
If we examine the tangent values, the cosine and sines are swapped, and one value becomes negative. Adding a right angle, \u03c0/2, yields the negative reciprocal of the original. Therefore:
tan(x + \u03c0/2) = tan(x - \u03c0/2)
More succinctly:
tan(x + \u03c0/2) = -cot(x)
These relationships showcase the beauty of trigonometric functions.
"]},"the-unit-circle-iii":{"title":"The Unit Circle III","sections":["
The Unit Circle III
Alexander De Carlo
While we are acquainted with the representations of cos and sin on the unit circle, it\'s essential to explore how all trigonometric functions manifest and derive identities from these visual connections.
Acknowledging that cos and sin correspond to our x and y values, the question arises: how do we express tan as a length measure? Leveraging the unit circle\'s triangle with a hypotenuse of 1, we can establish a right angle from the (cos, sin) point.
Upon closer inspection, we observe that tan represents that line, given that tan is opp/adj, and adj is 1. This elucidates the name \\"tan,\\" signifying its tangent relationship to the circle. Additionally, it visually explains why tan(\u03c0/2) and tan(\u03c0/4) are undefined, extending infinitely without intersecting the x-axis. To determine sec, representing adj/opp, we leverage the triangle formed with tan, realizing that sec must denote the hypotenuse of that triangle.
Applying a similar process, we can ascertain cotangent.
Similarly, the derivation of cosecant follows a comparable procedure.
Exploring the relationships between trigonometric functions through the Pythagorean theorem, we unveil crucial connections:
These relationships are commonly known as the Pythagorean Identities. Among them, the most significant is sin2(x) + cos2(x) = 1, a fundamental identity that merits in-depth study.
"]},"using-the-pythagorean-identity-sine-and-cosine":{"title":"Using the Pythagorean Identity: Sine and Cosine","sections":["
Then we can get the adjacent side from how we calculate cosine:
cos(\u03b8) = adj/hyp -sqrt(5) / 3 = x/3 x = -sqrt(5)"]},"trigonometric-functions-graphs-and-problems-i":{"title":"Trigonometric Functions Graphs and Problems I","sections":["
Trigonometric Functions Graphs and Problems I
Alexander De Carlo
While we are familiar with the six trigonometric functions through the unit circle, let\'s explore what these functions look like when graphed. Our x value will represent the angle value in radians, and the y value will be the corresponding return value.
Let\'s examine the special values in the sin function that are rational: 0 - 0, \u03c0/2 - 1, \u03c0 - 0, 3\u03c0/2 - -1, and 2\u03c0 - 0. We observe this cycling motion every 2\u03c0 radians, known as the period. Let\'s visualize this graphically.
Upon graphing and examination, a beautiful cycling pattern emerges.
Since sin(x) = cos(x - \u03c0/2), we anticipate a \u03c0/2 shift to the right for the cosine graph.
","
Now, let\'s apply standard function transformations to see how these functions change with applications such as f(x) = A sin[B(x - h)] + k. This applies to both sin and cos.
Let\'s examine each transformation individually:
When looking at sin and cos, which cycle between 1 and -1, we notice a change by a value from 1 in opposite directions from the average line, 0. Therefore, |A| is the amplitude, representing the distance the maximum and minimum values deviate from the midline point.
Note: A negative sign on A could flip everything across the x-axis. A = (max - min) / 2, as this finds the average distance from the midline.
B represents the period. In a normal wave, it repeats every 2\u03c0. When B increases, the entire thing compresses, resulting in a period of 2\u03c0 / B.
h denotes the horizontal shift, and k is the vertical shift. To find k, you can find the actual average for the midline: (max + min) / 2.
k is 1, making the midline 1, and A is 3, so it will extend up and down by 3. The 2x means this will happen during the period \u03c0.
For g(x):
A is 2. No reflection is needed. B is \u03c0, so the period is 2\u03c0 / \u03c0, which is just 2. x is shifted left by 1, and the midline is -3.
What problems can we model with our knowledge of the graphs of sines and cosines?
Consider the word problem: A Ferris wheel with a diameter of 520 feet has a low point of 30 feet. Every 30 minutes, it completes 1 rotation (starting at the bottom). Let f(x) model the height of a point on the Ferris wheel.
We begin at the lowest point, and since a cos graph always begins at 1, a negative cos must start at the bottom.
If the diameter is 520 feet, then we go 260 feet up and down from the midline, meaning a = 260.
If a full rotation is 30 minutes, then 30 = 2\u03c0 / B. B = \u03c0 / 15.
To start at 30, instead of the current -260, we need to add 290.
So, f(x) = -260cos(\u03c0 / 15 * x) + 290.
"]},"trigonometric-functions-graphs-and-problems-ii":{"title":"Trigonometric Functions Graphs and Problems II","sections":["
Trigonometric Functions Graphs and Problems II
Alexander De Carlo
We graphed cos and sin, but how would we graph their reciprocal functions, csc and sec? Remember that csc(x) = 1/sin(x) and sec(x) = 1/cos(x). This is very important. Realizing that sin and cos are in the denominator, we can also see that whenever they return zero, the fraction would be 1/0, which is undefined. These occur every \u03c0 period and are known as asymptotes.
Here for the csc graph, these are all the points we know are undefined. What happens when we approach those points? They must either approach infinity or negative infinity, as when the denominator gets smaller, it returns a larger and larger value. When the sin graph is in the positive half of the coordinate plane, it must blow up to infinity, and when the sin graph is negative, it will expand to negative infinity. Let\'s use a sine graph to visualize this:
As we see, some parts stay between 0 and 1 and others go from 0 to -1. Our csc graph will touch the sin graph when it peaks at 1 or -1, as 1/1 = 1, but will diverge from there. In the negative parts, to negative infinity. With this information, we can create our cosecant graph in orange.
The secant graph is done the exact same way. This process helps you graph any sec or csc graph, as you simply graph the sin or cos graph and then apply the asymptote concept.
tan(x) and cot(x) graphs:
tan(x) is undefined at \u03c0/2 + 2\u03c0k and 3\u03c0/2 + 2\u03c0k. Graphing tan is quite simple for this reason. We know when tan(x) goes negative and goes positive, so we can make a graph easily.
With cot(x), we notice that the 0\'s will become the asymptotes, BUT the asymptotes in tan(x) will become the 0\'s for cot(x). These ideas are common to see with reciprocal functions.
Comparing the two graphs, they almost look reflected. This comes from the identity that we previously derived that states tan(x + \u03c0/2) = -cot(x). Also, take note that for these graphs, the period is not 2\u03c0, just \u03c0.
y = 3cot(3\u03c0/2(x)).
Here we would have a stretch of 3, and a period of 3/2. Given that cot(x) has asymptotes at 0, \u03c0, 2\u03c0, etc.., we can solve for the asymptotes by taking both of these base asymptotes and then applying the proper transformations. :
3\u03c0/2(x) = 2\u03c0 x = 4/3
3\u03c0/2(x) = \u03c0 x = 2/3
We can see here that the steps in the asymptotes are in increments of 2/3, something we could also have calculated by knowing that the function repeats every \u03c0. Now, you could also calculate the 0\'s by directly applying transformations, but a much simpler way is to realize that the asymptotes of any trigonometric function are equidistant from the x intercept on the base function. This way, we can see that the 0\'s would be 1/3, 1, etc\u2026 Since this is a cot function, the function will continuously decrease. Given all of these baby steps calculated, try graphing this for yourself, since the challenging parts are out of the way."]},"the-inverse-trigonometric-functions":{"title":"The Inverse Trigonometric Functions","sections":["
The Inverse Trigonometric Functions
Alexander De Carlo
The inverse trig functions, instead of returning ratios, give the angle that provides that ratio. The naming of these is quite easy, and simply just adds an \'arc\' to the functions. They can also be named with the inverse sign, what we are familiar with from the right and general angle trigonometry.
arccos(x) OR cos-1(x) arcsin(x) OR sin-1(x) arctan(x) OR tan-1(x) arccsc(x) OR csc-1(x) arcsec(x) OR sec-1(x) arccot(x) OR cot-1(x)
For example, arcsin(1/2) = \u03c0/6
Why not also 5\u03c0/6? To make it a function, we have to restrict its domain to the right half of the unit circle, [-\u03c0/2, \u03c0/2]. arctan(x) is also only between [-\u03c0/2, \u03c0/2]. cos however must be defined as the top half of the unit circle,[0, \u03c0]. Similar to the square root, the domain restriction only exists when these functions are used alone, however in trigonometric equations, every angle will be allowed.
","
Composition
Consider sin(arccos(x)). How would we solve a composite function like this?
sin(arccos(x)) = \u221a(1 - x2)
We can resemble x/1 as a triangle since these are the lengths of the hypotenuse and adjacent.
sin is opp / hyp. So, what is opp? x2 + opp2 = 1. opp = \u221a(1 - x2)
Therefore, sin(arccos(x)) = \u221a(1 - x2)
We can also notice the same process would work for: cos(arcsin(x)).
The arccos and cos would cancel out, leaving \u03c0/2 - x.
\u2234 arccos(sin(x)) = \u03c0/2 - x arcsin(cos(x)) = \u03c0/2 - x
"]},"introduction-to-trigonometric-equations":{"title":"Introduction to Trigonometric Equations","sections":["
Introduction to Trigonometric Equations
Alexander De Carlo
With trigonometric equations, domain restrictions no longer apply; we consider every possible angle unless specified otherwise. For instance, 3\u03c0 / 4 + 2\u03c0k indicates solutions at 3\u03c0 / 4 and all angles obtained by adding or subtracting multiples of 2\u03c0, where k belongs to the set of integers (k \u220a \u2124).
","
2cos(\u1340) = sqrt(3) cos(\u1340) = sqrt(3) / 2 \u1340 = \u03c0 / 6, but can also be 11\u03c0 / 6 because of the unit circle. You can also go fully 2\u03c0 around the circle, allowing the following (in all these cases k \u220a \u2124). \u1340 = \u03c0 / 6 + 2\u03c0k, 11\u03c0 / 6 + 2\u03c0k
This derivation yields the formula: cos(\u03b1 - \u03b2) = cos(\u03b1)cos(\u03b2) + sin(\u03b1)sin(\u03b2), a valuable identity in sum and difference identities.
Now, let\'s explore other identities by setting \u03b2 = -\u03b3:
The product to sum identities are particularly useful anytime we want to convert multiplication of trigonometric functions into addition and subtraction. This will become useful when we eventually put these new identities to use. Now that we know product -> sum identities, how would we go upon getting sum -> product identities? The key is in rearranging \u03b1 and \u03b2 in our product to sum identities.
let u = \u03b1 + \u03b2, v = \u03b1 - \u03b2. We must get u and v in terms of \u03b1 and \u03b2 so we can change the entire equation to be in terms of u and v.
"]},"using-the-sum-and-difference-identity":{"title":"Using the Sum and Difference Identity","sections":["
Using the Sum and Difference Identity
Alexander De Carlo
The sum and difference identities, the most fundamental of the more complex trigonometric identities, can be used in many ways, including finding exact values to previously unknown angles.
We can find unknown angle values by adding or subtracting known ones. For example, let\'s consider \u03c0/6 and \u03c0/4. Summing these together, we get 5\u03c0/12, an angle that we currently have no clue what the exact value is.
What about a base fraction, like \u03c0/12? Knowing the exact value of that could help us find the exact value of a lot more angles. Here, we can use the difference formula.
What subtracts to equal \u03c0/12? \u03c0/3 and \u03c0/4 do!
Certainly isn\'t as nice looking as something like sin(\u03c0/3), which is why these values usually do not need to be memorized. This all goes to show that any trigonometric function will yield an exact value if the angle is in terms of \u03c0, due to the transcendentality of the trigonometric functions and the transcendentality of \u03c0.
sin(5 radians) will never be in exact value (it would be transcendental), but sin(3\u03c0/19) does have an exact value.
","
Simplification of the Sum and Difference Identities
Oftentimes, we find that the sum and difference identities are a bit too general for what we need to deal with. These can simplify to what are known as the double angle identities.
Notice that there may be a relationship between the overall degree of the relationship and the inner coefficient of sin(3\u03b1). We will dive deeper into this in later articles.
"]},"using-identities-single-sine-expression":{"title":"Using Identities: Single Sine Expression","sections":["
Using Identities: Single Sine Expression
Alexander De Carlo
As we journey through mathematics, we may find it very useful to rewrite trigonometric expressions as sine functions. This becomes helpful when solving trigonometric equations in some cases, but also with calculus and graphing. Let\'s see what we can do with a general sine expression.
The general form is A * sin(Bx + C) where A, B and C are constants. Let\'s try to separate this out. We want to work backwards from the answer because we are trying to convert complex expressions to this general form.
Using the sine addition identity, we can expand to A * [sin(Bx) * cos(C) + cos(Bx) * sin(C)]. Notice that sin(C) and cos(C) are constants. Carrying on, let\'s distribute out constant A and move the constants in front.
A * cos(C)sin(Bx) + A * sin(C)cos(Bx)
Since A * cos(C) and A * sin(B) are constant values, let\'s rewrite them as M and N respectively.
M * sin(Bx) + N * cos(Bx) M = A * cos(C) N = A * sin(C)
Hold on a second. We can use one of our identities to find a relationship between M, N, and A to remove those nasty sine and cosine functions.
Now that we have derived some identities and found some tricks within trigonometry, let\'s solve some equations that we could not previously solve before. Let\'s start with a problem involving sine and cosine.
2sin(4x) - 8cos(4x) = sin(2x) * cos(2x) + 3
This looks challenging, but it\'s quite simple with our knowledge of identities. First, we can use the single sine trick, then the product to sum identity. We could also multiply both sides of this equation by 2, and use the double angle identity. Both work here, but generally try to find the fastest solution for when things get harder.
Let\'s solve another, simpler equation for some exact values.
sin(x) * sin(2x) + cos(x) * cos(2x) = sqrt(3) / 2
Here, use the difference formula for cosine. When solving for equations, we should condense without losing a possible root.
cos(x - 2x) = sqrt(3) / 2 cos(-x) = sqrt(3) / 2
Remember that cosine isn\'t affected by that negative value, you can memorize this for efficiency, but it never hurts to sketch the unit circle to see this simple relationship.
cos(x) = sqrt(3) / 2
Here x = \u03c0/6, 11\u03c0/6.
Let\'s solve one more.
sin(x) + sin(3x) = cos(x) [0, 2\u03c0]
We can\'t use previous methods in this article, but we luck out because there isn\'t a constant term. This means if we can get a product form and factor, we can solve it. Good thing there\'s an identity for that!
x = \u03c0/12, 5\u03c0/12 (add \u03c0 to navigate through the interval) x = 13\u03c0/12, 17\u03c0/12
In the end, x = \u03c0/2, 3\u03c0/2, \u03c0/12, 5\u03c0/12, 13\u03c0/12, 17\u03c0/12.
"]},"power-reduction-and-half-angle-identities":{"title":"Power Reduction and Half Angle Identities","sections":["
Power Reduction and Half Angle Identities
Alexander De Carlo
We\'ve moved beyond fundamental trigonometric identities, which are essential for most trigonometry topics, to more advanced identities useful in calculus. The \\"Power Reduction\\" identity simplifies trigonometric functions raised to high powers, making them easier to solve or graph, using the Pythagorean identity:
sin2(\u03b8) + cos2(\u03b8) = 1
For higher-degree functions, we use the double angle identity for cosine:
cos(2\u03b8) = cos2(\u03b8) - sin2(\u03b8)
From the Pythagorean identity, we know:
cos2(\u03b8) = 1 - sin2(\u03b8)
Substituting, we get:
cos(2\u03b8) = 1 - 2sin2(\u03b8)
Solving for sin2(\u03b8):
sin2(\u03b8) = (1 - cos(2\u03b8)) / 2
Similarly, solving for cos2(\u03b8):
cos2(\u03b8) = (1 + cos(2\u03b8)) / 2
Using these, we derive the half-angle identities. Let \ud835\udf03 = \u03b1 / 2 to substitute and simplify:
sin2(\u03b8) = (1 - cos(2\u03b8)) / 2 cos2(\u03b8) = (1 + cos(2\u03b8)) / 2 cos(\u03b1 / 2) = sqrt((1 + cos(\u03b1)) / 2) sin(\u03b1 / 2) = sqrt((1 - cos(\u03b1)) / 2)"]}}')}}]);
\ No newline at end of file
diff --git a/static/js/573.7035b724.chunk.js b/static/js/573.7035b724.chunk.js
new file mode 100644
index 0000000..6e6ffa4
--- /dev/null
+++ b/static/js/573.7035b724.chunk.js
@@ -0,0 +1 @@
+"use strict";(self.webpackChunkmathinfo=self.webpackChunkmathinfo||[]).push([[573],{573:function(s){s.exports=JSON.parse('{"Summary":{"text":"
Algebra
Algebra involves arithemitic, variables, and various rules under its name. However, at its heart, Algebra is the study of equality. Equations, which encompass the study of any relationship, are at the centerpiece of Algebra\'s vast domain. Topics include the study of linear relationships, quadratics, polynomials, rational expressions, radicals, and ways to simplify. Many consider it to be mathematics most important branch of study, as Algebra is one of the few branches that cross its roots with every other topic in math. Its expansiveness allows it be taught from the middle school level to graduate school lectures.
"},"the-basis-of-quadratics":{"title":"The Basis of Quadratics","sections":["
The Basis of Quadratics
Alexander De Carlo
In mathematics, specifically algebra, solving equations for certain variables has been at the forefront in the exploration of new ideas for thousands of years. Quadratic equations have been known to mathematicians since ancient times, however it is believed that a formula to solve all quadratic equations was discovered much later by Arab mathematicians in 9th Century A.D. But what exactly is a quadratic equation?
To be considered quadratic, the highest degree term in a polynomial must be x2. Some examples of this are: x2, 9x2 + 6x + 7, y = 3x2 - 9, etc. Each of these are considered quadratic expressions / equations. But, how do we solve them?
First, there is solving linear equations: which is something we may be more familiar with. Linear equations are always in the form: y = mx + b. Solving these usually consists of using simple operations to isolate the variable x. Let\'s walk through an example.
9x + 15 = 6 9x = -11 (subtract 15 from both sides) x = -11/9 (divide 9 from both sides)
Quite simple! Moving on, it is standard for linear and quadratic equations like these to first be made equal to 0 for simplicity.
ax + b = 0 {a \u2260 0} (if a was 0, there would be no x in the equation!)
How do we solve this? There are no numbers for coefficients! Before we panic, we\'ll simply use the same process we did before to solve this general case.
ax + b = 0 ax = -b (subtract b from both sides) x = -b/a (divide by a on both sides)
We have just created an equation that can be used to solve any linear equation when equal to 0! Try it yourself with the first example, just make sure the equation equals 0.
While this is great and all, we are still trying to discover how to solve quadratic equations, not linear ones. When looking at quadratics, we can see that they either come in 1, 2 or 3 terms.
1 term: x2 = 0 2 terms: x2 + 9 = 0, 3x2 + 5x = 0, etc. 3 terms: 3x2 + 5x + 7 = 0, 11x2 + 6x + 9 = 15 (remember, you can subtract 15 from both sides), etc.
Let\'s attempt to solve all three kinds of these quadratics. One term quadratics are the easiest because they always are 0!
11x2 = 0 x2 = 0 (divide by 11 on both sides) x = 0 (square root both sides)
Something slightly more challenging would be the two term quadratics, however they are very similar to the first.
5x2 + 7 = 11 5x2 = 4 (subtract 7 from both sides) x2 = 4/5 (divide by 5 on both sides) x = +2/\u221a5, -2/\u221a5 (take the square root on both sides, remember this gives us a positive and negative answer)
5x2 + 7x = 0 x(5x + 7) = 0 (factor out an x from the left)
Here we can use the 0 product law, where we know that anything times 0 is 0! So, if we have two separate terms, x and (5x+7) multiplying to equal 0, we know that two cases can arise, where the first term x is 0 OR the second term (5x+7) is 0. This can give us two solutions by breaking up the equation.
x = 0 AND 5x + 7 = 0 x = -7/5 Therefore, x = {-7/5, 0}
For many, all of this is simply review, however understanding these simpler cases is integral to understanding the most general case: the three term quadratic. ALL quadratics can be expressed like this: ax2 + bx + c = 0 {a \u2260 0}. To get the one and two term quadratics, b and c can both be 0. So how do we solve the generic 3 term?
2x2 + 11x - 2 = 0
We can try to move things around and factor things out, but nothing seems to help us. Try things out for yourself! You\'ll see that you won\'t be able to isolate x without a different method of factoring.
To solve this equation, we need to condense it where we have two terms to easily solve, but how do we do that? Let\'s look at the expression (x + y)2 and multiply it out. (x + y)(x + y) using FOIL, gives us x2 + 2yx + y2. An expression like (x + 2)2 becomes x2 + 4x + 4, a three term quadratic that can be factored and condensed! So, if we get any quadratic into x2 + 2yx + y2 form, we might be able to solve quadratics in all cases.
2x2 + 11x - 2 = 0
Let\'s rearrange this into something that could be put in x2 + 2yx + y2 form.
x2 + (11/2)x = 1 (added 2 on both sides, then divided everything by 2).
Looking at the coefficient (11/4), it can be expressed in 2y form: 2 * (11/4) where y = 11/4.
x2 + (2)(11/4)x = 1
We are getting close to the form we want, we just need to take y2 and add it to this equation. We know y = 11/4, so we should square this and add it to both sides!
2 + (2)(11/4)x + (11/4)2 = 1 + (11/4)2
Now, we can factor the left side. Since it\'s in x2 + 2yx + y2 form, let\'s turn it into (x + y)2.
And those are our answers. If we look at the process, we brought the constant term to the other side, divided by the x2 coefficient, and then added a specific constant term to both sides of the equation that allowed us to factor. Considering ax2 + bx + c = 0, it was half of b and then squared: (b/2)2 or b2/4. Adding that then allows us to factor to (x + b/2)2.
This process is called "completing the square" and can be used in all cases. Here is a visualization of the process "completing" the square in a sense.
The large square and the two rectangular areas represent x2 + bx when all added together. With only that, the square isn\'t finished. However, adding the small square area of (b/2)2 completes it into the entire square of the area (x + b/2)2. An example of completeing the square:
3x2 + 6x - 8 = 0 (Step 1: Get x2 + bx alone.) 3x2 + 6x = 8 x2 + 2x = 8/3 (Step 2: Add b2/4 to both sides.) x2 + 2x + 1 = 11/3 (Step 3: Factor to (x + b/2)2 and solve.) (x + 1)2 = 11/3 x + 1 = +/- \u221a(11/3) x = -1 +/- \u221a(11/3)
Perhaps there is a formula to solve all quadratics like linear equations. Now that we know the process, let\'s see if we can solve the classic ax2 + bx + c = 0.
ax2 + bx + c = 0 ax2 + bx = -c x2 + (b/a)x = -c/a (The b/a coefficient will act like the "b" in our method.) x2 + (b/a)x + b2/4a2 = -c/a + b2/4a2 [x + (b/2a)]2 = -c/a + b2/4a2 (To make things neater on the right, turn it into a fraction by multiplying -c/a * 4a in the numerator and denominator to get the common base 4a2.) [x + (b/2a)]2 = (b2 - 4ac)/4a2 x + (b/2a) = \u221a(b2 - 4ac)/\u221a4a2 x + (b/2a) = +/- \u221a[(b2 - 4ac)/2a] x = -b/2a +/- \u221a[(b2 - 4ac)/2a] (Don\'t forget the common base: 2a.) x = [-b +/- \u221a(b2 - 4ac)]/2a
We\'ve finally arrived at the formula to solve any quadratic we would like! It\'s called the quadratic formula and it solves for x in ANY quadratic.
4x2 + 9x - 8 = 0 (a = 4, b = 9 and c = -8.)
2x2 - 14 = 0 (a = 2, b = 0 and c = -14.)
x2 = 0 (a = 1, b = 0 and c = 0.)
The quadratic formula is often overkill for many problems, as a lot can be solved with simpler methods such as factoring. However, the quadratic formula is the only one that can solve ALL quadratics.
\\\\[ x = \\\\frac{{-b \\\\pm \\\\sqrt{{b^2 - 4ac}}}}{{2a}} \\\\]"]},"studying-the-quadratic-formula":{"title":"Studying the Quadratic Formula","sections":["
Studying the Quadratic Formula
Alexander De Carlo
In the quadratic formula, the solution equations for the quadratic polynomials of ax2 + bx + c = 0, a != 0, is expressed as x = [-b \xb1 \u221a(b2 - 4ac)] / 2a. At its core, there are the most important parts: \xb1 and b2 - 4ac. The \xb1 gives us the ability for 2 possible solutions, however what lies in the radical: b2 - 4ac, determines the nature of these roots. This is why we refer to it as the "discriminant". When b2 - 4ac = 0, there is always 1 solution to the quadratic, as you either add or subtract 0 in the equation. This is considered the "double root", and is graphically represented with a parabola that touches the x axis only once. When b2 - 4ac > 0, there are always 2 solutions, and when b2 - 4ac < 0, there are 2 complex solutions intuitively. These complex solutions always come in conjugates, as the complex component is either added or subtracted from the real component -b. For example, if given 3 - 6i as a root, you immediately know that 3 + 6i must be the other root.
","
It immediately can be noticed that finding the sums or products of roots from polynomial quadratics must always yield real results, as conjugates when added or multiplied must always have their complex components disappear.
(a + bi) + (a - bi) = 2a (a + bi)(a - bi) = a2 - abi + abi - b2i2 (a + bi)(a - bi) = a2 + b2
What does this mean about the sums and products of all quadratic roots?
It is quite fascinating to see that by adding and multiplying the roots of a quadratic, very simple equations arise from them. These formulas can be used to build certain quadratic equations with certain roots and properties by using the sum and product of the roots. Perhaps the biggest takeaway could be the relationship shown between the coefficient terms and constant terms of a quadratic equation and how exactly they relate to the problem at hand.
"]}}')}}]);
\ No newline at end of file
diff --git a/static/js/573.a627a8b7.chunk.js b/static/js/573.a627a8b7.chunk.js
deleted file mode 100644
index b5f9b8d..0000000
--- a/static/js/573.a627a8b7.chunk.js
+++ /dev/null
@@ -1 +0,0 @@
-"use strict";(self.webpackChunkmathinfo=self.webpackChunkmathinfo||[]).push([[573],{573:function(s){s.exports=JSON.parse('{"Summary":{"text":"
Algebra
Algebra involves arithemitic, variables, and various rules under its name. However, at its heart, Algebra is the study of equality. Equations, which encompass the study of any relationship, are at the centerpiece of Algebra\'s vast domain. Topics include the study of linear relationships, quadratics, polynomials, rational expressions, radicals, and ways to simplify. Many consider it to be mathematics most important branch of study, as Algebra is one of the few branches that cross its roots with every other topic in math. Its expansiveness allows it be taught from the middle school level to graduate school lectures.
","image":"
Algebra graphic.
"},"the-basis-of-quadratics":{"title":"The Basis of Quadratics","sections":[{"text":"
The Basis of Quadratics
Alexander De Carlo
In mathematics, specifically algebra, solving equations for certain variables has been at the forefront in the exploration of new ideas for thousands of years. Quadratic equations have been known to mathematicians since ancient times, however it is believed that a formula to solve all quadratic equations was discovered much later by Arab mathematicians in 9th Century A.D. But what exactly is a quadratic equation?
To be considered quadratic, the highest degree term in a polynomial must be x2. Some examples of this are: x2, 9x2 + 6x + 7, y = 3x2 - 9, etc. Each of these are considered quadratic expressions / equations. But, how do we solve them?
First, there is solving linear equations: which is something we may be more familiar with. Linear equations are always in the form: y = mx + b. Solving these usually consists of using simple operations to isolate the variable x. Let\'s walk through an example.
9x + 15 = 6 9x = -11 (subtract 15 from both sides) x = -11/9 (divide 9 from both sides)
Quite simple! Moving on, it is standard for linear and quadratic equations like these to first be made equal to 0 for simplicity.
ax + b = 0 {a \u2260 0} (if a was 0, there would be no x in the equation!)
How do we solve this? There are no numbers for coefficients! Before we panic, we\'ll simply use the same process we did before to solve this general case.
ax + b = 0 ax = -b (subtract b from both sides) x = -b/a (divide by a on both sides)
We have just created an equation that can be used to solve any linear equation when equal to 0! Try it yourself with the first example, just make sure the equation equals 0.
While this is great and all, we are still trying to discover how to solve quadratic equations, not linear ones. When looking at quadratics, we can see that they either come in 1, 2 or 3 terms.
1 term: x2 = 0 2 terms: x2 + 9 = 0, 3x2 + 5x = 0, etc. 3 terms: 3x2 + 5x + 7 = 0, 11x2 + 6x + 9 = 15 (remember, you can subtract 15 from both sides), etc.
Let\'s attempt to solve all three kinds of these quadratics. One term quadratics are the easiest because they always are 0!
11x2 = 0 x2 = 0 (divide by 11 on both sides) x = 0 (square root both sides)
Something slightly more challenging would be the two term quadratics, however they are very similar to the first.
5x2 + 7 = 11 5x2 = 4 (subtract 7 from both sides) x2 = 4/5 (divide by 5 on both sides) x = +2/\u221a5, -2/\u221a5 (take the square root on both sides, remember this gives us a positive and negative answer)
5x2 + 7x = 0 x(5x + 7) = 0 (factor out an x from the left)
Here we can use the 0 product law, where we know that anything times 0 is 0! So, if we have two separate terms, x and (5x+7) multiplying to equal 0, we know that two cases can arise, where the first term x is 0 OR the second term (5x+7) is 0. This can give us two solutions by breaking up the equation.
x = 0 AND 5x + 7 = 0 x = -7/5 Therefore, x = {-7/5, 0}
For many, all of this is simply review, however understanding these simpler cases is integral to understanding the most general case: the three term quadratic. ALL quadratics can be expressed like this: ax2 + bx + c = 0 {a \u2260 0}. To get the one and two term quadratics, b and c can both be 0. So how do we solve the generic 3 term?
2x2 + 11x - 2 = 0
We can try to move things around and factor things out, but nothing seems to help us. Try things out for yourself! You\'ll see that you won\'t be able to isolate x without a different method of factoring.
To solve this equation, we need to condense it where we have two terms to easily solve, but how do we do that? Let\'s look at the expression (x + y)2 and multiply it out. (x + y)(x + y) using FOIL, gives us x2 + 2yx + y2. An expression like (x + 2)2 becomes x2 + 4x + 4, a three term quadratic that can be factored and condensed! So, if we get any quadratic into x2 + 2yx + y2 form, we might be able to solve quadratics in all cases.
2x2 + 11x - 2 = 0
Let\'s rearrange this into something that could be put in x2 + 2yx + y2 form.
x2 + (11/2)x = 1 (added 2 on both sides, then divided everything by 2).
Looking at the coefficient (11/4), it can be expressed in 2y form: 2 * (11/4) where y = 11/4.
x2 + (2)(11/4)x = 1
We are getting close to the form we want, we just need to take y2 and add it to this equation. We know y = 11/4, so we should square this and add it to both sides!
2 + (2)(11/4)x + (11/4)2 = 1 + (11/4)2
Now, we can factor the left side. Since it\'s in x2 + 2yx + y2 form, let\'s turn it into (x + y)2.
And those are our answers. If we look at the process, we brought the constant term to the other side, divided by the x2 coefficient, and then added a specific constant term to both sides of the equation that allowed us to factor. Considering ax2 + bx + c = 0, it was half of b and then squared: (b/2)2 or b2/4. Adding that then allows us to factor to (x + b/2)2.
This process is called "completing the square" and can be used in all cases. Here is a visualization of the process "completing" the square in a sense.
The large square and the two rectangular areas represent x2 + bx when all added together. With only that, the square isn\'t finished. However, adding the small square area of (b/2)2 completes it into the entire square of the area (x + b/2)2. An example of completeing the square:
3x2 + 6x - 8 = 0 (Step 1: Get x2 + bx alone.) 3x2 + 6x = 8 x2 + 2x = 8/3 (Step 2: Add b2/4 to both sides.) x2 + 2x + 1 = 11/3 (Step 3: Factor to (x + b/2)2 and solve.) (x + 1)2 = 11/3 x + 1 = +/- \u221a(11/3) x = -1 +/- \u221a(11/3)
Perhaps there is a formula to solve all quadratics like linear equations. Now that we know the process, let\'s see if we can solve the classic ax2 + bx + c = 0.
ax2 + bx + c = 0 ax2 + bx = -c x2 + (b/a)x = -c/a (The b/a coefficient will act like the "b" in our method.) x2 + (b/a)x + b2/4a2 = -c/a + b2/4a2 [x + (b/2a)]2 = -c/a + b2/4a2 (To make things neater on the right, turn it into a fraction by multiplying -c/a * 4a in the numerator and denominator to get the common base 4a2.) [x + (b/2a)]2 = (b2 - 4ac)/4a2 x + (b/2a) = \u221a(b2 - 4ac)/\u221a4a2 x + (b/2a) = +/- \u221a[(b2 - 4ac)/2a] x = -b/2a +/- \u221a[(b2 - 4ac)/2a] (Don\'t forget the common base: 2a.) x = [-b +/- \u221a(b2 - 4ac)]/2a
We\'ve finally arrived at the formula to solve any quadratic we would like! It\'s called the quadratic formula and it solves for x in ANY quadratic.
4x2 + 9x - 8 = 0 (a = 4, b = 9 and c = -8.)
2x2 - 14 = 0 (a = 2, b = 0 and c = -14.)
x2 = 0 (a = 1, b = 0 and c = 0.)
The quadratic formula is often overkill for many problems, as a lot can be solved with simpler methods such as factoring. However, the quadratic formula is the only one that can solve ALL quadratics.
\\\\[ x = \\\\frac{{-b \\\\pm \\\\sqrt{{b^2 - 4ac}}}}{{2a}} \\\\]","image":"
The graph of y = x2.
"}]},"studying-the-quadratic-formula":{"title":"Studying the Quadratic Formula","sections":[{"text":"
Studying the Quadratic Formula
Alexander De Carlo
In the quadratic formula, the solution equations for the quadratic polynomials of ax2 + bx + c = 0, a != 0, is expressed as x = [-b \xb1 \u221a(b2 - 4ac)] / 2a. At its core, there are the most important parts: \xb1 and b2 - 4ac. The \xb1 gives us the ability for 2 possible solutions, however what lies in the radical: b2 - 4ac, determines the nature of these roots. This is why we refer to it as the "discriminant". When b2 - 4ac = 0, there is always 1 solution to the quadratic, as you either add or subtract 0 in the equation. This is considered the "double root", and is graphically represented with a parabola that touches the x axis only once. When b2 - 4ac > 0, there are always 2 solutions, and when b2 - 4ac < 0, there are 2 complex solutions intuitively. These complex solutions always come in conjugates, as the complex component is either added or subtracted from the real component -b. For example, if given 3 - 6i as a root, you immediately know that 3 + 6i must be the other root.
","image":"
The graph of y = x2. If this was inputted into the quadratic formula, b2 - 4ac would equal 0, giving this equation a double root.
"},{"text":"
It immediately can be noticed that finding the sums or products of roots from polynomial quadratics must always yield real results, as conjugates when added or multiplied must always have their complex components disappear.
(a + bi) + (a - bi) = 2a (a + bi)(a - bi) = a2 - abi + abi - b2i2 (a + bi)(a - bi) = a2 + b2
What does this mean about the sums and products of all quadratic roots?
It is quite fascinating to see that by adding and multiplying the roots of a quadratic, very simple equations arise from them. These formulas can be used to build certain quadratic equations with certain roots and properties by using the sum and product of the roots. Perhaps the biggest takeaway could be the relationship shown between the coefficient terms and constant terms of a quadratic equation and how exactly they relate to the problem at hand.
","image":"
The graph of y = x2 + 3x - 4, and its roots: x = -4 and x = 1.
"}]}}')}}]);
\ No newline at end of file
diff --git a/static/js/775.362ff3f8.chunk.js b/static/js/775.362ff3f8.chunk.js
deleted file mode 100644
index fb6d8e7..0000000
--- a/static/js/775.362ff3f8.chunk.js
+++ /dev/null
@@ -1 +0,0 @@
-"use strict";(self.webpackChunkmathinfo=self.webpackChunkmathinfo||[]).push([[775],{775:function(e){e.exports=JSON.parse('{"Summary":{"text":"
Proofs
Proofs provide the backbone of mathematics, as they rigorously explain why things are the way they are. While much of math has real life application which betters our lives, proofs usually consist of pure logic and mathematical ideas to solely explain and prove some mathematical topic. Given proofs uphold logical soundness, it is considered to be the most complex and difficult branch of mathematics. Proofs serve to better provide a deeper understanding to mathematical concepts as the idea goes against simply providing a formula or statement without evidence for why that it is true.
","image":"
Proofs graphic.
"},"pythagorean-theorem-and-distance-formula-derivation":{"title":"The Pythagorean Theorem and The Distance Formula Derivation","sections":[{"text":"
The Pythagorean Theorem and The Distance Formula Derivation
Alexander De Carlo
The Pythagorean Theorem is one of the most fundamental theorems in all of Mathematics, but little are familiar with the proof of such a formula. This theorem is used to find the length of any diagonal line which gives us a formula for how to calculate the distance between two points in space. But how was it proven?
Let\'s throw away any knowledge of the Pythagorean Theorem and simply try to answer this question: What is the length of this line on the coordinate plane?
Looking at the image, we can see that A = (-6, -3), and B = (7, 6). We can easily calculate distance when the points are on the same dimension, but how do we calculate distance when two dimensions are involved? Looking at A and B, we can see they are 13 units away horizontally and 12 units away vertically. Let\'s represent this as a triangle.
","image":"
Diagram of right triangle.
"},{"text":"
Here, c is our unknown, b is 13 and a is 12. This creates a right triangle. Right triangles are special for many reasons - a big factor being their properties of similarity.
Above shows how each component angle of each triangle must be all equivalent when Triangle 3 is bisected to where it internally creates Triangle 1 and Triangle 2. What similarity tells us is that the ratios of the side lengths must be equivalent. This is because while the triangles can have different sizes overall, their interior angles dictate the side length proportions. Therefore, when two angles are shown to be equal, the third must also be equal, creating these proportions. For example, in the previous model: a / b = e / d. Right triangles are the only triangles where you can have this and its two components similar, due to the creation of two equal degree angles when bisecting the hypotenuse. You can mess around with these ratios and find many relationships, so let\'s try to expand and discover something we didn\'t expect.
","image":"
A triangle representation of the distance between points A and B.
"},{"text":"
Consider two ratios: a / c = e / a and b / c = f / b. These ratios are particularly interesting because they contain a and b side lengths of the main triangle twice in their respective equations. By cross multiplying, a2 = ce and b2 = cf. Since c is in both of these equations, let\'s add them together to simplify things: a2 + b2 = ce + cf. It is clear that we can factor out c: a2 + b2 = c(e + f). Looking at the diagram: e + f = c, meaning we can substitute e + f out for c and get a2 + b2 = c2. This relationship is not obvious at all, but can beautifully be derived from very simple similarity relationships. Some good news here is that since c is isolated, we can solve for the diagonal side length of a triangle given only vertical and horizontal lengths, being c = sqrt(a2 + b2). We can finally get that equation for the distance of any two points on a plane.
let D be the distance C = sqrt(a2 + b2) (Substitute out hypotenuse length c.) D = sqrt(a2 + b2) (a is our vertical length and b is our horizontal length.) a = | y1 - y2 | b = | x1 - x2 | D = sqrt((y2 - y1)2 + (x2 - x1)2) (The absolute values are removed because the whole thing is squared, and since negative * negative = positive, the order of variables doesn\'t matter.)
Now we can find the distance between A = (-6, -3) and B = (7, 6). With proper substitution we get:
D = sqrt((-3 - 6)2 + (-6 - 7)2) D = sqrt(81 + 169) D = sqrt(250) = \u224815.811","image":""}]},"three-dimensional-pythagorean-theorem":{"title":"Three Dimensional Pythagorean Theorem","sections":[{"text":"
Three Dimensional Pythagorean Theorem
Alexander De Carlo
Deriving the Pythagorean Theorem in three dimensions is surprisingly simple, as it builds on the 2D Pythagorean Theorem. The Pythagorean Theorem in two dimensions states: a2 + b2 = c2, where a and b are the legs of a right triangle, and c is the hypotenuse.
From this three dimensional illustration, we want to find what s is from non-angled lines: x, y and z.With help from the Pythagorean Theorem, we can deduce: c2 = x2 + y2. We can also notice that on the rectangular prism, z stays completely vertical. With that information, it is time to use the Pythagorean Theorem on the triangle containing s.
z2 + (sqrt(x2 + y2))2 = s2 x2 + y2 + z2 = s2 (Absolute value signs are not needed when canceling out due to x and y both squared.)
With changing the variable names to match the Pythagorean Theorem, we get: a2 + b2 + c2 = d2. There is a clear pattern when increasing one dimension, and it\'s by simply adding a new variable. To find a distance formula in 3D space, the process is exactly the same:
D = sqrt((x2 - x1)2 + (y2 - y1)2 + (z2 - z1)2)
When going up a dimension, all you need to do is add the subtracted coordinates squared. You can see this in action when going down dimensions. If D = sqrt((x2 - x1)2 + (y2 - y1)2 + (z2 - z1)2), decreasing to a single dimension gives us D = sqrt(x2 - x1)2), which when canceled is: D = | x2 - x1 |, the formula for finding the distance between two points on the number line.
Therefore, for n dimensions, the Pythagorean Theorem theoretically would be: a12 + \u2026 + an2 = c2, where n is the amount of dimensions, each term is the distance in its dimension, and c is the hypotenuse.
","image":"
Diagram of right triangle.
"}]}}')}}]);
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+"use strict";(self.webpackChunkmathinfo=self.webpackChunkmathinfo||[]).push([[775],{775:function(e){e.exports=JSON.parse('{"Summary":{"text":"
Proofs
Proofs provide the backbone of mathematics, as they rigorously explain why things are the way they are. While much of math has real life application which betters our lives, proofs usually consist of pure logic and mathematical ideas to solely explain and prove some mathematical topic. Given proofs uphold logical soundness, it is considered to be the most complex and difficult branch of mathematics. Proofs serve to better provide a deeper understanding to mathematical concepts as the idea goes against simply providing a formula or statement without evidence for why that it is true.
"},"pythagorean-theorem-and-distance-formula-derivation":{"title":"The Pythagorean Theorem and The Distance Formula Derivation","sections":["
The Pythagorean Theorem and The Distance Formula Derivation
Alexander De Carlo
The Pythagorean Theorem is one of the most fundamental theorems in all of Mathematics, but little are familiar with the proof of such a formula. This theorem is used to find the length of any diagonal line which gives us a formula for how to calculate the distance between two points in space. But how was it proven?
Let\'s throw away any knowledge of the Pythagorean Theorem and simply try to answer this question: What is the length of this line on the coordinate plane?
Looking at the image, we can see that A = (-6, -3), and B = (7, 6). We can easily calculate distance when the points are on the same dimension, but how do we calculate distance when two dimensions are involved? Looking at A and B, we can see they are 13 units away horizontally and 12 units away vertically. Let\'s represent this as a triangle.
","
Here, c is our unknown, b is 13 and a is 12. This creates a right triangle. Right triangles are special for many reasons - a big factor being their properties of similarity.
Above shows how each component angle of each triangle must be all equivalent when Triangle 3 is bisected to where it internally creates Triangle 1 and Triangle 2. What similarity tells us is that the ratios of the side lengths must be equivalent. This is because while the triangles can have different sizes overall, their interior angles dictate the side length proportions. Therefore, when two angles are shown to be equal, the third must also be equal, creating these proportions. For example, in the previous model: a / b = e / d. Right triangles are the only triangles where you can have this and its two components similar, due to the creation of two equal degree angles when bisecting the hypotenuse. You can mess around with these ratios and find many relationships, so let\'s try to expand and discover something we didn\'t expect.
","
Consider two ratios: a / c = e / a and b / c = f / b. These ratios are particularly interesting because they contain a and b side lengths of the main triangle twice in their respective equations. By cross multiplying, a2 = ce and b2 = cf. Since c is in both of these equations, let\'s add them together to simplify things: a2 + b2 = ce + cf. It is clear that we can factor out c: a2 + b2 = c(e + f). Looking at the diagram: e + f = c, meaning we can substitute e + f out for c and get a2 + b2 = c2. This relationship is not obvious at all, but can beautifully be derived from very simple similarity relationships. Some good news here is that since c is isolated, we can solve for the diagonal side length of a triangle given only vertical and horizontal lengths, being c = sqrt(a2 + b2). We can finally get that equation for the distance of any two points on a plane.
let D be the distance C = sqrt(a2 + b2) (Substitute out hypotenuse length c.) D = sqrt(a2 + b2) (a is our vertical length and b is our horizontal length.) a = | y1 - y2 | b = | x1 - x2 | D = sqrt((y2 - y1)2 + (x2 - x1)2) (The absolute values are removed because the whole thing is squared, and since negative * negative = positive, the order of variables doesn\'t matter.)
Now we can find the distance between A = (-6, -3) and B = (7, 6). With proper substitution we get:
D = sqrt((-3 - 6)2 + (-6 - 7)2) D = sqrt(81 + 169) D = sqrt(250) = \u224815.811"]},"three-dimensional-pythagorean-theorem":{"title":"Three Dimensional Pythagorean Theorem","sections":["
Three Dimensional Pythagorean Theorem
Alexander De Carlo
Deriving the Pythagorean Theorem in three dimensions is surprisingly simple, as it builds on the 2D Pythagorean Theorem. The Pythagorean Theorem in two dimensions states: a2 + b2 = c2, where a and b are the legs of a right triangle, and c is the hypotenuse.
From this three dimensional illustration, we want to find what s is from non-angled lines: x, y and z.With help from the Pythagorean Theorem, we can deduce: c2 = x2 + y2. We can also notice that on the rectangular prism, z stays completely vertical. With that information, it is time to use the Pythagorean Theorem on the triangle containing s.
z2 + (sqrt(x2 + y2))2 = s2 x2 + y2 + z2 = s2 (Absolute value signs are not needed when canceling out due to x and y both squared.)
With changing the variable names to match the Pythagorean Theorem, we get: a2 + b2 + c2 = d2. There is a clear pattern when increasing one dimension, and it\'s by simply adding a new variable. To find a distance formula in 3D space, the process is exactly the same:
D = sqrt((x2 - x1)2 + (y2 - y1)2 + (z2 - z1)2)
When going up a dimension, all you need to do is add the subtracted coordinates squared. You can see this in action when going down dimensions. If D = sqrt((x2 - x1)2 + (y2 - y1)2 + (z2 - z1)2), decreasing to a single dimension gives us D = sqrt(x2 - x1)2), which when canceled is: D = | x2 - x1 |, the formula for finding the distance between two points on the number line.
Therefore, for n dimensions, the Pythagorean Theorem theoretically would be: a12 + \u2026 + an2 = c2, where n is the amount of dimensions, each term is the distance in its dimension, and c is the hypotenuse.
"]}}')}}]);
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similarity index 70%
rename from static/js/main.9371a6fb.js
rename to static/js/main.d54655ad.js
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customers finds 320 are repeat buyers.","A dairy company claims that only 5% of its milk cartons are defective; an inspection of 300 cartons finds 20 defects.","A phone manufacturer claims that 99% of its phones pass quality tests; out of 1,000 phones tested, 985 pass.","A survey suggests that 75% of people prefer online shopping; a sample of 600 people finds 450 prefer online shopping.","A restaurant claims that 80% of its customers are satisfied with their service; a survey of 250 customers shows 200 are satisfied.","A university states that 70% of applicants are accepted into their programs; out of 400 applicants, 280 are accepted.","A survey claims that 65% of adults exercise regularly; a sample of 500 adults finds 320 who exercise regularly.","A social media company asserts that 90% of users are satisfied with their privacy settings; a poll of 1,000 users finds 850 satisfied.","A retailer claims that 30% of customers use their loyalty program; out of 800 customers surveyed, 240 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employees.","A research study compares the effectiveness of two diets; Diet A results in weight loss for 70 out of 100 participants, and Diet B results in weight loss for 65 out of 95 participants.","A researcher compares the participation rates in two extracurricular activities; Activity 1 has 50 participants out of 200 students, and Activity 2 has 60 participants out of 210 students.","A company compares the success rates of two sales teams; Team A closes 120 out of 300 deals, and Team B closes 110 out of 280 deals.","A hospital compares the infection rates after two types of surgeries; Surgery A has 5 infections out of 200 patients, and Surgery B has 10 infections out of 250 patients.","A health study compares the vaccination rates between two regions; Region 1 has 500 vaccinated out of 1,000 residents, and Region 2 has 450 vaccinated out of 950 residents.","A university compares the acceptance rates of two departments; Department A accepts 200 out of 400 applicants, and Department B accepts 180 out of 370 applicants."],"2":["A researcher wants to test if the average weight of a sample of 30 apples is different from the known population mean of 150 grams.","A teacher wants to test if the average score of a class of 25 students is higher than the school\'s average score of 75.","A factory manager wants to determine if the average length of a sample of 20 rods is different from the standard length of 10 meters.","A nutritionist wants to test if the average calorie intake of a sample of 50 people is lower than the recommended 2,000 calories per day.","A biologist wants to know if the average height of a sample of 15 plants differs from the expected 30 centimeters.","A car manufacturer wants to determine if the average fuel efficiency of a sample of 40 cars is higher than the industry standard of 25 miles per gallon.","A company wants to test if the average lifespan of a sample of 35 batteries is different from the advertised 500 hours.","A scientist wants to test if 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known concentration of 0.5 mol/L.","A sports analyst wants to test if the average points scored by a sample of 15 basketball players is different from the league average of 20 points.","A zoologist wants to determine if the average weight of a sample of 12 elephants is higher than the known average of 5,000 kilograms.","A pharmacist wants to test if the average dissolution time of a sample of 20 pills is different from the standard 30 minutes.","A financial analyst wants to determine if the average return of a sample of 25 investments is different from the expected 8%.","A meteorologist wants to test if the average rainfall of a sample of 40 days is lower than the historical average of 3 inches."],"3":["A researcher compares the average test scores of two different teaching methods; Method A with a mean score of 85 from 30 students, and Method B with a mean score of 80 from 28 students.","A company wants to determine if there is a difference in average productivity between two shifts; Shift 1 has an average of 50 units from 40 workers, and Shift 2 has an average of 55 units from 35 workers.","A nutritionist compares the average weight loss of two diets; Diet A results in an average loss of 5 kg for 25 people, and Diet B results in an average loss of 6 kg for 30 people.","A biologist wants to test if there is a difference in average growth between two species of plants; Species X grows an average of 10 cm for 20 plants, and Species Y grows an average of 12 cm for 18 plants.","A car manufacturer compares the average fuel efficiency of two car models; Model A has an average efficiency of 25 mpg from 50 cars, and Model B has an average efficiency of 27 mpg from 45 cars.","A school wants to determine if there is a difference in average math scores between two classes; Class 1 has an average score of 78 from 30 students, and Class 2 has an average score of 82 from 28 students.","A health researcher compares the average cholesterol levels of two diets; Diet A has an average level of 190 mg/dL for 20 people, and Diet B has an average level of 185 mg/dL for 25 people.","A sports analyst wants to test if there is a difference in average points scored by two basketball teams; Team A has an average of 90 points from 15 games, and Team B has an average of 88 points from 18 games.","A sociologist compares the average income of two neighborhoods; Neighborhood A has an average income of $50,000 from 40 households, and Neighborhood B has an average income of $55,000 from 35 households.","A psychologist wants to determine if there is a difference in average stress levels between two groups; Group A has an average stress score of 30 from 25 participants, and Group B has an average stress score of 28 from 30 participants.","A marketing team compares the average sales of two advertising campaigns; Campaign 1 generates an average of $10,000 from 20 runs, and Campaign 2 generates an average of $12,000 from 18 runs.","A teacher compares the average reading scores of two different reading programs; Program A has an average score of 85 from 25 students, and Program B has an average score of 88 from 28 students.","A dietitian wants to test if there is a difference in average blood sugar levels between two diets; Diet A results in an average level of 95 mg/dL for 20 people, and Diet B results in an average level of 90 mg/dL for 22 people.","A researcher compares the average time spent on social media by two age groups; Group 1 spends an average of 2 hours from 30 participants, and Group 2 spends an average of 1.5 hours from 25 participants.","A university wants to determine if there is a difference in average starting salaries between graduates of two programs; Program A has an average salary of $50,000 from 20 graduates, and Program B has an average salary of $55,000 from 22 graduates.","A health researcher compares the average blood pressure levels of two different exercise programs; Program A results in an average level of 120 mmHg for 15 participants, and Program B results in an average level of 115 mmHg for 18 participants.","A company wants to test if there is a difference in average customer satisfaction between two products; Product A has an average satisfaction score of 4.5 from 50 customers, and Product B has an average satisfaction score of 4.2 from 55 customers.","A researcher compares the average time to complete a task between two software tools; Tool A takes an average of 30 minutes from 20 users, and Tool B takes an average of 25 minutes from 22 users.","A university wants to determine if there is a difference in average GPA between two majors; Major A has an average GPA of 3.2 from 30 students, and Major B has an average GPA of 3.5 from 28 students.","A scientist wants to test if there is a difference in average growth rates between two fertilizers; Fertilizer A results in an average growth of 5 cm for 25 plants, and Fertilizer B results in an average growth of 6 cm for 30 plants."],"4":["A researcher tests if there is a difference in blood pressure before and after a treatment in the same group of 30 patients.","A teacher wants to determine if students\' test scores improve after a new teaching method by comparing scores from the same 25 students before and after the method is implemented.","A company wants to test if a new training program improves employee productivity by comparing productivity levels of the same 20 employees before and after the training.","A doctor wants to know if a new medication affects cholesterol levels by comparing the cholesterol levels of the same 15 patients before and after taking the medication.","A coach wants to test if a new exercise routine improves athletic performance by comparing the performance metrics of the same 18 athletes before and after the routine.","A scientist wants to determine if a chemical treatment affects plant growth by comparing the heights of the same 25 plants before and after the treatment.","A nutritionist wants to test if a dietary supplement affects weight by comparing the weights of the same 30 individuals before and after taking the supplement.","A psychologist wants to determine if a therapy method reduces anxiety by comparing the anxiety levels of the same 20 patients before and after the therapy.","A researcher wants to test if a new study technique improves memory retention by comparing test scores of the same 15 students before and after using the technique.","A company wants to determine if a new software update improves processing speed by comparing the speed of the same 25 computers before and after the update.","A fitness trainer wants to test if a new workout plan increases strength by comparing the lifting weights of the same 20 clients before and after the plan.","A scientist wants to test if a new fertilizer affects crop yield by comparing the yields of the same 10 plots before and after using the fertilizer.","A researcher wants to determine if a new teaching strategy affects engagement by comparing the engagement scores of the same 30 students before and after the strategy.","A company wants to test if a new customer service protocol improves satisfaction by comparing satisfaction scores of the same 40 customers before and after the protocol.","A scientist wants to test if a new diet plan affects blood sugar levels by comparing levels of the same 18 individuals before and after the diet.","A researcher wants to determine if a new therapy reduces depression symptoms by comparing the symptoms of the same 22 patients before and after the therapy.","A fitness coach wants to test if a new running program improves speed by comparing the race times of the same 15 runners before and after the program.","A school wants to determine if a new curriculum improves reading skills by comparing test scores of the same 28 students before and after the curriculum.","A company wants to test if a new marketing strategy increases sales by comparing the sales figures of the same 35 products before and after the strategy.","A researcher wants to test if a new learning app improves language skills by comparing language test scores of the same 20 students before and after using the app."],"5":["A geneticist wants to test if the observed frequencies of different flower colors match the expected ratios in a Mendelian inheritance experiment.","A dice manufacturer wants to test if a die is fair by comparing the observed frequencies of each face to the expected equal frequencies.","A researcher wants to determine if the observed distribution of blood types in a sample matches the expected distribution in the general population.","A company wants to test if customer preferences for product colors match the expected market research proportions.","A teacher wants to test if the observed grades in a class match the expected distribution based on historical data.","A biologist wants to test if the observed frequencies of different insect species match the expected proportions in an ecosystem.","A marketing team wants to test if the observed preferences for different advertisements match the expected preferences based on a survey.","A researcher wants to determine if the observed frequencies of different genetic traits match the expected ratios in a population.","A game developer wants to test if the observed frequencies of outcomes in a game match the expected probabilities.","A sociologist wants to test if the observed distribution of household sizes matches the expected distribution in a city.","A quality control manager wants to test if the observed defect types in a production process match the expected proportions.","A psychologist wants to determine if the observed frequencies of different personality types match the expected distribution in a sample.","A company wants to test if the observed sales of products in different regions match the expected sales based on market analysis.","A researcher wants to test if the observed distribution of responses to a survey question matches the expected distribution.","A zoologist wants to test if the observed frequencies of different animal behaviors match the expected proportions in a study.","A retail store wants to determine if the observed purchase frequencies of different product categories match the expected frequencies.","A researcher wants to test if the observed distribution of academic majors matches the expected distribution based on enrollment data.","A pollster wants to test if the observed voting preferences match the expected proportions based on pre-election polls.","A scientist wants to determine if the observed frequencies of different chemical reactions match the expected ratios.","A researcher wants to test if the observed distribution of car colors in a parking lot matches the expected distribution."],"6":["A researcher wants to test if there is an association between gender and voting preference in a sample of 500 people.","A sociologist wants to determine if there is a relationship between education level and income category in a survey of 1,000 individuals.","A company wants to test if there is a link between customer age group and product preference in a market analysis.","A health researcher wants to determine if there is an association between smoking status and incidence of lung disease in a study of 2,000 participants.","A teacher wants to test if there is a relationship between study habits and academic performance in a class of 100 students.","A political scientist wants to determine if there is an association between political affiliation and support for a policy in a survey of 800 voters.","A marketing team wants to test if there is a link between advertisement type and purchase decision in an experiment with 300 consumers.","A psychologist wants to determine if there is a relationship between personality type and career choice in a sample of 500 adults.","A company wants to test if there is an association between training program and employee retention in a study of 1,200 employees.","A sociologist wants to determine if there is a link between marital status and happiness level in a survey of 700 people.","A researcher wants to test if there is a relationship between diet type and weight loss success in a study of 400 participants.","A health organization wants to determine if there is an association between exercise frequency and heart disease incidence in a study of 1,500 individuals.","A school wants to test if there is a link between extracurricular activity participation and academic success in a sample of 600 students.","A company wants to determine if there is a relationship between job role and job satisfaction in a survey of 1,000 employees.","A sociologist wants to test if there is an association between neighborhood type and crime rate in a study of 300 neighborhoods.","A researcher wants to determine if there is a relationship between internet usage and social interaction levels in a survey of 800 individuals.","A health researcher wants to test if there is a link between alcohol consumption and liver disease in a study of 1,200 participants.","A political analyst wants to determine if there is an association between region and voting turnout in an analysis of 50 regions.","A company wants to test if there is a relationship between product type and return rate in a study of 500 product returns.","A researcher wants to determine if there is an association between sleep patterns and productivity levels in a study of 300 participants."],"7":["A researcher wants to test if the distribution of blood types is the same across different ethnic groups in a sample of 1,000 people.","A company wants to determine if customer satisfaction levels are consistent across different store locations in a survey of 800 customers.","A school wants to test if the distribution of grades is similar across different classes in a sample of 200 students.","A health researcher wants to determine if the incidence of a disease is the same across different age groups in a study of 1,500 participants.","A marketing team wants to test if the preferences for a new product are consistent across different regions in a market analysis.","A political scientist wants to determine if voting patterns are similar across different districts in an election study.","A sociologist wants to test if the distribution of household sizes is the same across different neighborhoods in a survey of 600 households.","A researcher wants to determine if the frequency of internet usage is consistent across different age groups in a sample of 500 people.","A health organization wants to test if the rates of vaccination are the same across different communities in a study of 1,200 participants.","A company wants to determine if employee retention rates are consistent across different departments in a survey of 1,000 employees.","A researcher wants to test if the distribution of leisure activities is the same across different age groups in a study of 300 participants.","A school wants to determine if participation in extracurricular activities is consistent across different grade levels in a sample of 400 students.","A marketing team wants to test if brand loyalty is the same across different customer segments in a survey of 700 consumers.","A researcher wants to determine if the frequency of exercise is consistent across different income groups in a study of 500 participants.","A health researcher wants to test if the incidence of diabetes is the same across different ethnic groups in a study of 1,000 individuals.","A political analyst wants to determine if the distribution of political affiliations is consistent across different regions in a survey of 800 voters.","A company wants to test if the return rates are the same across different product categories in an analysis of 1,200 returns.","A researcher wants to determine if the distribution of reading habits is consistent across different education levels in a survey of 600 people.","A health organization wants to test if the frequency of health check-ups is the same across different age groups in a study of 1,500 participants.","A school wants to determine if the distribution of test scores is similar across different subjects in a sample of 200 students."],"8":["A researcher wants to test if there is a significant relationship between hours studied and test scores in a sample of 50 students.","A health researcher wants to determine if there is a significant association between daily exercise time and cholesterol levels in a study of 200 individuals.","A company wants to test if there is a significant relationship between advertising expenditure and sales revenue in a sample of 30 months of data.","A biologist wants to determine if there is a significant association between sunlight exposure and plant growth in a study of 40 plants.","A sociologist wants to test if there is a significant relationship between education level and income in a survey of 1,000 people.","A researcher wants to determine if there is a significant association between age and blood pressure in a study of 150 participants.","A company wants to test if there is a significant relationship between product price and sales volume in an analysis of 25 products.","A health researcher wants to determine if there is a significant association between sleep duration and cognitive performance in a study of 100 individuals.","A teacher wants to test if there is a significant relationship between attendance and academic performance in a class of 30 students.","A marketing team wants to determine if there is a significant association between social media engagement and brand awareness in a study of 50 campaigns.","A scientist wants to test if there is a significant relationship between fertilizer amount and crop yield in an experiment with 60 plots.","A financial analyst wants to determine if there is a significant association between investment amount and return in a study of 80 portfolios.","A researcher wants to test if there is a significant relationship between screen time and eye strain in a study of 90 participants.","A health organization wants to determine if there is a significant association between diet quality and BMI in a study of 300 individuals.","A company wants to test if there is a significant relationship between training hours and employee productivity in a survey of 120 employees.","A sociologist wants to determine if there is a significant association between urbanization and pollution levels in a study of 50 cities.","A researcher wants to test if there is a significant relationship between stress levels and job performance in a study of 70 employees.","A school wants to determine if there is a significant association between homework time and test scores in a sample of 60 students.","A company wants to test if there is a significant relationship between customer service training and customer satisfaction in a survey of 40 employees.","A health researcher wants to determine if there is a significant association between alcohol consumption and liver enzyme levels in a study of 250 participants."],"9":["A pollster wants to estimate the proportion of voters who support a particular candidate in an election based on a sample of 1,000 voters.","A researcher wants to estimate the proportion of people who favor a new policy based on a survey of 800 respondents.","A company wants to estimate the proportion of customers who are satisfied with their product based on a survey of 500 customers.","A school wants to estimate the proportion of students who pass a particular exam based on a sample of 200 students.","A health organization wants to estimate the proportion of people who smoke in a population based on a survey of 1,500 individuals.","A marketing team wants to estimate the proportion of consumers who recognize their brand based on a sample of 600 consumers.","A researcher wants to estimate the proportion of people who exercise regularly based on a survey of 700 participants.","A political analyst wants to estimate the proportion of people who approve of a government policy based on a survey of 1,200 individuals.","A company wants to estimate the proportion of employees who are satisfied with their job based on a survey of 300 employees.","A researcher wants to estimate the proportion of households that recycle based on a survey of 400 households.","A health researcher wants to estimate the proportion of people with a specific health condition based on a sample of 1,000 participants.","A university wants to estimate the proportion of students who graduate on time based on a sample of 500 students.","A pollster wants to estimate the proportion of voters who are undecided in an election based on a survey of 800 voters.","A company wants to estimate the proportion of customers who would recommend their product based on a survey of 600 customers.","A researcher wants to estimate the proportion of people who volunteer regularly based on a survey of 700 respondents.","A health organization wants to estimate the proportion of people who have received a vaccination based on a survey of 1,200 individuals.","A school wants to estimate the proportion of students who participate in extracurricular activities based on a sample of 400 students.","A company wants to estimate the proportion of employees who have received a promotion in the last year based on a survey of 300 employees.","A researcher wants to estimate the proportion of people who use public transportation based on a survey of 500 participants.","A political analyst wants to estimate the proportion of people who have donated to a political campaign based on a survey of 1,000 individuals."],"10":["A researcher wants to estimate the difference in the proportion of males and females who favor a new policy based on a survey of 500 males and 500 females.","A company wants to estimate the difference in the proportion of customers satisfied with two different products based on surveys of 300 customers for each product.","A health researcher wants to estimate the difference in the proportion of smokers and non-smokers who develop lung disease based on a study of 1,000 smokers and 1,000 non-smokers.","A political analyst wants to estimate the difference in the proportion of urban and rural residents who support a candidate based on surveys of 800 urban and 800 rural residents.","A school wants to estimate the difference in the proportion of students passing two different exams based on samples of 200 students for each exam.","A marketing team wants to estimate the difference in the proportion of consumers who recognize two different brands based on surveys of 500 consumers for each brand.","A researcher wants to estimate the difference in the proportion of people who exercise regularly in two different age groups based on surveys of 400 people in each group.","A company wants to estimate the difference in the proportion of employees satisfied with their jobs in two different departments based on surveys of 300 employees in each department.","A health organization wants to estimate the difference in the proportion of vaccinated and unvaccinated people who develop a disease based on a study of 1,000 vaccinated and 1,000 unvaccinated individuals.","A school wants to estimate the difference in the proportion of students who participate in sports in two different grades based on samples of 200 students in each grade.","A researcher wants to estimate the difference in the proportion of people who use public transportation in two different cities based on surveys of 500 people in each city.","A political analyst wants to estimate the difference in the proportion of supporters of two different political parties based on surveys of 1,000 supporters for each party.","A company wants to estimate the difference in the proportion of customers who would recommend two different services based on surveys of 300 customers for each service.","A health researcher wants to estimate the difference in the proportion of people with a health condition in two different regions based on studies of 1,000 people in each region.","A marketing team wants to estimate the difference in the proportion of consumers who prefer two different advertising campaigns based on surveys of 500 consumers for each campaign.","A researcher wants to estimate the difference in the proportion of volunteers in two different communities based on surveys of 400 people in each community.","A company wants to estimate the difference in the proportion of employees who have received a promotion in two different branches based on surveys of 300 employees in each branch.","A health organization wants to estimate the difference in the proportion of people with high blood pressure in two different age groups based on studies of 1,000 people in each group.","A school wants to estimate the difference in the proportion of students who pass two different subjects based on samples of 200 students for each subject.","A researcher wants to estimate the difference in the proportion of people who donate to charity in two different income groups based on surveys of 500 people in each group."],"11":["A researcher wants to estimate the average height of a population based on a sample of 50 people.","A company wants to estimate the average satisfaction score of their customers based on a survey of 300 customers.","A health researcher wants to estimate the average cholesterol level in a population based on a study of 200 individuals.","A teacher wants to estimate the average test score of a class based on a sample of 30 students.","A nutritionist wants to estimate the average daily calorie intake of a population based on a survey of 150 people.","A biologist wants to estimate the average growth rate of a plant species based on a study of 40 plants.","A company wants to estimate the average productivity of their employees based on a survey of 100 employees.","A sociologist wants to estimate the average household income in a city based on a survey of 400 households.","A health organization wants to estimate the average BMI in a population based on a study of 500 participants.","A university wants to estimate the average GPA of their students based on a sample of 200 students.","A pollster wants to estimate the average approval rating of a politician based on a survey of 1,000 voters.","A researcher wants to estimate the average number of hours worked per week by a population based on a survey of 300 people.","A marketing team wants to estimate the average brand recognition score based on a survey of 600 consumers.","A health researcher wants to estimate the average blood pressure in a population based on a study of 250 individuals.","A company wants to estimate the average time to complete a task based on a study of 50 employees.","A researcher wants to estimate the average concentration of a solution based on a sample of 30 measurements.","A teacher wants to estimate the average reading speed of students based on a sample of 25 students.","A scientist wants to estimate the average temperature of a region based on a study of 1,000 daily temperatures.","A sociologist wants to estimate the average family size in a community based on a survey of 300 families.","A health organization wants to estimate the average number of doctor visits per year based on a study of 500 individuals."],"12":["A researcher wants to estimate the difference in average test scores between two classes based on samples of 30 students from each class.","A health researcher wants to estimate the difference in average cholesterol levels between men and women based on a study of 200 men and 200 women.","A company wants to estimate the difference in average productivity between two departments based on surveys of 50 employees from each department.","A biologist wants to estimate the difference in average growth rates between two plant species based on a study of 40 plants from each species.","A teacher wants to estimate the difference in average reading speeds between two grades based on samples of 25 students from each grade.","A health organization wants to estimate the difference in average BMI between two age groups based on studies of 300 individuals from each group.","A researcher wants to estimate the difference in average income between two cities based on surveys of 400 households from each city.","A company wants to estimate the difference in average satisfaction scores between two products based on surveys of 300 customers for each product.","A university wants to estimate the difference in average GPAs between two programs based on samples of 50 students from each program.","A health researcher wants to estimate the difference in average blood pressure between two ethnic groups based on a study of 250 individuals from each group.","A teacher wants to estimate the difference in average math scores between two teaching methods based on samples of 30 students for each method.","A company wants to estimate the difference in average training times between two training programs based on a study of 50 employees for each program.","A scientist wants to estimate the difference in average temperatures between two regions based on a study of 1,000 daily temperatures from each region.","A health organization wants to estimate the difference in average exercise times between two communities based on studies of 200 people from each community.","A researcher wants to estimate the difference in average sleep durations between two age groups based on surveys of 300 individuals from each group.","A teacher wants to estimate the difference in average writing scores between two classrooms based on samples of 25 students from each classroom.","A company wants to estimate the difference in average time to market between two product development processes based on a study of 30 products from each process.","A researcher wants to estimate the difference in average stress levels between two occupations based on surveys of 50 individuals from each occupation.","A health organization wants to estimate the difference in average vaccination rates between two regions based on studies of 500 individuals from each region.","A sociologist wants to estimate the difference in average family sizes between two neighborhoods based on surveys of 300 families from each neighborhood."],"13":["A researcher wants to estimate the difference in average test scores before and after implementing a new teaching method on the same 30 students.","A health researcher wants to estimate the difference in average cholesterol levels before and after a dietary intervention in the same 50 individuals.","A company wants to estimate the difference in average productivity before and after using new software on the same 40 employees.","A teacher wants to estimate the difference in average exam scores before and after a study strategy is used by the same 20 students.","A researcher wants to estimate the difference in average blood pressure before and after a medication is administered to the same 60 patients.","A university wants to estimate the difference in average GPA before and after a tutoring program on the same 50 students.","A company wants to estimate the difference in average customer satisfaction before and after a service change based on surveys from the same 100 customers.","A scientist wants to estimate the difference in average plant growth before and after a fertilizer is applied to the same 30 plants.","A health organization wants to estimate the difference in average weight before and after a fitness program in the same 200 individuals.","A researcher wants to estimate the difference in average reading speed before and after a training program in the same 25 students.","A marketing team wants to estimate the difference in average brand recognition before and after an advertising campaign on the same 100 consumers.","A health researcher wants to estimate the difference in average blood sugar levels before and after a treatment in the same 70 patients.","A teacher wants to estimate the difference in average homework completion rates before and after a policy change on the same 30 students.","A company wants to estimate the difference in average sales performance before and after a new incentive program in the same 50 sales representatives.","A researcher wants to estimate the difference in average cognitive performance before and after a mental training program in the same 80 individuals.","A health organization wants to estimate the difference in average physical activity levels before and after a community intervention in the same 150 participants.","A scientist wants to estimate the difference in average air quality measurements before and after a policy implementation in the same 20 locations.","A researcher wants to estimate the difference in average stress levels before and after a mindfulness program in the same 90 employees.","A company wants to estimate the difference in average project completion time before and after a process change in the same 40 teams.","A teacher wants to estimate the difference in average attendance rates before and after a new attendance policy on the same 60 students."],"14":["A researcher wants to estimate the slope of the relationship between hours studied and exam scores in a sample of 100 students.","A health researcher wants to estimate the slope of the relationship between daily calorie intake and weight loss in a sample of 200 individuals.","A company wants to estimate the slope of the relationship between training hours and employee productivity based on a sample of 50 employees.","A scientist wants to estimate the slope of the relationship between fertilizer amount and plant growth in an experiment with 60 plants.","A marketing team wants to estimate the slope of the relationship between advertising expenditure and sales revenue based on data from 30 campaigns.","A researcher wants to estimate the slope of the relationship between sleep duration and cognitive performance in a study of 80 participants.","A health organization wants to estimate the slope of the relationship between physical activity and blood pressure in a sample of 300 individuals.","A teacher wants to estimate the slope of the relationship between class attendance and grades in a sample of 100 students.","A company wants to estimate the slope of the relationship between customer satisfaction and repeat purchase rate based on a survey of 200 customers.","A scientist wants to estimate the slope of the relationship between temperature and reaction rate in a chemical experiment with 40 trials.","A researcher wants to estimate the slope of the relationship between study time and GPA in a sample of 150 students.","A health researcher wants to estimate the slope of the relationship between medication dosage and symptom reduction in a study of 90 patients.","A marketing team wants to estimate the slope of the relationship between social media engagement and brand awareness in a study of 50 campaigns.","A company wants to estimate the slope of the relationship between work experience and salary in a survey of 300 employees.","A researcher wants to estimate the slope of the relationship between exercise frequency and fitness level in a study of 100 participants.","A health organization wants to estimate the slope of the relationship between diet quality and cholesterol levels in a sample of 150 individuals.","A scientist wants to estimate the slope of the relationship between light exposure and plant growth in an experiment with 60 plants.","A teacher wants to estimate the slope of the relationship between homework completion and test scores in a sample of 80 students.","A company wants to estimate the slope of the 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customers finds 320 are repeat buyers.","A dairy company claims that only 5% of its milk cartons are defective; an inspection of 300 cartons finds 20 defects.","A phone manufacturer claims that 99% of its phones pass quality tests; out of 1,000 phones tested, 985 pass.","A survey suggests that 75% of people prefer online shopping; a sample of 600 people finds 450 prefer online shopping.","A restaurant claims that 80% of its customers are satisfied with their service; a survey of 250 customers shows 200 are satisfied.","A university states that 70% of applicants are accepted into their programs; out of 400 applicants, 280 are accepted.","A survey claims that 65% of adults exercise regularly; a sample of 500 adults finds 320 who exercise regularly.","A social media company asserts that 90% of users are satisfied with their privacy settings; a poll of 1,000 users finds 850 satisfied.","A retailer claims that 30% of customers use their loyalty program; out of 800 customers surveyed, 240 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employees.","A research study compares the effectiveness of two diets; Diet A results in weight loss for 70 out of 100 participants, and Diet B results in weight loss for 65 out of 95 participants.","A researcher compares the participation rates in two extracurricular activities; Activity 1 has 50 participants out of 200 students, and Activity 2 has 60 participants out of 210 students.","A company compares the success rates of two sales teams; Team A closes 120 out of 300 deals, and Team B closes 110 out of 280 deals.","A hospital compares the infection rates after two types of surgeries; Surgery A has 5 infections out of 200 patients, and Surgery B has 10 infections out of 250 patients.","A health study compares the vaccination rates between two regions; Region 1 has 500 vaccinated out of 1,000 residents, and Region 2 has 450 vaccinated out of 950 residents.","A university compares the acceptance rates of two departments; Department A accepts 200 out of 400 applicants, and Department B accepts 180 out of 370 applicants."],"2":["A researcher wants to test if the average weight of a sample of 30 apples is different from the known population mean of 150 grams.","A teacher wants to test if the average score of a class of 25 students is higher than the school\'s average score of 75.","A factory manager wants to determine if the average length of a sample of 20 rods is different from the standard length of 10 meters.","A nutritionist wants to test if the average calorie intake of a sample of 50 people is lower than the recommended 2,000 calories per day.","A biologist wants to know if the average height of a sample of 15 plants differs from the expected 30 centimeters.","A car manufacturer wants to determine if the average fuel efficiency of a sample of 40 cars is higher than the industry standard of 25 miles per gallon.","A company wants to test if the average lifespan of a sample of 35 batteries is different from the advertised 500 hours.","A scientist wants to test if 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known concentration of 0.5 mol/L.","A sports analyst wants to test if the average points scored by a sample of 15 basketball players is different from the league average of 20 points.","A zoologist wants to determine if the average weight of a sample of 12 elephants is higher than the known average of 5,000 kilograms.","A pharmacist wants to test if the average dissolution time of a sample of 20 pills is different from the standard 30 minutes.","A financial analyst wants to determine if the average return of a sample of 25 investments is different from the expected 8%.","A meteorologist wants to test if the average rainfall of a sample of 40 days is lower than the historical average of 3 inches."],"3":["A researcher compares the average test scores of two different teaching methods; Method A with a mean score of 85 from 30 students, and Method B with a mean score of 80 from 28 students.","A company wants to determine if there is a difference in average productivity between two shifts; Shift 1 has an average of 50 units from 40 workers, and Shift 2 has an average of 55 units from 35 workers.","A nutritionist compares the average weight loss of two diets; Diet A results in an average loss of 5 kg for 25 people, and Diet B results in an average loss of 6 kg for 30 people.","A biologist wants to test if there is a difference in average growth between two species of plants; Species X grows an average of 10 cm for 20 plants, and Species Y grows an average of 12 cm for 18 plants.","A car manufacturer compares the average fuel efficiency of two car models; Model A has an average efficiency of 25 mpg from 50 cars, and Model B has an average efficiency of 27 mpg from 45 cars.","A school wants to determine if there is a difference in average math scores between two classes; Class 1 has an average score of 78 from 30 students, and Class 2 has an average score of 82 from 28 students.","A health researcher compares the average cholesterol levels of two diets; Diet A has an average level of 190 mg/dL for 20 people, and Diet B has an average level of 185 mg/dL for 25 people.","A sports analyst wants to test if there is a difference in average points scored by two basketball teams; Team A has an average of 90 points from 15 games, and Team B has an average of 88 points from 18 games.","A sociologist compares the average income of two neighborhoods; Neighborhood A has an average income of $50,000 from 40 households, and Neighborhood B has an average income of $55,000 from 35 households.","A psychologist wants to determine if there is a difference in average stress levels between two groups; Group A has an average stress score of 30 from 25 participants, and Group B has an average stress score of 28 from 30 participants.","A marketing team compares the average sales of two advertising campaigns; Campaign 1 generates an average of $10,000 from 20 runs, and Campaign 2 generates an average of $12,000 from 18 runs.","A teacher compares the average reading scores of two different reading programs; Program A has an average score of 85 from 25 students, and Program B has an average score of 88 from 28 students.","A dietitian wants to test if there is a difference in average blood sugar levels between two diets; Diet A results in an average level of 95 mg/dL for 20 people, and Diet B results in an average level of 90 mg/dL for 22 people.","A researcher compares the average time spent on social media by two age groups; Group 1 spends an average of 2 hours from 30 participants, and Group 2 spends an average of 1.5 hours from 25 participants.","A university wants to determine if there is a difference in average starting salaries between graduates of two programs; Program A has an average salary of $50,000 from 20 graduates, and Program B has an average salary of $55,000 from 22 graduates.","A health researcher compares the average blood pressure levels of two different exercise programs; Program A results in an average level of 120 mmHg for 15 participants, and Program B results in an average level of 115 mmHg for 18 participants.","A company wants to test if there is a difference in average customer satisfaction between two products; Product A has an average satisfaction score of 4.5 from 50 customers, and Product B has an average satisfaction score of 4.2 from 55 customers.","A researcher compares the average time to complete a task between two software tools; Tool A takes an average of 30 minutes from 20 users, and Tool B takes an average of 25 minutes from 22 users.","A university wants to determine if there is a difference in average GPA between two majors; Major A has an average GPA of 3.2 from 30 students, and Major B has an average GPA of 3.5 from 28 students.","A scientist wants to test if there is a difference in average growth rates between two fertilizers; Fertilizer A results in an average growth of 5 cm for 25 plants, and Fertilizer B results in an average growth of 6 cm for 30 plants."],"4":["A researcher tests if there is a difference in blood pressure before and after a treatment in the same group of 30 patients.","A teacher wants to determine if students\' test scores improve after a new teaching method by comparing scores from the same 25 students before and after the method is implemented.","A company wants to test if a new training program improves employee productivity by comparing productivity levels of the same 20 employees before and after the training.","A doctor wants to know if a new medication affects cholesterol levels by comparing the cholesterol levels of the same 15 patients before and after taking the medication.","A coach wants to test if a new exercise routine improves athletic performance by comparing the performance metrics of the same 18 athletes before and after the routine.","A scientist wants to determine if a chemical treatment affects plant growth by comparing the heights of the same 25 plants before and after the treatment.","A nutritionist wants to test if a dietary supplement affects weight by comparing the weights of the same 30 individuals before and after taking the supplement.","A psychologist wants to determine if a therapy method reduces anxiety by comparing the anxiety levels of the same 20 patients before and after the therapy.","A researcher wants to test if a new study technique improves memory retention by comparing test scores of the same 15 students before and after using the technique.","A company wants to determine if a new software update improves processing speed by comparing the speed of the same 25 computers before and after the update.","A fitness trainer wants to test if a new workout plan increases strength by comparing the lifting weights of the same 20 clients before and after the plan.","A scientist wants to test if a new fertilizer affects crop yield by comparing the yields of the same 10 plots before and after using the fertilizer.","A researcher wants to determine if a new teaching strategy affects engagement by comparing the engagement scores of the same 30 students before and after the strategy.","A company wants to test if a new customer service protocol improves satisfaction by comparing satisfaction scores of the same 40 customers before and after the protocol.","A scientist wants to test if a new diet plan affects blood sugar levels by comparing levels of the same 18 individuals before and after the diet.","A researcher wants to determine if a new therapy reduces depression symptoms by comparing the symptoms of the same 22 patients before and after the therapy.","A fitness coach wants to test if a new running program improves speed by comparing the race times of the same 15 runners before and after the program.","A school wants to determine if a new curriculum improves reading skills by comparing test scores of the same 28 students before and after the curriculum.","A company wants to test if a new marketing strategy increases sales by comparing the sales figures of the same 35 products before and after the strategy.","A researcher wants to test if a new learning app improves language skills by comparing language test scores of the same 20 students before and after using the app."],"5":["A geneticist wants to test if the observed frequencies of different flower colors match the expected ratios in a Mendelian inheritance experiment.","A dice manufacturer wants to test if a die is fair by comparing the observed frequencies of each face to the expected equal frequencies.","A researcher wants to determine if the observed distribution of blood types in a sample matches the expected distribution in the general population.","A company wants to test if customer preferences for product colors match the expected market research proportions.","A teacher wants to test if the observed grades in a class match the expected distribution based on historical data.","A biologist wants to test if the observed frequencies of different insect species match the expected proportions in an ecosystem.","A marketing team wants to test if the observed preferences for different advertisements match the expected preferences based on a survey.","A researcher wants to determine if the observed frequencies of different genetic traits match the expected ratios in a population.","A game developer wants to test if the observed frequencies of outcomes in a game match the expected probabilities.","A sociologist wants to test if the observed distribution of household sizes matches the expected distribution in a city.","A quality control manager wants to test if the observed defect types in a production process match the expected proportions.","A psychologist wants to determine if the observed frequencies of different personality types match the expected distribution in a sample.","A company wants to test if the observed sales of products in different regions match the expected sales based on market analysis.","A researcher wants to test if the observed distribution of responses to a survey question matches the expected distribution.","A zoologist wants to test if the observed frequencies of different animal behaviors match the expected proportions in a study.","A retail store wants to determine if the observed purchase frequencies of different product categories match the expected frequencies.","A researcher wants to test if the observed distribution of academic majors matches the expected distribution based on enrollment data.","A pollster wants to test if the observed voting preferences match the expected proportions based on pre-election polls.","A scientist wants to determine if the observed frequencies of different chemical reactions match the expected ratios.","A researcher wants to test if the observed distribution of car colors in a parking lot matches the expected distribution."],"6":["A researcher wants to test if there is an association between gender and voting preference in a sample of 500 people.","A sociologist wants to determine if there is a relationship between education level and income category in a survey of 1,000 individuals.","A company wants to test if there is a link between customer age group and product preference in a market analysis.","A health researcher wants to determine if there is an association between smoking status and incidence of lung disease in a study of 2,000 participants.","A teacher wants to test if there is a relationship between study habits and academic performance in a class of 100 students.","A political scientist wants to determine if there is an association between political affiliation and support for a policy in a survey of 800 voters.","A marketing team wants to test if there is a link between advertisement type and purchase decision in an experiment with 300 consumers.","A psychologist wants to determine if there is a relationship between personality type and career choice in a sample of 500 adults.","A company wants to test if there is an association between training program and employee retention in a study of 1,200 employees.","A sociologist wants to determine if there is a link between marital status and happiness level in a survey of 700 people.","A researcher wants to test if there is a relationship between diet type and weight loss success in a study of 400 participants.","A health organization wants to determine if there is an association between exercise frequency and heart disease incidence in a study of 1,500 individuals.","A school wants to test if there is a link between extracurricular activity participation and academic success in a sample of 600 students.","A company wants to determine if there is a relationship between job role and job satisfaction in a survey of 1,000 employees.","A sociologist wants to test if there is an association between neighborhood type and crime rate in a study of 300 neighborhoods.","A researcher wants to determine if there is a relationship between internet usage and social interaction levels in a survey of 800 individuals.","A health researcher wants to test if there is a link between alcohol consumption and liver disease in a study of 1,200 participants.","A political analyst wants to determine if there is an association between region and voting turnout in an analysis of 50 regions.","A company wants to test if there is a relationship between product type and return rate in a study of 500 product returns.","A researcher wants to determine if there is an association between sleep patterns and productivity levels in a study of 300 participants."],"7":["A researcher wants to test if the distribution of blood types is the same across different ethnic groups in a sample of 1,000 people.","A company wants to determine if customer satisfaction levels are consistent across different store locations in a survey of 800 customers.","A school wants to test if the distribution of grades is similar across different classes in a sample of 200 students.","A health researcher wants to determine if the incidence of a disease is the same across different age groups in a study of 1,500 participants.","A marketing team wants to test if the preferences for a new product are consistent across different regions in a market analysis.","A political scientist wants to determine if voting patterns are similar across different districts in an election study.","A sociologist wants to test if the distribution of household sizes is the same across different neighborhoods in a survey of 600 households.","A researcher wants to determine if the frequency of internet usage is consistent across different age groups in a sample of 500 people.","A health organization wants to test if the rates of vaccination are the same across different communities in a study of 1,200 participants.","A company wants to determine if employee retention rates are consistent across different departments in a survey of 1,000 employees.","A researcher wants to test if the distribution of leisure activities is the same across different age groups in a study of 300 participants.","A school wants to determine if participation in extracurricular activities is consistent across different grade levels in a sample of 400 students.","A marketing team wants to test if brand loyalty is the same across different customer segments in a survey of 700 consumers.","A researcher wants to determine if the frequency of exercise is consistent across different income groups in a study of 500 participants.","A health researcher wants to test if the incidence of diabetes is the same across different ethnic groups in a study of 1,000 individuals.","A political analyst wants to determine if the distribution of political affiliations is consistent across different regions in a survey of 800 voters.","A company wants to test if the return rates are the same across different product categories in an analysis of 1,200 returns.","A researcher wants to determine if the distribution of reading habits is consistent across different education levels in a survey of 600 people.","A health organization wants to test if the frequency of health check-ups is the same across different age groups in a study of 1,500 participants.","A school wants to determine if the distribution of test scores is similar across different subjects in a sample of 200 students."],"8":["A researcher wants to test if there is a significant relationship between hours studied and test scores in a sample of 50 students.","A health researcher wants to determine if there is a significant association between daily exercise time and cholesterol levels in a study of 200 individuals.","A company wants to test if there is a significant relationship between advertising expenditure and sales revenue in a sample of 30 months of data.","A biologist wants to determine if there is a significant association between sunlight exposure and plant growth in a study of 40 plants.","A sociologist wants to test if there is a significant relationship between education level and income in a survey of 1,000 people.","A researcher wants to determine if there is a significant association between age and blood pressure in a study of 150 participants.","A company wants to test if there is a significant relationship between product price and sales volume in an analysis of 25 products.","A health researcher wants to determine if there is a significant association between sleep duration and cognitive performance in a study of 100 individuals.","A teacher wants to test if there is a significant relationship between attendance and academic performance in a class of 30 students.","A marketing team wants to determine if there is a significant association between social media engagement and brand awareness in a study of 50 campaigns.","A scientist wants to test if there is a significant relationship between fertilizer amount and crop yield in an experiment with 60 plots.","A financial analyst wants to determine if there is a significant association between investment amount and return in a study of 80 portfolios.","A researcher wants to test if there is a significant relationship between screen time and eye strain in a study of 90 participants.","A health organization wants to determine if there is a significant association between diet quality and BMI in a study of 300 individuals.","A company wants to test if there is a significant relationship between training hours and employee productivity in a survey of 120 employees.","A sociologist wants to determine if there is a significant association between urbanization and pollution levels in a study of 50 cities.","A researcher wants to test if there is a significant relationship between stress levels and job performance in a study of 70 employees.","A school wants to determine if there is a significant association between homework time and test scores in a sample of 60 students.","A company wants to test if there is a significant relationship between customer service training and customer satisfaction in a survey of 40 employees.","A health researcher wants to determine if there is a significant association between alcohol consumption and liver enzyme levels in a study of 250 participants."],"9":["A pollster wants to estimate the proportion of voters who support a particular candidate in an election based on a sample of 1,000 voters.","A researcher wants to estimate the proportion of people who favor a new policy based on a survey of 800 respondents.","A company wants to estimate the proportion of customers who are satisfied with their product based on a survey of 500 customers.","A school wants to estimate the proportion of students who pass a particular exam based on a sample of 200 students.","A health organization wants to estimate the proportion of people who smoke in a population based on a survey of 1,500 individuals.","A marketing team wants to estimate the proportion of consumers who recognize their brand based on a sample of 600 consumers.","A researcher wants to estimate the proportion of people who exercise regularly based on a survey of 700 participants.","A political analyst wants to estimate the proportion of people who approve of a government policy based on a survey of 1,200 individuals.","A company wants to estimate the proportion of employees who are satisfied with their job based on a survey of 300 employees.","A researcher wants to estimate the proportion of households that recycle based on a survey of 400 households.","A health researcher wants to estimate the proportion of people with a specific health condition based on a sample of 1,000 participants.","A university wants to estimate the proportion of students who graduate on time based on a sample of 500 students.","A pollster wants to estimate the proportion of voters who are undecided in an election based on a survey of 800 voters.","A company wants to estimate the proportion of customers who would recommend their product based on a survey of 600 customers.","A researcher wants to estimate the proportion of people who volunteer regularly based on a survey of 700 respondents.","A health organization wants to estimate the proportion of people who have received a vaccination based on a survey of 1,200 individuals.","A school wants to estimate the proportion of students who participate in extracurricular activities based on a sample of 400 students.","A company wants to estimate the proportion of employees who have received a promotion in the last year based on a survey of 300 employees.","A researcher wants to estimate the proportion of people who use public transportation based on a survey of 500 participants.","A political analyst wants to estimate the proportion of people who have donated to a political campaign based on a survey of 1,000 individuals."],"10":["A researcher wants to estimate the difference in the proportion of males and females who favor a new policy based on a survey of 500 males and 500 females.","A company wants to estimate the difference in the proportion of customers satisfied with two different products based on surveys of 300 customers for each product.","A health researcher wants to estimate the difference in the proportion of smokers and non-smokers who develop lung disease based on a study of 1,000 smokers and 1,000 non-smokers.","A political analyst wants to estimate the difference in the proportion of urban and rural residents who support a candidate based on surveys of 800 urban and 800 rural residents.","A school wants to estimate the difference in the proportion of students passing two different exams based on samples of 200 students for each exam.","A marketing team wants to estimate the difference in the proportion of consumers who recognize two different brands based on surveys of 500 consumers for each brand.","A researcher wants to estimate the difference in the proportion of people who exercise regularly in two different age groups based on surveys of 400 people in each group.","A company wants to estimate the difference in the proportion of employees satisfied with their jobs in two different departments based on surveys of 300 employees in each department.","A health organization wants to estimate the difference in the proportion of vaccinated and unvaccinated people who develop a disease based on a study of 1,000 vaccinated and 1,000 unvaccinated individuals.","A school wants to estimate the difference in the proportion of students who participate in sports in two different grades based on samples of 200 students in each grade.","A researcher wants to estimate the difference in the proportion of people who use public transportation in two different cities based on surveys of 500 people in each city.","A political analyst wants to estimate the difference in the proportion of supporters of two different political parties based on surveys of 1,000 supporters for each party.","A company wants to estimate the difference in the proportion of customers who would recommend two different services based on surveys of 300 customers for each service.","A health researcher wants to estimate the difference in the proportion of people with a health condition in two different regions based on studies of 1,000 people in each region.","A marketing team wants to estimate the difference in the proportion of consumers who prefer two different advertising campaigns based on surveys of 500 consumers for each campaign.","A researcher wants to estimate the difference in the proportion of volunteers in two different communities based on surveys of 400 people in each community.","A company wants to estimate the difference in the proportion of employees who have received a promotion in two different branches based on surveys of 300 employees in each branch.","A health organization wants to estimate the difference in the proportion of people with high blood pressure in two different age groups based on studies of 1,000 people in each group.","A school wants to estimate the difference in the proportion of students who pass two different subjects based on samples of 200 students for each subject.","A researcher wants to estimate the difference in the proportion of people who donate to charity in two different income groups based on surveys of 500 people in each group."],"11":["A researcher wants to estimate the average height of a population based on a sample of 50 people.","A company wants to estimate the average satisfaction score of their customers based on a survey of 300 customers.","A health researcher wants to estimate the average cholesterol level in a population based on a study of 200 individuals.","A teacher wants to estimate the average test score of a class based on a sample of 30 students.","A nutritionist wants to estimate the average daily calorie intake of a population based on a survey of 150 people.","A biologist wants to estimate the average growth rate of a plant species based on a study of 40 plants.","A company wants to estimate the average productivity of their employees based on a survey of 100 employees.","A sociologist wants to estimate the average household income in a city based on a survey of 400 households.","A health organization wants to estimate the average BMI in a population based on a study of 500 participants.","A university wants to estimate the average GPA of their students based on a sample of 200 students.","A pollster wants to estimate the average approval rating of a politician based on a survey of 1,000 voters.","A researcher wants to estimate the average number of hours worked per week by a population based on a survey of 300 people.","A marketing team wants to estimate the average brand recognition score based on a survey of 600 consumers.","A health researcher wants to estimate the average blood pressure in a population based on a study of 250 individuals.","A company wants to estimate the average time to complete a task based on a study of 50 employees.","A researcher wants to estimate the average concentration of a solution based on a sample of 30 measurements.","A teacher wants to estimate the average reading speed of students based on a sample of 25 students.","A scientist wants to estimate the average temperature of a region based on a study of 1,000 daily temperatures.","A sociologist wants to estimate the average family size in a community based on a survey of 300 families.","A health organization wants to estimate the average number of doctor visits per year based on a study of 500 individuals."],"12":["A researcher wants to estimate the difference in average test scores between two classes based on samples of 30 students from each class.","A health researcher wants to estimate the difference in average cholesterol levels between men and women based on a study of 200 men and 200 women.","A company wants to estimate the difference in average productivity between two departments based on surveys of 50 employees from each department.","A biologist wants to estimate the difference in average growth rates between two plant species based on a study of 40 plants from each species.","A teacher wants to estimate the difference in average reading speeds between two grades based on samples of 25 students from each grade.","A health organization wants to estimate the difference in average BMI between two age groups based on studies of 300 individuals from each group.","A researcher wants to estimate the difference in average income between two cities based on surveys of 400 households from each city.","A company wants to estimate the difference in average satisfaction scores between two products based on surveys of 300 customers for each product.","A university wants to estimate the difference in average GPAs between two programs based on samples of 50 students from each program.","A health researcher wants to estimate the difference in average blood pressure between two ethnic groups based on a study of 250 individuals from each group.","A teacher wants to estimate the difference in average math scores between two teaching methods based on samples of 30 students for each method.","A company wants to estimate the difference in average training times between two training programs based on a study of 50 employees for each program.","A scientist wants to estimate the difference in average temperatures between two regions based on a study of 1,000 daily temperatures from each region.","A health organization wants to estimate the difference in average exercise times between two communities based on studies of 200 people from each community.","A researcher wants to estimate the difference in average sleep durations between two age groups based on surveys of 300 individuals from each group.","A teacher wants to estimate the difference in average writing scores between two classrooms based on samples of 25 students from each classroom.","A company wants to estimate the difference in average time to market between two product development processes based on a study of 30 products from each process.","A researcher wants to estimate the difference in average stress levels between two occupations based on surveys of 50 individuals from each occupation.","A health organization wants to estimate the difference in average vaccination rates between two regions based on studies of 500 individuals from each region.","A sociologist wants to estimate the difference in average family sizes between two neighborhoods based on surveys of 300 families from each neighborhood."],"13":["A researcher wants to estimate the difference in average test scores before and after implementing a new teaching method on the same 30 students.","A health researcher wants to estimate the difference in average cholesterol levels before and after a dietary intervention in the same 50 individuals.","A company wants to estimate the difference in average productivity before and after using new software on the same 40 employees.","A teacher wants to estimate the difference in average exam scores before and after a study strategy is used by the same 20 students.","A researcher wants to estimate the difference in average blood pressure before and after a medication is administered to the same 60 patients.","A university wants to estimate the difference in average GPA before and after a tutoring program on the same 50 students.","A company wants to estimate the difference in average customer satisfaction before and after a service change based on surveys from the same 100 customers.","A scientist wants to estimate the difference in average plant growth before and after a fertilizer is applied to the same 30 plants.","A health organization wants to estimate the difference in average weight before and after a fitness program in the same 200 individuals.","A researcher wants to estimate the difference in average reading speed before and after a training program in the same 25 students.","A marketing team wants to estimate the difference in average brand recognition before and after an advertising campaign on the same 100 consumers.","A health researcher wants to estimate the difference in average blood sugar levels before and after a treatment in the same 70 patients.","A teacher wants to estimate the difference in average homework completion rates before and after a policy change on the same 30 students.","A company wants to estimate the difference in average sales performance before and after a new incentive program in the same 50 sales representatives.","A researcher wants to estimate the difference in average cognitive performance before and after a mental training program in the same 80 individuals.","A health organization wants to estimate the difference in average physical activity levels before and after a community intervention in the same 150 participants.","A scientist wants to estimate the difference in average air quality measurements before and after a policy implementation in the same 20 locations.","A researcher wants to estimate the difference in average stress levels before and after a mindfulness program in the same 90 employees.","A company wants to estimate the difference in average project completion time before and after a process change in the same 40 teams.","A teacher wants to estimate the difference in average attendance rates before and after a new attendance policy on the same 60 students."],"14":["A researcher wants to estimate the slope of the relationship between hours studied and exam scores in a sample of 100 students.","A health researcher wants to estimate the slope of the relationship between daily calorie intake and weight loss in a sample of 200 individuals.","A company wants to estimate the slope of the relationship between training hours and employee productivity based on a sample of 50 employees.","A scientist wants to estimate the slope of the relationship between fertilizer amount and plant growth in an experiment with 60 plants.","A marketing team wants to estimate the slope of the relationship between advertising expenditure and sales revenue based on data from 30 campaigns.","A researcher wants to estimate the slope of the relationship between sleep duration and cognitive performance in a study of 80 participants.","A health organization wants to estimate the slope of the relationship between physical activity and blood pressure in a sample of 300 individuals.","A teacher wants to estimate the slope of the relationship between class attendance and grades in a sample of 100 students.","A company wants to estimate the slope of the relationship between customer satisfaction and repeat purchase rate based on a survey of 200 customers.","A scientist wants to estimate the slope of the relationship between temperature and reaction rate in a chemical experiment with 40 trials.","A researcher wants to estimate the slope of the relationship between study time and GPA in a sample of 150 students.","A health researcher wants to estimate the slope of the relationship between medication dosage and symptom reduction in a study of 90 patients.","A marketing team wants to estimate the slope of the relationship between social media engagement and brand awareness in a study of 50 campaigns.","A company wants to estimate the slope of the relationship between work experience and salary in a survey of 300 employees.","A researcher wants to estimate the slope of the relationship between exercise frequency and fitness level in a study of 100 participants.","A health organization wants to estimate the slope of the relationship between diet quality and cholesterol levels in a sample of 150 individuals.","A scientist wants to estimate the slope of the relationship between light exposure and plant growth in an experiment with 60 plants.","A teacher wants to estimate the slope of the relationship between homework completion and test scores in a sample of 80 students.","A company wants to estimate the slope of the 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similarity index 100%
rename from static/js/main.9371a6fb.js.LICENSE.txt
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diff --git a/static/js/main.9371a6fb.js.map b/static/js/main.d54655ad.js.map
similarity index 89%
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promise);\n\n\t\t\t\t\t// start chunk loading\n\t\t\t\t\tvar url = __webpack_require__.p + __webpack_require__.u(chunkId);\n\t\t\t\t\t// create error before stack unwound to get useful stacktrace later\n\t\t\t\t\tvar error = new Error();\n\t\t\t\t\tvar loadingEnded = function(event) {\n\t\t\t\t\t\tif(__webpack_require__.o(installedChunks, chunkId)) {\n\t\t\t\t\t\t\tinstalledChunkData = installedChunks[chunkId];\n\t\t\t\t\t\t\tif(installedChunkData !== 0) installedChunks[chunkId] = undefined;\n\t\t\t\t\t\t\tif(installedChunkData) {\n\t\t\t\t\t\t\t\tvar errorType = event && (event.type === 'load' ? 'missing' : event.type);\n\t\t\t\t\t\t\t\tvar realSrc = event && event.target && event.target.src;\n\t\t\t\t\t\t\t\terror.message = 'Loading chunk ' + chunkId + ' failed.\\n(' + errorType + ': ' + realSrc + ')';\n\t\t\t\t\t\t\t\terror.name = 'ChunkLoadError';\n\t\t\t\t\t\t\t\terror.type = errorType;\n\t\t\t\t\t\t\t\terror.request = realSrc;\n\t\t\t\t\t\t\t\tinstalledChunkData[1](error);\n\t\t\t\t\t\t\t}\n\t\t\t\t\t\t}\n\t\t\t\t\t};\n\t\t\t\t\t__webpack_require__.l(url, loadingEnded, \"chunk-\" + chunkId, chunkId);\n\t\t\t\t}\n\t\t\t}\n\t\t}\n};\n\n// no prefetching\n\n// no preloaded\n\n// no HMR\n\n// no HMR manifest\n\n// no on chunks loaded\n\n// install a JSONP callback for chunk loading\nvar webpackJsonpCallback = function(parentChunkLoadingFunction, data) {\n\tvar chunkIds = data[0];\n\tvar moreModules = data[1];\n\tvar runtime = data[2];\n\t// add \"moreModules\" to the modules object,\n\t// then flag all \"chunkIds\" as loaded and fire callback\n\tvar moduleId, chunkId, i = 0;\n\tif(chunkIds.some(function(id) { return installedChunks[id] !== 0; })) {\n\t\tfor(moduleId in moreModules) {\n\t\t\tif(__webpack_require__.o(moreModules, moduleId)) {\n\t\t\t\t__webpack_require__.m[moduleId] = moreModules[moduleId];\n\t\t\t}\n\t\t}\n\t\tif(runtime) var result = runtime(__webpack_require__);\n\t}\n\tif(parentChunkLoadingFunction) parentChunkLoadingFunction(data);\n\tfor(;i < chunkIds.length; i++) {\n\t\tchunkId = chunkIds[i];\n\t\tif(__webpack_require__.o(installedChunks, chunkId) && installedChunks[chunkId]) {\n\t\t\tinstalledChunks[chunkId][0]();\n\t\t}\n\t\tinstalledChunks[chunkId] = 0;\n\t}\n\n}\n\nvar chunkLoadingGlobal = self[\"webpackChunkmathinfo\"] = self[\"webpackChunkmathinfo\"] || [];\nchunkLoadingGlobal.forEach(webpackJsonpCallback.bind(null, 0));\nchunkLoadingGlobal.push = webpackJsonpCallback.bind(null, chunkLoadingGlobal.push.bind(chunkLoadingGlobal));","////////////////////////////////////////////////////////////////////////////////\n//#region Types and Constants\n////////////////////////////////////////////////////////////////////////////////\n\n/**\n * Actions represent the type of change to a location value.\n */\nexport enum Action {\n /**\n * A POP indicates a change to an arbitrary index in the history stack, such\n * as a back or forward navigation. It does not describe the direction of the\n * navigation, only that the current index changed.\n *\n * Note: This is the default action for newly created history objects.\n */\n Pop = \"POP\",\n\n /**\n * A PUSH indicates a new entry being added to the history stack, such as when\n * a link is clicked and a new page loads. When this happens, all subsequent\n * entries in the stack are lost.\n */\n Push = \"PUSH\",\n\n /**\n * A REPLACE indicates the entry at the current index in the history stack\n * being replaced by a new one.\n */\n Replace = \"REPLACE\",\n}\n\n/**\n * The pathname, search, and hash values of a URL.\n */\nexport interface Path {\n /**\n * A URL pathname, beginning with a /.\n */\n pathname: string;\n\n /**\n * A URL search string, beginning with a ?.\n */\n search: string;\n\n /**\n * A URL fragment identifier, beginning with a #.\n */\n hash: string;\n}\n\n/**\n * An entry in a history stack. A location contains information about the\n * URL path, as well as possibly some arbitrary state and a key.\n */\nexport interface Location extends Path {\n /**\n * A value of arbitrary data associated with this location.\n */\n state: any;\n\n /**\n * A unique string associated with this location. May be used to safely store\n * and retrieve data in some other storage API, like `localStorage`.\n *\n * Note: This value is always \"default\" on the initial location.\n */\n key: string;\n}\n\n/**\n * A change to the current location.\n */\nexport interface Update {\n /**\n * The action that triggered the change.\n */\n action: Action;\n\n /**\n * The new location.\n */\n location: Location;\n\n /**\n * The delta between this location and the former location in the history stack\n */\n delta: number | null;\n}\n\n/**\n * A function that receives notifications about location changes.\n */\nexport interface Listener {\n (update: Update): void;\n}\n\n/**\n * Describes a location that is the destination of some navigation, either via\n * `history.push` or `history.replace`. May be either a URL or the pieces of a\n * URL path.\n */\nexport type To = string | Partial;\n\n/**\n * A history is an interface to the navigation stack. The history serves as the\n * source of truth for the current location, as well as provides a set of\n * methods that may be used to change it.\n *\n * It is similar to the DOM's `window.history` object, but with a smaller, more\n * focused API.\n */\nexport interface History {\n /**\n * The last action that modified the current location. This will always be\n * Action.Pop when a history instance is first created. This value is mutable.\n */\n readonly action: Action;\n\n /**\n * The current location. This value is mutable.\n */\n readonly location: Location;\n\n /**\n * Returns a valid href for the given `to` value that may be used as\n * the value of an attribute.\n *\n * @param to - The destination URL\n */\n createHref(to: To): string;\n\n /**\n * Returns a URL for the given `to` value\n *\n * @param to - The destination URL\n */\n createURL(to: To): URL;\n\n /**\n * Encode a location the same way window.history would do (no-op for memory\n * history) so we ensure our PUSH/REPLACE navigations for data routers\n * behave the same as POP\n *\n * @param to Unencoded path\n */\n encodeLocation(to: To): Path;\n\n /**\n * Pushes a new location onto the history stack, increasing its length by one.\n * If there were any entries in the stack after the current one, they are\n * lost.\n *\n * @param to - The new URL\n * @param state - Data to associate with the new location\n */\n push(to: To, state?: any): void;\n\n /**\n * Replaces the current location in the history stack with a new one. The\n * location that was replaced will no longer be available.\n *\n * @param to - The new URL\n * @param state - Data to associate with the new location\n */\n replace(to: To, state?: any): void;\n\n /**\n * Navigates `n` entries backward/forward in the history stack relative to the\n * current index. For example, a \"back\" navigation would use go(-1).\n *\n * @param delta - The delta in the stack index\n */\n go(delta: number): void;\n\n /**\n * Sets up a listener that will be called whenever the current location\n * changes.\n *\n * @param listener - A function that will be called when the location changes\n * @returns unlisten - A function that may be used to stop listening\n */\n listen(listener: Listener): () => void;\n}\n\ntype HistoryState = {\n usr: any;\n key?: string;\n idx: number;\n};\n\nconst PopStateEventType = \"popstate\";\n//#endregion\n\n////////////////////////////////////////////////////////////////////////////////\n//#region Memory History\n////////////////////////////////////////////////////////////////////////////////\n\n/**\n * A user-supplied object that describes a location. Used when providing\n * entries to `createMemoryHistory` via its `initialEntries` option.\n */\nexport type InitialEntry = string | Partial;\n\nexport type MemoryHistoryOptions = {\n initialEntries?: InitialEntry[];\n initialIndex?: number;\n v5Compat?: boolean;\n};\n\n/**\n * A memory history stores locations in memory. This is useful in stateful\n * environments where there is no web browser, such as node tests or React\n * Native.\n */\nexport interface MemoryHistory extends History {\n /**\n * The current index in the history stack.\n */\n readonly index: number;\n}\n\n/**\n * Memory history stores the current location in memory. It is designed for use\n * in stateful non-browser environments like tests and React Native.\n */\nexport function createMemoryHistory(\n options: MemoryHistoryOptions = {}\n): MemoryHistory {\n let { initialEntries = [\"/\"], initialIndex, v5Compat = false } = options;\n let entries: Location[]; // Declare so we can access from createMemoryLocation\n entries = initialEntries.map((entry, index) =>\n createMemoryLocation(\n entry,\n typeof entry === \"string\" ? null : entry.state,\n index === 0 ? \"default\" : undefined\n )\n );\n let index = clampIndex(\n initialIndex == null ? entries.length - 1 : initialIndex\n );\n let action = Action.Pop;\n let listener: Listener | null = null;\n\n function clampIndex(n: number): number {\n return Math.min(Math.max(n, 0), entries.length - 1);\n }\n function getCurrentLocation(): Location {\n return entries[index];\n }\n function createMemoryLocation(\n to: To,\n state: any = null,\n key?: string\n ): Location {\n let location = createLocation(\n entries ? getCurrentLocation().pathname : \"/\",\n to,\n state,\n key\n );\n warning(\n location.pathname.charAt(0) === \"/\",\n `relative pathnames are not supported in memory history: ${JSON.stringify(\n to\n )}`\n );\n return location;\n }\n\n function createHref(to: To) {\n return typeof to === \"string\" ? to : createPath(to);\n }\n\n let history: MemoryHistory = {\n get index() {\n return index;\n },\n get action() {\n return action;\n },\n get location() {\n return getCurrentLocation();\n },\n createHref,\n createURL(to) {\n return new URL(createHref(to), \"http://localhost\");\n },\n encodeLocation(to: To) {\n let path = typeof to === \"string\" ? parsePath(to) : to;\n return {\n pathname: path.pathname || \"\",\n search: path.search || \"\",\n hash: path.hash || \"\",\n };\n },\n push(to, state) {\n action = Action.Push;\n let nextLocation = createMemoryLocation(to, state);\n index += 1;\n entries.splice(index, entries.length, nextLocation);\n if (v5Compat && listener) {\n listener({ action, location: nextLocation, delta: 1 });\n }\n },\n replace(to, state) {\n action = Action.Replace;\n let nextLocation = createMemoryLocation(to, state);\n entries[index] = nextLocation;\n if (v5Compat && listener) {\n listener({ action, location: nextLocation, delta: 0 });\n }\n },\n go(delta) {\n action = Action.Pop;\n let nextIndex = clampIndex(index + delta);\n let nextLocation = entries[nextIndex];\n index = nextIndex;\n if (listener) {\n listener({ action, location: nextLocation, delta });\n }\n },\n listen(fn: Listener) {\n listener = fn;\n return () => {\n listener = null;\n };\n },\n };\n\n return history;\n}\n//#endregion\n\n////////////////////////////////////////////////////////////////////////////////\n//#region Browser History\n////////////////////////////////////////////////////////////////////////////////\n\n/**\n * A browser history stores the current location in regular URLs in a web\n * browser environment. This is the standard for most web apps and provides the\n * cleanest URLs the browser's address bar.\n *\n * @see https://github.com/remix-run/history/tree/main/docs/api-reference.md#browserhistory\n */\nexport interface BrowserHistory extends UrlHistory {}\n\nexport type BrowserHistoryOptions = UrlHistoryOptions;\n\n/**\n * Browser history stores the location in regular URLs. This is the standard for\n * most web apps, but it requires some configuration on the server to ensure you\n * serve the same app at multiple URLs.\n *\n * @see https://github.com/remix-run/history/tree/main/docs/api-reference.md#createbrowserhistory\n */\nexport function createBrowserHistory(\n options: BrowserHistoryOptions = {}\n): BrowserHistory {\n function createBrowserLocation(\n window: Window,\n globalHistory: Window[\"history\"]\n ) {\n let { pathname, search, hash } = window.location;\n return createLocation(\n \"\",\n { pathname, search, hash },\n // state defaults to `null` because `window.history.state` does\n (globalHistory.state && globalHistory.state.usr) || null,\n (globalHistory.state && globalHistory.state.key) || \"default\"\n );\n }\n\n function createBrowserHref(window: Window, to: To) {\n return typeof to === \"string\" ? to : createPath(to);\n }\n\n return getUrlBasedHistory(\n createBrowserLocation,\n createBrowserHref,\n null,\n options\n );\n}\n//#endregion\n\n////////////////////////////////////////////////////////////////////////////////\n//#region Hash History\n////////////////////////////////////////////////////////////////////////////////\n\n/**\n * A hash history stores the current location in the fragment identifier portion\n * of the URL in a web browser environment.\n *\n * This is ideal for apps that do not control the server for some reason\n * (because the fragment identifier is never sent to the server), including some\n * shared hosting environments that do not provide fine-grained controls over\n * which pages are served at which URLs.\n *\n * @see https://github.com/remix-run/history/tree/main/docs/api-reference.md#hashhistory\n */\nexport interface HashHistory extends UrlHistory {}\n\nexport type HashHistoryOptions = UrlHistoryOptions;\n\n/**\n * Hash history stores the location in window.location.hash. This makes it ideal\n * for situations where you don't want to send the location to the server for\n * some reason, either because you do cannot configure it or the URL space is\n * reserved for something else.\n *\n * @see https://github.com/remix-run/history/tree/main/docs/api-reference.md#createhashhistory\n */\nexport function createHashHistory(\n options: HashHistoryOptions = {}\n): HashHistory {\n function createHashLocation(\n window: Window,\n globalHistory: Window[\"history\"]\n ) {\n let {\n pathname = \"/\",\n search = \"\",\n hash = \"\",\n } = parsePath(window.location.hash.substr(1));\n\n // Hash URL should always have a leading / just like window.location.pathname\n // does, so if an app ends up at a route like /#something then we add a\n // leading slash so all of our path-matching behaves the same as if it would\n // in a browser router. This is particularly important when there exists a\n // root splat route () since that matches internally against\n // \"/*\" and we'd expect /#something to 404 in a hash router app.\n if (!pathname.startsWith(\"/\") && !pathname.startsWith(\".\")) {\n pathname = \"/\" + pathname;\n }\n\n return createLocation(\n \"\",\n { pathname, search, hash },\n // state defaults to `null` because `window.history.state` does\n (globalHistory.state && globalHistory.state.usr) || null,\n (globalHistory.state && globalHistory.state.key) || \"default\"\n );\n }\n\n function createHashHref(window: Window, to: To) {\n let base = window.document.querySelector(\"base\");\n let href = \"\";\n\n if (base && base.getAttribute(\"href\")) {\n let url = window.location.href;\n let hashIndex = url.indexOf(\"#\");\n href = hashIndex === -1 ? url : url.slice(0, hashIndex);\n }\n\n return href + \"#\" + (typeof to === \"string\" ? to : createPath(to));\n }\n\n function validateHashLocation(location: Location, to: To) {\n warning(\n location.pathname.charAt(0) === \"/\",\n `relative pathnames are not supported in hash history.push(${JSON.stringify(\n to\n )})`\n );\n }\n\n return getUrlBasedHistory(\n createHashLocation,\n createHashHref,\n validateHashLocation,\n options\n );\n}\n//#endregion\n\n////////////////////////////////////////////////////////////////////////////////\n//#region UTILS\n////////////////////////////////////////////////////////////////////////////////\n\n/**\n * @private\n */\nexport function invariant(value: boolean, message?: string): asserts value;\nexport function invariant(\n value: T | null | undefined,\n message?: string\n): asserts value is T;\nexport function invariant(value: any, message?: string) {\n if (value === false || value === null || typeof value === \"undefined\") {\n throw new Error(message);\n }\n}\n\nexport function warning(cond: any, message: string) {\n if (!cond) {\n // eslint-disable-next-line no-console\n if (typeof console !== \"undefined\") console.warn(message);\n\n try {\n // Welcome to debugging history!\n //\n // This error is thrown as a convenience so you can more easily\n // find the source for a warning that appears in the console by\n // enabling \"pause on exceptions\" in your JavaScript debugger.\n throw new Error(message);\n // eslint-disable-next-line no-empty\n } catch (e) {}\n }\n}\n\nfunction createKey() {\n return Math.random().toString(36).substr(2, 8);\n}\n\n/**\n * For browser-based histories, we combine the state and key into an object\n */\nfunction getHistoryState(location: Location, index: number): HistoryState {\n return {\n usr: location.state,\n key: location.key,\n idx: index,\n };\n}\n\n/**\n * Creates a Location object with a unique key from the given Path\n */\nexport function createLocation(\n current: string | Location,\n to: To,\n state: any = null,\n key?: string\n): Readonly {\n let location: Readonly = {\n pathname: typeof current === \"string\" ? current : current.pathname,\n search: \"\",\n hash: \"\",\n ...(typeof to === \"string\" ? parsePath(to) : to),\n state,\n // TODO: This could be cleaned up. push/replace should probably just take\n // full Locations now and avoid the need to run through this flow at all\n // But that's a pretty big refactor to the current test suite so going to\n // keep as is for the time being and just let any incoming keys take precedence\n key: (to && (to as Location).key) || key || createKey(),\n };\n return location;\n}\n\n/**\n * Creates a string URL path from the given pathname, search, and hash components.\n */\nexport function createPath({\n pathname = \"/\",\n search = \"\",\n hash = \"\",\n}: Partial) {\n if (search && search !== \"?\")\n pathname += search.charAt(0) === \"?\" ? search : \"?\" + search;\n if (hash && hash !== \"#\")\n pathname += hash.charAt(0) === \"#\" ? hash : \"#\" + hash;\n return pathname;\n}\n\n/**\n * Parses a string URL path into its separate pathname, search, and hash components.\n */\nexport function parsePath(path: string): Partial {\n let parsedPath: Partial = {};\n\n if (path) {\n let hashIndex = path.indexOf(\"#\");\n if (hashIndex >= 0) {\n parsedPath.hash = path.substr(hashIndex);\n path = path.substr(0, hashIndex);\n }\n\n let searchIndex = path.indexOf(\"?\");\n if (searchIndex >= 0) {\n parsedPath.search = path.substr(searchIndex);\n path = path.substr(0, searchIndex);\n }\n\n if (path) {\n parsedPath.pathname = path;\n }\n }\n\n return parsedPath;\n}\n\nexport interface UrlHistory extends History {}\n\nexport type UrlHistoryOptions = {\n window?: Window;\n v5Compat?: boolean;\n};\n\nfunction getUrlBasedHistory(\n getLocation: (window: Window, globalHistory: Window[\"history\"]) => Location,\n createHref: (window: Window, to: To) => string,\n validateLocation: ((location: Location, to: To) => void) | null,\n options: UrlHistoryOptions = {}\n): UrlHistory {\n let { window = document.defaultView!, v5Compat = false } = options;\n let globalHistory = window.history;\n let action = Action.Pop;\n let listener: Listener | null = null;\n\n let index = getIndex()!;\n // Index should only be null when we initialize. If not, it's because the\n // user called history.pushState or history.replaceState directly, in which\n // case we should log a warning as it will result in bugs.\n if (index == null) {\n index = 0;\n globalHistory.replaceState({ ...globalHistory.state, idx: index }, \"\");\n }\n\n function getIndex(): number {\n let state = globalHistory.state || { idx: null };\n return state.idx;\n }\n\n function handlePop() {\n action = Action.Pop;\n let nextIndex = getIndex();\n let delta = nextIndex == null ? null : nextIndex - index;\n index = nextIndex;\n if (listener) {\n listener({ action, location: history.location, delta });\n }\n }\n\n function push(to: To, state?: any) {\n action = Action.Push;\n let location = createLocation(history.location, to, state);\n if (validateLocation) validateLocation(location, to);\n\n index = getIndex() + 1;\n let historyState = getHistoryState(location, index);\n let url = history.createHref(location);\n\n // try...catch because iOS limits us to 100 pushState calls :/\n try {\n globalHistory.pushState(historyState, \"\", url);\n } catch (error) {\n // If the exception is because `state` can't be serialized, let that throw\n // outwards just like a replace call would so the dev knows the cause\n // https://html.spec.whatwg.org/multipage/nav-history-apis.html#shared-history-push/replace-state-steps\n // https://html.spec.whatwg.org/multipage/structured-data.html#structuredserializeinternal\n if (error instanceof DOMException && error.name === \"DataCloneError\") {\n throw error;\n }\n // They are going to lose state here, but there is no real\n // way to warn them about it since the page will refresh...\n window.location.assign(url);\n }\n\n if (v5Compat && listener) {\n listener({ action, location: history.location, delta: 1 });\n }\n }\n\n function replace(to: To, state?: any) {\n action = Action.Replace;\n let location = createLocation(history.location, to, state);\n if (validateLocation) validateLocation(location, to);\n\n index = getIndex();\n let historyState = getHistoryState(location, index);\n let url = history.createHref(location);\n globalHistory.replaceState(historyState, \"\", url);\n\n if (v5Compat && listener) {\n listener({ action, location: history.location, delta: 0 });\n }\n }\n\n function createURL(to: To): URL {\n // window.location.origin is \"null\" (the literal string value) in Firefox\n // under certain conditions, notably when serving from a local HTML file\n // See https://bugzilla.mozilla.org/show_bug.cgi?id=878297\n let base =\n window.location.origin !== \"null\"\n ? window.location.origin\n : window.location.href;\n\n let href = typeof to === \"string\" ? to : createPath(to);\n invariant(\n base,\n `No window.location.(origin|href) available to create URL for href: ${href}`\n );\n return new URL(href, base);\n }\n\n let history: History = {\n get action() {\n return action;\n },\n get location() {\n return getLocation(window, globalHistory);\n },\n listen(fn: Listener) {\n if (listener) {\n throw new Error(\"A history only accepts one active listener\");\n }\n window.addEventListener(PopStateEventType, handlePop);\n listener = fn;\n\n return () => {\n window.removeEventListener(PopStateEventType, handlePop);\n listener = null;\n };\n },\n createHref(to) {\n return createHref(window, to);\n },\n createURL,\n encodeLocation(to) {\n // Encode a Location the same way window.location would\n let url = createURL(to);\n return {\n pathname: url.pathname,\n search: url.search,\n hash: url.hash,\n };\n },\n push,\n replace,\n go(n) {\n return globalHistory.go(n);\n },\n };\n\n return history;\n}\n\n//#endregion\n","export default function _arrayWithHoles(arr) {\n if (Array.isArray(arr)) return arr;\n}","export default function _arrayLikeToArray(arr, len) {\n if (len == null || len > arr.length) len = arr.length;\n for (var i = 0, arr2 = new Array(len); i < len; i++) arr2[i] = arr[i];\n return arr2;\n}","import arrayLikeToArray from \"./arrayLikeToArray.js\";\nexport default function _unsupportedIterableToArray(o, minLen) {\n if (!o) return;\n if (typeof o === \"string\") return arrayLikeToArray(o, minLen);\n var n = Object.prototype.toString.call(o).slice(8, -1);\n if (n === \"Object\" && o.constructor) n = o.constructor.name;\n if (n === \"Map\" || n === \"Set\") return Array.from(o);\n if (n === \"Arguments\" || /^(?:Ui|I)nt(?:8|16|32)(?:Clamped)?Array$/.test(n)) return arrayLikeToArray(o, minLen);\n}","export default function _nonIterableRest() {\n throw new TypeError(\"Invalid attempt to destructure non-iterable instance.\\nIn order to be iterable, non-array objects must have a [Symbol.iterator]() method.\");\n}","import arrayWithHoles from \"./arrayWithHoles.js\";\nimport iterableToArrayLimit from \"./iterableToArrayLimit.js\";\nimport unsupportedIterableToArray from \"./unsupportedIterableToArray.js\";\nimport nonIterableRest from \"./nonIterableRest.js\";\nexport default function _slicedToArray(arr, i) {\n return arrayWithHoles(arr) || iterableToArrayLimit(arr, i) || unsupportedIterableToArray(arr, i) || nonIterableRest();\n}","export default function _iterableToArrayLimit(arr, i) {\n var _i = null == arr ? null : \"undefined\" != typeof Symbol && arr[Symbol.iterator] || arr[\"@@iterator\"];\n if (null != _i) {\n var _s,\n _e,\n _x,\n _r,\n _arr = [],\n _n = !0,\n _d = !1;\n try {\n if (_x = (_i = _i.call(arr)).next, 0 === i) {\n if (Object(_i) !== _i) return;\n _n = !1;\n } else for (; !(_n = (_s = _x.call(_i)).done) && (_arr.push(_s.value), _arr.length !== i); _n = !0);\n } catch (err) {\n _d = !0, _e = err;\n } finally {\n try {\n if (!_n && null != _i[\"return\"] && (_r = _i[\"return\"](), Object(_r) !== _r)) return;\n } finally {\n if (_d) throw _e;\n }\n }\n return _arr;\n }\n}","export default function _iterableToArray(iter) {\n if (typeof Symbol !== \"undefined\" && iter[Symbol.iterator] != null || iter[\"@@iterator\"] != null) return Array.from(iter);\n}","import arrayWithoutHoles from \"./arrayWithoutHoles.js\";\nimport iterableToArray from \"./iterableToArray.js\";\nimport unsupportedIterableToArray from \"./unsupportedIterableToArray.js\";\nimport nonIterableSpread from \"./nonIterableSpread.js\";\nexport default function _toConsumableArray(arr) {\n return arrayWithoutHoles(arr) || iterableToArray(arr) || unsupportedIterableToArray(arr) || nonIterableSpread();\n}","import arrayLikeToArray from \"./arrayLikeToArray.js\";\nexport default function _arrayWithoutHoles(arr) {\n if (Array.isArray(arr)) return arrayLikeToArray(arr);\n}","export default function _nonIterableSpread() {\n throw new TypeError(\"Invalid attempt to spread non-iterable instance.\\nIn order to be iterable, non-array objects must have a [Symbol.iterator]() method.\");\n}","export default function _classCallCheck(instance, Constructor) {\n if (!(instance instanceof Constructor)) {\n throw new TypeError(\"Cannot call a class as a function\");\n }\n}","export default function _typeof(obj) {\n \"@babel/helpers - typeof\";\n\n return _typeof = \"function\" == typeof Symbol && \"symbol\" == typeof Symbol.iterator ? function (obj) {\n return typeof obj;\n } : function (obj) {\n return obj && \"function\" == typeof Symbol && obj.constructor === Symbol && obj !== Symbol.prototype ? \"symbol\" : typeof obj;\n }, _typeof(obj);\n}","import _typeof from \"./typeof.js\";\nimport toPrimitive from \"./toPrimitive.js\";\nexport default function _toPropertyKey(arg) {\n var key = toPrimitive(arg, \"string\");\n return _typeof(key) === \"symbol\" ? key : String(key);\n}","import _typeof from \"./typeof.js\";\nexport default function _toPrimitive(input, hint) {\n if (_typeof(input) !== \"object\" || input === null) return input;\n var prim = input[Symbol.toPrimitive];\n if (prim !== undefined) {\n var res = prim.call(input, hint || \"default\");\n if (_typeof(res) !== \"object\") return res;\n throw new TypeError(\"@@toPrimitive must return a primitive value.\");\n }\n return (hint === \"string\" ? String : Number)(input);\n}","import toPropertyKey from \"./toPropertyKey.js\";\nfunction _defineProperties(target, props) {\n for (var i = 0; i < props.length; i++) {\n var descriptor = props[i];\n descriptor.enumerable = descriptor.enumerable || false;\n descriptor.configurable = true;\n if (\"value\" in descriptor) descriptor.writable = true;\n Object.defineProperty(target, toPropertyKey(descriptor.key), descriptor);\n }\n}\nexport default function _createClass(Constructor, protoProps, staticProps) {\n if (protoProps) _defineProperties(Constructor.prototype, protoProps);\n if (staticProps) _defineProperties(Constructor, staticProps);\n Object.defineProperty(Constructor, \"prototype\", {\n writable: false\n });\n return Constructor;\n}","export default function _setPrototypeOf(o, p) {\n _setPrototypeOf = Object.setPrototypeOf ? Object.setPrototypeOf.bind() : function _setPrototypeOf(o, p) {\n o.__proto__ = p;\n return o;\n };\n return _setPrototypeOf(o, p);\n}","import setPrototypeOf from \"./setPrototypeOf.js\";\nexport default function _inherits(subClass, superClass) {\n if (typeof superClass !== \"function\" && superClass !== null) {\n throw new TypeError(\"Super expression must either be null or a function\");\n }\n subClass.prototype = Object.create(superClass && superClass.prototype, {\n constructor: {\n value: subClass,\n writable: true,\n configurable: true\n }\n });\n Object.defineProperty(subClass, \"prototype\", {\n writable: false\n });\n if (superClass) setPrototypeOf(subClass, superClass);\n}","export default function _getPrototypeOf(o) {\n _getPrototypeOf = Object.setPrototypeOf ? Object.getPrototypeOf.bind() : function _getPrototypeOf(o) {\n return o.__proto__ || Object.getPrototypeOf(o);\n };\n return _getPrototypeOf(o);\n}","export default function _isNativeReflectConstruct() {\n if (typeof Reflect === \"undefined\" || !Reflect.construct) return false;\n if (Reflect.construct.sham) return false;\n if (typeof Proxy === \"function\") return true;\n try {\n Boolean.prototype.valueOf.call(Reflect.construct(Boolean, [], function () {}));\n return true;\n } catch (e) {\n return false;\n }\n}","import _typeof from \"./typeof.js\";\nimport assertThisInitialized from \"./assertThisInitialized.js\";\nexport default function _possibleConstructorReturn(self, call) {\n if (call && (_typeof(call) === \"object\" || typeof call === \"function\")) {\n return call;\n } else if (call !== void 0) {\n throw new TypeError(\"Derived constructors may only return object or undefined\");\n }\n return assertThisInitialized(self);\n}","export default function _assertThisInitialized(self) {\n if (self === void 0) {\n throw new ReferenceError(\"this hasn't been initialised - super() hasn't been called\");\n }\n return self;\n}","import getPrototypeOf from \"./getPrototypeOf.js\";\nimport isNativeReflectConstruct from \"./isNativeReflectConstruct.js\";\nimport possibleConstructorReturn from \"./possibleConstructorReturn.js\";\nexport default function _createSuper(Derived) {\n var hasNativeReflectConstruct = isNativeReflectConstruct();\n return function _createSuperInternal() {\n var Super = getPrototypeOf(Derived),\n result;\n if (hasNativeReflectConstruct) {\n var NewTarget = getPrototypeOf(this).constructor;\n result = Reflect.construct(Super, arguments, NewTarget);\n } else {\n result = Super.apply(this, arguments);\n }\n return possibleConstructorReturn(this, result);\n };\n}","import setPrototypeOf from \"./setPrototypeOf.js\";\nimport isNativeReflectConstruct from \"./isNativeReflectConstruct.js\";\nexport default function _construct(Parent, args, Class) {\n if (isNativeReflectConstruct()) {\n _construct = Reflect.construct.bind();\n } else {\n _construct = function _construct(Parent, args, Class) {\n var a = [null];\n a.push.apply(a, args);\n var Constructor = Function.bind.apply(Parent, a);\n var instance = new Constructor();\n if (Class) setPrototypeOf(instance, Class.prototype);\n return instance;\n };\n }\n return _construct.apply(null, arguments);\n}","import getPrototypeOf from \"./getPrototypeOf.js\";\nimport setPrototypeOf from \"./setPrototypeOf.js\";\nimport isNativeFunction from \"./isNativeFunction.js\";\nimport construct from \"./construct.js\";\nexport default function _wrapNativeSuper(Class) {\n var _cache = typeof Map === \"function\" ? new Map() : undefined;\n _wrapNativeSuper = function _wrapNativeSuper(Class) {\n if (Class === null || !isNativeFunction(Class)) return Class;\n if (typeof Class !== \"function\") {\n throw new TypeError(\"Super expression must either be null or a function\");\n }\n if (typeof _cache !== \"undefined\") {\n if (_cache.has(Class)) return _cache.get(Class);\n _cache.set(Class, Wrapper);\n }\n function Wrapper() {\n return construct(Class, arguments, getPrototypeOf(this).constructor);\n }\n Wrapper.prototype = Object.create(Class.prototype, {\n constructor: {\n value: Wrapper,\n enumerable: false,\n writable: true,\n configurable: true\n }\n });\n return setPrototypeOf(Wrapper, Class);\n };\n return _wrapNativeSuper(Class);\n}","export default function _isNativeFunction(fn) {\n return Function.toString.call(fn).indexOf(\"[native code]\") !== -1;\n}","import type { Location, Path, To } from \"./history\";\nimport { warning, invariant, parsePath } from \"./history\";\n\n/**\n * Map of routeId -> data returned from a loader/action/error\n */\nexport interface RouteData {\n [routeId: string]: any;\n}\n\nexport enum ResultType {\n data = \"data\",\n deferred = \"deferred\",\n redirect = \"redirect\",\n error = \"error\",\n}\n\n/**\n * Successful result from a loader or action\n */\nexport interface SuccessResult {\n type: ResultType.data;\n data: any;\n statusCode?: number;\n headers?: Headers;\n}\n\n/**\n * Successful defer() result from a loader or action\n */\nexport interface DeferredResult {\n type: ResultType.deferred;\n deferredData: DeferredData;\n statusCode?: number;\n headers?: Headers;\n}\n\n/**\n * Redirect result from a loader or action\n */\nexport interface RedirectResult {\n type: ResultType.redirect;\n status: number;\n location: string;\n revalidate: boolean;\n reloadDocument?: boolean;\n}\n\n/**\n * Unsuccessful result from a loader or action\n */\nexport interface ErrorResult {\n type: ResultType.error;\n error: any;\n headers?: Headers;\n}\n\n/**\n * Result from a loader or action - potentially successful or unsuccessful\n */\nexport type DataResult =\n | SuccessResult\n | DeferredResult\n | RedirectResult\n | ErrorResult;\n\ntype LowerCaseFormMethod = \"get\" | \"post\" | \"put\" | \"patch\" | \"delete\";\ntype UpperCaseFormMethod = Uppercase;\n\n/**\n * Users can specify either lowercase or uppercase form methods on
![\\"\\"](\\"/imgs/Trigonometry/0.png\\")
","image":"![\\"\\"](\\"/imgs/Trigonometry/1_2.png\\")
![\\"\\"](\\"/imgs/Trigonometry/1_3.png\\")
![\\"\\"](\\"/imgs/Trigonometry/1_5.png\\")
"},{"text":"
![\\"\\"](\\"/imgs/Algebra/0.png\\")
![\\"\\"](\\"/imgs/Algebra/1_2.png\\")
![\\"\\"](\\"/imgs/Algebra/1_3.png\\")
![\\"\\"](\\"/imgs/Proofs/0.png\\")
![\\"\\"](\\"/imgs/Proofs/1_3.png\\")
