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Submitting this to see if I am doing it correctly
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@tdegeorge
tdegeorge committed Sep 23, 2024
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91 changes: 91 additions & 0 deletions source/precalculus/source/06-TR/03.ptx
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<subsection>
<title>Activities</title>

<remark>
<p>
In this section, we will learn how to use right triangles to evaluate trigonometric functions. Before doing that, however, let's review some key concepts of right triangles that can also come in handy when solving.
</p>
</remark>
<definition xml:id="def-pythagorean-theorem">
<statement>
<p>
The <term>Pythagorean Theorem</term> is <me>a^2+b^2=c^2</me>, where <m>a</m> and <m>b</m> are lengths of the legs of a right triangle and <m>c</m> is the length of the hypotenuse. In other words, the theorem tells us that if we know the lengths of any two sides of a right triangle, we can find out the length of the third side.
</p>
</statement>
</definition>

<activity>
Suppose the hypotenuse of a right triangle is 13 cm long and one of the legs is 5 cm long.
<task>
<statement>
<p>
Draw a picture of this right triangle and label the sides. Use <m>x</m> to refer to the missing side.
</p>
</statement>
<answer>
<p>
<figure xml:id="drawing-of-5-12-13-triangle">
<image source="Images/5-12-13-triangle.png"/>
<caption>One example of how a student can draw the triangle.</caption>
</figure>
</p>
</answer>
</task>
<task>
<statement>
<p>
What is the value of <m>x</m> (i.e., the length of the third side)?
<ol marker= "A." cols="1">
<li> <m>144</m> </li>
<li> <m>12</m></li>
<li> <m>194</m> </li>
<li> <m>14</m> </li></ol>
</p>
</statement>
<answer>
<p>
B
</p>
</answer>
</task>
</activity>

<remark>
The Pythagorean Theorem can be very helpful in finding the third side of a right triangle as long as we know the length of two other sides. What if we are given the side length of only one side of a given right triangle? How could we find the lengths of the other two sides?
</remark>

<definition xml:id="def-sides-of-right-triangles">
<statement>
<p>
Before we can answer the question posed, we must first make sure we know how to call specific sides of a right triangle.
</p>
<ul>
<li>
<p>
The <term>hypotenuse</term> of a right triangle is always the side opposite the right angle. This side also happens to be the longest side of the triangle.
</p>
</li>
<li>
<p>
The <term>opposite side</term> is across from a given angle.
</p>
</li>
<li>
<p>
The <term>adjacent side</term> is the non-hypotenuse side that is next to a given angle.
</p>
</li>
</ul>
</p>
<p>
When given an angle, all sides of a triangle can be labeled. For example, suppose angle <m>A</m> is given, then the sides of a right triangle can be labeled as:
<figure xml:id="labeling-sides-of-right-triangle">
<image source="Images/Defining-sides-of-a-triangle.png"/>
<caption>From the perspective of angle A, all sides of a right triangle can be labeled.</caption>
</figure>
</p>
</statement>
</definition>

<remark>
<p>
Right triangles can be used to evaluate trigonometric functions, such as sine and cosine. Let's explore how to evaluate all six trigonometric functions!
</p>
</remark>
</subsection>

<exercises>
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128 changes: 128 additions & 0 deletions source/precalculus/source/06-TR/04.ptx
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<subsection>
<title>Activities</title>

<remark>
<p>
Recall from the previous section that we can find values of trigonometric functions of right triangles. In this section, we can take what we know about right triangles and use the same idea to find exact values of trigonometric functions of special angles.
</p>
</remark>
<definition xml:id="def-special-right-triangles">
<statement>
<p>
There are two special right triangle relationships that will continually appear throughout the study of mathematics.
<ul>
<li>
<p>
<m>45-45-90</m> triangle
</p>
</li>
<li>
<p>
<m>30-60-90</m> triangle
</p>
</li>
</ul>
</p>
</statement>
</definition>

<activity xml:id="exploring-45-45-90-triangle">
<introduction>
<p>
In this activity, let's explore the relationship of the <m>45-45-90</m> special right triangle.
</introduction>
<task>
<statement>
<p>
Draw a right triangle and label the angles to have <m>45</m><degree/>, <m>45</m> <degree/>, and <m>90</m><degree/>.
</p>
</statement>
<answer>
<p>
Students should be able to draw a right triangle with one <m>90</m><degree/> angle (indicated by a box) and two <m>45</m><degree/> angles.
</p>
</answer>
</task>
<task>
<statement>
<p>
In a <m>45-45-90</m> triangle, two angles are the same size. If those two angles are the same size, what do we know about the sides opposite those angles?
</p>
</statement>
<answer>
<p>
Students should recognize that the two legs of the triangle must be the same size.
</p>
</answer>
</task>
<task>
<statement>
<p>
Suppose one of the legs of the right triangle is of length <m>1</m>, how long is the other leg?
</p>
</statement>
<answer>
<p>
Students may notice that because the two legs are equal, then the two legs should both be equal to <m>1</m>.
</p>
</answer>
</task>
<task>
<statement>
<p>
Now that we know two sides of the right triangle, use that information and the Pythagorean Theorem to find the length of the third side.
</p>
</statement>
<answer>
<p>
<m>1^2+1^2=c^2</m>
</p>
<p>
<m>2=c^2</m>
</p>
<p>
<m>\sqrt{2}=c</m>
</p>
</answer>
</task>
</activity>

<remark>
<p>
From <xref ref="exploring-45-45-90-triangle"/>, we saw that a <m>45-45-90</m> triangle is an isosceles right triangle, which means that two of the sides of the triangle are equal. The ratio of its legs and hypotenuse is expressed as follows: <me>Leg:Leg:Hypotenuse=1:1:\sqrt{2}</me>. In terms of <m>x</m>, this ratio can be expressed as <me>x:x:\sqrt{2}</me>.
</p>
</remark>

<activity xml:id="exploring-30-60-90-triangle">
<introduction>
<p>
Suppose you are given a <m>30-60-90</m> triangle, where you know the angle opposite the <m>30</m><degree/> angle is of length <m>1</m>.
</p>
</introduction>
<task>
<statement>
<p>
Draw a right triangle and label the angles to have <m>30</m><degree/>, <m>60</m> <degree/>, and <m>90</m><degree/> and the side opposite of the <m>30</m><degree/> angle as having a length of <m>1</m>.
</p>
</statement>
<answer>
<p>
Students should be able to draw a right triangle with one <m>90</m><degree/> angle (indicated by a box), one <m>30</m><degree/> angle and one <m>45</m><degree/> angle. THe side opposite the <m>30</m><degree/> angle should have a side length of <m>1</m>.
</p>
</answer>
</task>
<task>
<statement>
<p>
Unlike the <m>45-45-90</m> triangle, we cannot easily find another side length with the information given. Using trigonometry, which trigonometric function would help us find the length of the hypotenuse with the information given?
<ol marker= "A." cols="1">
<li> <m>\sin$\theta$</m> </li>
<li> <m>\cos$\theta$</m></li>
<li> <m>\tan$\theta$</m> </li></ol>
</p>
</statement>
<answer>
<p>
<m>\sin$\theta$</m>
</p>
</answer>
</task>

</activity>
</subsection>

<exercises>
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