diff --git a/source/precalculus/assets/Images/5-12-13-triangle.png b/source/precalculus/assets/Images/5-12-13-triangle.png new file mode 100644 index 00000000..43cdc567 Binary files /dev/null and b/source/precalculus/assets/Images/5-12-13-triangle.png differ diff --git a/source/precalculus/assets/Images/Defining-sides-of-a-triangle.png b/source/precalculus/assets/Images/Defining-sides-of-a-triangle.png new file mode 100644 index 00000000..28bf0320 Binary files /dev/null and b/source/precalculus/assets/Images/Defining-sides-of-a-triangle.png differ diff --git a/source/precalculus/source/06-TR/03.ptx b/source/precalculus/source/06-TR/03.ptx index a27aee81..6bf39f0b 100644 --- a/source/precalculus/source/06-TR/03.ptx +++ b/source/precalculus/source/06-TR/03.ptx @@ -12,6 +12,611 @@ Activities + +

+ 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. +

+ + + + +

+ The Pythagorean Theorem is a^2+b^2=c^2, where a and b are lengths of the legs of a right triangle and c is the length of the hypotenuse. If we know the lengths of any two sides of a right triangle, we can find the length of the third side. +

+ + + + + +

+ Suppose two legs of a right triangle measure 3 inches and 4 inches. +

+ + + + +

+ Draw a picture of this right triangle and label the sides. Use x to refer to the missing side. +

+ + +

+ Students should draw a triangle with x representing the hypotenuse and the two legs 3 and 4 inches respectively. +

+ + + + + +

+ What is the value of x (i.e., the length of the third side)? +

    +
  1. 5 inches
    2. +
  2. \sqrt{5} inches
    3. +
  3. 25 inches
    4. +
  4. 16 inches
+

+ + +

+ A +

+ + + + + + +

+ Suppose the hypotenuse of a right triangle is 13 cm long and one of the legs is 5 cm long. +

+ + + + +

+ Draw a picture of this right triangle and label the sides. Use x to refer to the missing side. +

+ + +

+

+ + One example of how a student can draw the triangle. +
+

+ + + + + +

+ What is the value of x (i.e., the length of the third side)? +

    +
  1. 144
    2. +
  2. 12
    3. +
  3. 194
    4. +
  4. 14
+

+ + +

+ B +

+ + + + + + +

+ Pythagorean triples are integers a, b, and c that satisfy the Pythagorean theorem. and highlight some of the most common types of Pythagorean triples: 3-4-5 and 5-12-13. All triangles similar to the 3-4-5 triangle will also have side lengths that are multiples of 3-4-5 (like 6-8-10). Similarly, this is true for all triangles similar to the 5-12-13 triangle. +

+ + + + + +

+ When working with right triangles, it is often helpful to refer to specific angles and sides. One way this is done is by using letters, such as A and a to show that these are an angle-side pair because every angle has a side opposite the angle in a triangle. Note that the capital letter indicates the angle, and the lower case letter indicates the side. Another way to label the sides of a triangle is to use the relationships between a given angle within a triangle and the sides. +

+
    +
  • +

    + The hypotenuse of a right triangle is always the side opposite the right angle. This side also happens to be the longest side of the triangle. +

    +
    • +
  • +

    + The opposite side is the non-hypotenuse side across from a given angle. +

    +
    • +
  • +

    + The adjacent side is the non-hypotenuse side that is next to a given angle. +

    +
    • +
+

+ When given an angle, all sides of a triangle can be labeled. For example, suppose angle A is given, then the sides of a right triangle can be labeled as: +

+ + From the perspective of angle A, all sides of a right triangle can be labeled. +
+

+ + + + + +

+ Suppose you are given a right triangle where the hypotenuse is 11 cm long and and one of the interior angles is 60 degrees. +

+ + + + +

+ Draw a picture of this right triangle and label the sides (refer back to to help you label the sides). +

+ + + + + +

+ What is the measure of the third angle? +

    +
  1. 90 degrees
    2. +
  2. 60 degrees
    3. +
  3. 30 degrees
    4. +
  4. 180 degrees
+

+ + +

+ C +

+ + + + + +

+ Suppose you are asked to find one of the sides of the right triangle. What additional information would you need to find the length of another side of the triangle? +

+ + +

+ Students will probably notice that the Pythagorean Theorem is not helpful in this case because they only know the length of one side. This is a great opportunity to discuss how the Pythagorean Theorem is useful in finding side lengths when at least two sides are known. +

+ + + + + +

+ 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. In , only one side and one angle were given. In this case, the Pythagorean Theorem is not enough to help us find another side of the right triangle (unless it is one of the triples!). +

+ + + + +

+ Trigonometric ratios such as sine, cosine, and tangent are based on the relationships between a given angle \theta and side lengths in a right triangle. Three trigonometric functions, sine, cosine, and tangent, are often used to understand the relationships between a given angle of a triangle and its sides. For acute angles, such as \theta, these functions can be defined as ratios between the sides of a right triangle. + + + \sin\theta=\frac{opposite}{hypotenuse} + + + \cos\theta=\frac{adjacent}{hypotenuse} + + + \tan\theta=\frac{opposite}{adjacent} + + +

+

+ Notice that these are defined according to the sides of a triangle - which is why it is important to be able to label correctly! +

+ + + + + +

The top of the Eiffel Tower is seen from a distance of d = 500 meters at an angle of \alpha=31 degrees. +

+ + + + +

+ Draw a diagram to represent the situation. Use x to refer to the missing side. +

+ + +

+ Students should draw a right triangle where \theta is the angle formed with the ground, 500 meters as the side adjacent to \theta, and the height of the Eiffel Tower (side opposite \theta) as x. +

+ + + + + +

+ Which trig function could we use to find the height of the tower? +

    +
  1. sine
    2. +
  2. cosine
    3. +
  3. tangent

+ + +

+ C +

+ + + + + +

+ How could we correctly set up the trigonometric ratio to find the height of the Eiffel Tower? +

    +
  1. \sin{31}=\frac{x}{500}
    2. +
  2. \cos{31}=\frac{500}{x}
    3. +
  3. \cos{31}=\frac{500}{x}
    4. +
  4. \tan{31}=\frac{x}{500}

+ + +

+ D +

+ + + + + +

+ Find the height of the tower. +

    +
  1. 428.58 meters
    2. +
  2. 300.43 meters
    3. +
  3. 220.85 meters
    4. +
  4. 257.52 meters

+ + +

+ B +

+ + + + + + +

+ For each triangle given, determine which trigonometric ratio would be the most helpful in determining the length of the side of a triangle. Be sure to draw a picture of the triangle to help you determine the relationship between the given angle and sides. +

+ + + + +

+ In triangle ABC, B=37 degrees and a=11. Which trigonometric function will best help determine the length of side c? +

    +
  1. sine
    2. +
  2. cosine
    3. +
  3. tangent

+ + +

+ B +

+ + + + + +

+ In triangle ABC, A=32 degrees and b=13. Which trigonometric function will best help determine the length of side a? +

    +
  1. sine
    2. +
  2. cosine
    3. +
  3. tangent

+ + +

+ C +

+ + + + + +

+ In triangle ABC, angle A=24 degrees and the hypotenuse is 14. Which trigonometric function will best help determine the length of side a? +

    +
  1. sine
    2. +
  2. cosine
    3. +
  3. tangent

+ + +

+ A +

+ + + + + + +

+ Suppose you are given triangle ABC, where a=35, b=12, and c=37, with c being the hypotenuse of the triangle. +

+ + + + +

+ Find the ratio of \tan{B}. +

    +
  1. \frac{35}{12}
    2. +
  2. \frac{35}{37}
    3. +
  3. \frac{12}{35}
    4. +
  4. \frac{12}{37}

+ + +

+ C +

+ + + + + +

+ Find the ratio of \cos{A}. +

    +
  1. \frac{35}{12}
    2. +
  2. \frac{35}{37}
    3. +
  3. \frac{12}{35}
    4. +
  4. \frac{12}{37}

+ + +

+ D +

+ + + + + +

+ Find the ratio of \sin{B}. +

    +
  1. \frac{35}{12}
    2. +
  2. \frac{35}{37}
    3. +
  3. \frac{12}{35}
    4. +
  4. \frac{12}{37}

+ + +

+ D +

+ + + + + +

+ Suppose we want to know the measure of angle A. We can find the measure of angle A in three different ways by using either sine, cosine, or tangent (since all side lengths are given). For each trigonometric function, write the trigonometric ratio that can be used to find the measure of angle A. +

+ + +

+ Students should be able to write all three trigonometric functions: \cos{A}=\frac{12}{37}, \sin{A}=\frac{35}{37}, and \tan{A}=\frac{35}{312}. +

+ + + + + +

+ Now that we have set up a trigonometric ratio to help find the measure of angle A, how can we use these ratios to determine how big A is? +

+ + +

+ Give students the opportunity to discuss with one another on how they would try to determine the measure of angle A. Instructors might want to give a hint about how to "undo" the trigonometric function. +

+ + + + + +

Sometimes you will need to use trigonometric functions to find the measure of an angle. In these cases, you will need to use the inverse trig function key on your calculator (such as \sin^(-1))to find the angle that yields that trig value. +

+

For example, the sine function \sin takes an angle and gives us the ratio \frac{opposite}{hypotenuse}, but \sin^{-1}(called "inverse sine") takes the ratio \frac{opposite}{hypotenuse} and gives us an angle. +

+ + + + +

+ Refer back to , where you were given all the sides of a right triangle, but no angle measures. (In triangle ABC, a=35, b=12, and c=37, with c being the hypotenuse of the triangle). +

+ + + + +

+ What is the trigonometric ratio for \cos{A}? +

    +
  1. \frac{35}{12}
    2. +
  2. \frac{35}{37}
    3. +
  3. \frac{12}{35}
    4. +
  4. \frac{12}{37}

+ + +

+ D +

+ + + + + +

+ Use the inverse trig function, \cos^{-1} to find the measure of angle A. (Make sure your calculator is in degree mode!) +

+ + +

+ Students should get approximately 71.08 degrees. +

+ + + + + +

+ What is the trigonometric ratio for \sin{A}? +

    +
  1. \frac{35}{12}
    2. +
  2. \frac{35}{37}
    3. +
  3. \frac{12}{35}
    4. +
  4. \frac{12}{37}

+ + +

+ B +

+ + + + + +

+ Use the inverse trig function, \sin^{-1} to find the measure of angle A. (Make sure your calculator is in degree mode!) +

+ + +

+ Students should get approximately 71.08 degrees. +

+ + + + + +

+ What is the trigonometric ratio for \tan{A}? +

    +
  1. \frac{35}{12}
    2. +
  2. \frac{35}{37}
    3. +
  3. \frac{12}{35}
    4. +
  4. \frac{12}{37}

+ + +

+ A +

+ + + + + +

+ Use the inverse trig function, \tan^{-1} to find the measure of angle A. (Make sure your calculator is in degree mode!) +

+ + +

+ Students should get approximately 71.08 degrees. +

+ + + + + +

+ Refer back to parts (b), (d), and (f). What do you notice about your answers from those parts? +

+ + +

+ Students should notice that they got the same angle measure for A regardless of which trigonometric function they used. +

+ + + + + +

+ Now that we know the measure of angle A, find the measure of angle B. +

+ + +

+ Angle B is approximately 18.92 degrees. +

+ + + + + +

+ Determining all of the side lengths and angle measures of a right triangle is known as solving a right triangle. In and , we were given all the sides of the triangle and used trigonometric ratios to determine the measure of the angles. +

+ + + + +

+ Solve the following triangles using your knowledge of right triangles, the Pythagorean Theorem and trigonometric functions. Be sure to draw a picture to help you determine the relationship between the angles and sides. +

+ + + + +

+ In triangle ABC, B=53 degrees and c=5 meters (with c being the hypotenuse). +

+ + +

+ A=37 degrees, C=90 degrees, a=3 meters, and b=4 meters. +

+ + + + + +

+ In triangle ABC, A=28 degrees and b=29.3 miles (with c being the hypotenuse). +

+ + +

+ B=62 degrees, C=90 degrees, a=15.6 miles, and c=33.2 miles. +

+ + + + + +

+ In triangle ABC, a=8 feet, b=17 feet, and c=15 feet (with b being the hypotenuse). +

+ + +

+ A=28.07 degrees, B=90 degrees, and C=61.93 degrees. +

+ + + + diff --git a/source/precalculus/source/06-TR/04.ptx b/source/precalculus/source/06-TR/04.ptx index 73427b24..eb716300 100644 --- a/source/precalculus/source/06-TR/04.ptx +++ b/source/precalculus/source/06-TR/04.ptx @@ -12,6 +12,136 @@ Activities + +

+ 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. +

+ + + + +

+ There are two special right triangle relationships that will continually appear throughout the study of mathematics. +

    +
  • +

    + 45-45-90 triangle +

    +
    • +
  • +

    + 30-60-90 triangle +

    +
    • +
+

+ + + + + +

+ In this activity, let's explore the relationship of the 45-45-90 special right triangle. +

+ + + +

+ Draw a right triangle and label the angles to have 45, 45, and 90. +

+ + +

+ Students should be able to draw a right triangle with one 90 angle (indicated by a box) and two 45 angles. +

+ + + + +

+ In a 45-45-90 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? +

+ + +

+ Students should recognize that the two legs of the triangle must be the same size. +

+ + + + +

+ Suppose one of the legs of the right triangle is of length 1, how long is the other leg? +

+ + +

+ Students may notice that because the two legs are equal, then the two legs should both be equal to 1. +

+ + + + +

+ 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. +

+ + +

+ 1^2+1^2=c^2 +

+

+ 2=c^2 +

+

+ \sqrt{2}=c +

+ + + + + +

+ From , we saw that a 45-45-90 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: Leg:Leg:Hypotenuse=1:1:\sqrt{2}. In terms of x, this ratio can be expressed as x:x:\sqrt{2}. +

+ + + + +

+ Suppose you are given a 30-60-90 triangle, where you know the angle opposite the 30 angle is of length 1. +

+ + + +

+ Draw a right triangle and label the angles to have 30, 60 , and 90 and the side opposite of the 30 angle as having a length of 1. +

+ + +

+ Students should be able to draw a right triangle with one 90 angle (indicated by a box), one 30 angle and one 45 angle. THe side opposite the 30 angle should have a side length of 1. +

+ + + + +

+ Unlike the 45-45-90 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? +

    +
  1. \sin\theta
    2. +
  2. \cos\theta
    3. +
  3. \tan\theta
+

+ + +

+ A +

+ + + +