Trigonometry

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photo taken by Joe Sueyoshi

Applications of Right Triangle Trigonometry

Learning objectives

A student will be able to:

Video: Solving Right Triangles - The Basics

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Introduction

In this lesson we will return to right triangle trigonometry. Many real situations involve right triangles. In your previous study of geometry you may have used right triangles to solve problems involving distances, using the Pythagorean Theorem. In this lesson you will solve problems involving right triangles, using your knowledge of angles and trigonometric functions. We will begin by solving right triangles, which means identifying all the measures of all three angles and the lengths of all three sides of a right triangle. Then we will turn to several kinds of problems.

Solving Right Triangles

You can use your knowledge of the Pythagorean Theorem and the six trigonometric functions to solve a right triangle. Because a right triangle is a triangle with a 90 degree angle, solving a right triangle requires that you find the measures of one or both of the other angles. How you solve will depend on how much information is given. The following examples show two situations: a triangle missing one side, and a triangle missing two sides.

Example 1: Solve the triangle shown below.

Solution:

We need to find the lengths of all sides and the measures of all angles. In this triangle, two of the three sides are given. We can find the length of the third side using the Pythagorean Theorem:

8^2 + b^2 & = 10^2 \ 64 + b^2 & = 100 \ b^2 & = 36 \ b & = \pm 6 \Rightarrow b = 6

(You may have also recognized the "Pythagorean Triple," 6, 8, 10, instead of carrying out the Pythagorean Theorem.)

You can also find the third side using a trigonometric ratio. Notice that the missing side, b, is adjacent to angle A, and the hypotenuse is given. Therefore we can use the cosine function to find the length of b:

\cos (53.13^\circ) & = \frac{\text{adjacent side}} {\text{hypotenuse}} = \frac{b} {10} \ .6 & = \frac{b} {10} \ b & = .6(10) = 6

We could also use the tangent function, as the opposite side was given. It may seem confusing that you can find the missing side in more than one way. The point is, however, not to create confusion, but to show that you must look at what information is missing, and choose a strategy. Overall, when you need to identify one side of the triangle, you can either use the Pythagorean Theorem, or you can use a trig ratio.

To solve the above triangle, we also have to identify the measures of all three angles. Two angles are given: 90\;\mathrm{degrees} and 53.13\;\mathrm{degrees}. We can find the third angle using the triangle angle sum:

180 - 90 - 53.13 = 36.87^\circ.

Now let's consider a triangle that has two missing sides.

Example 2: Solve the triangle shown below.

Solution:

In this triangle, we need to find the lengths of two sides. We can find the length of one side using a trig ratio. Then we can find the length of the third side either using a trig ratio, or the Pythagorean Theorem.

We are given the measure of angle A, and the length of the side adjacent to angle A. If we want to find the length of the hypotenuse, c, we can use the cosine ratio:

\cos (40^\circ) & = \frac{\text{adjacent}} {\text{hypotenuse}} = \frac{6} {c} \ \cos (40^\circ) & = \frac{6} {c} \\ c\ \cos (40^\circ) & = 6\ c & = \frac{6} {\cos (40^\circ)} \approx 7.83

If we want to find the length of the other leg of the triangle, we can use the tangent ratio. (Why is this a better idea than to use the sine?)

\tan (40^\circ) & = \frac{\text{opposite}} {\text{adjacent}} = \frac{a} {6} \ \tan (40^\circ) & = \frac{a} {c} \ a & = 6\ \tan (40^\circ) \approx 5.03

Now we know the lengths of all three sides of this triangle. In the review questions, you will verify the values of c and a using the Pythagorean Theorem. Here, to finish solving the triangle, we only need to find the measure of angle B:

180 - 90 - 40 = 50^\circ

Notice that in both examples, one of the two non-right angles was given. If neither of the two non-right angles is given, you will need new strategy to find the angles. You will learn this strategy in chapter 4.