Learning Objectives
In this section, you will:
7.2.1 – Use right triangles to evaluate trigonometric functions.
7.2.2 – Use the definitions of trigonometric functions of any angle.
7.2.3 – Use right-triangle trigonometry to solve applied problems.
Mt. Everest, which straddles the border between China and Nepal, is the tallest mountain in the world. Measuring its height is no easy task and, in fact, the actual measurement has been a source of controversy for hundreds of years. The measurement process involves the use of triangles and a branch of mathematics known as trigonometry. In this section, we will define a new group of functions known as trigonometric functions, and find out how they can be used to measure heights, such as those of the tallest mountains.
7.2.1 – Using Right Triangles to Evaluate Trigonometric Functions
(Figure) shows a right triangle with a vertical side of length [latex]\,y\,[/latex] and a horizontal side has length [latex]\,x.\,[/latex] Notice that the triangle is inscribed in a circle of radius 1. Such a circle, with a center at the origin and a radius of 1, is known as a unit circle.
We can define the trigonometric functions in terms an angle t and the lengths of the sides of the triangle. The adjacent side is the side closest to the angle, x. (Adjacent means “next to.”) The opposite side is the side across from the angle, y. The hypotenuse is the side of the triangle opposite the right angle, 1. These sides are labeled in (Figure).
Given a right triangle with an acute angle of [latex]\,t,[/latex] the first three trigonometric functions are listed.
A common mnemonic for remembering these relationships is SohCahToa, formed from the first letters of “Sine is opposite over hypotenuse, Cosine is adjacent over hypotenuse, Tangent is opposite over adjacent.”
For the triangle shown in (Figure), we have the following.
How To
Given the side lengths of a right triangle and one of the acute angles, find the sine, cosine, and tangent of that angle.
- Find the sine as the ratio of the opposite side to the hypotenuse.
- Find the cosine as the ratio of the adjacent side to the hypotenuse.
- Find the tangent as the ratio of the opposite side to the adjacent side.
Example 1 – Evaluating a Trigonometric Function of a Right Triangle
Given the triangle shown in (Figure), find the value of [latex]\,\mathrm{cos}\,\alpha .[/latex]
The side adjacent to the angle is 15, and the hypotenuse of the triangle is 17.
Try It
Given the triangle shown in (Figure), find the value of [latex]\,\text{sin}\,t.[/latex]
Show answer
[latex]$$\frac{7}{25}$$[/latex]
7.2.2 – Using Trigonometric Functions
In previous example, we evaluated the sine and cosine in triangles where we knew all three sides. But the real power of right-triangle trigonometry emerges when we look at triangles in which we know an angle but do not know all the sides.
How To
Given a right triangle, the length of one side, and the measure of one acute angle, find the remaining sides.
- For each