# 3 Right Triangle Trigonometry

### Learning Objectives

In this section you will:

- Use right triangles to evaluate trigonometric functions.
- Find function values for and
- Use equal cofunctions of complementary angles.
- Use the deﬁnitions of trigonometric functions of any angle.
- 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.

### Using Right Triangles to Evaluate Trigonometric Functions

(Figure) shows a right triangle with a vertical side of lengthand a horizontal side has lengthNotice 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 ofthe first three trigonometric functions are listed.

A common mnemonic for remembering these relationships is SohCahToa, formed from the first letters of “__ S__ine is

__pposite over__

**o**__ypotenuse,__

**h**__osine is__

**C**__djacent over__

**a**__ypotenuse,__

**h**__angent is__

**T**__pposite over__

**o****djacent.”**

__a__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.

### Evaluating a Trigonometric Function of a Right Triangle

Given the triangle shown in (Figure), find the value of

## Show Solution

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

## Show Solution

#### Reciprocal Functions

In addition to sine, cosine, and tangent, there are three more functions. These too are defined in terms of the sides of the triangle.

Take another look at these definitions. These functions are the reciprocals of the first three functions.

When working with right triangles, keep in mind that the same rules apply regardless of the orientation of the triangle. In fact, we can evaluate the six trigonometric functions of either of the two acute angles in the triangle in (Figure). The side opposite one acute angle is the side adjacent to the other acute angle, and vice versa.

Many problems ask for all six trigonometric functions for a given angle in a triangle. A possible strategy to use is to find the sine, cosine, and tangent of the angles first. Then, find the other trigonometric functions easily using the reciprocals.

### How To

**Given the side lengths of a right triangle, evaluate the six trigonometric functions of one of the acute angles.
**

- If needed, draw the right triangle and label the angle provided.
- Identify the angle, the adjacent side, the side opposite the angle, and the hypotenuse of the right triangle.
- Find the required function:
- sine as the ratio of the opposite side to the hypotenuse
- cosine as the ratio of the adjacent side to the hypotenuse
- tangent as the ratio of the opposite side to the adjacent side
- secant as the ratio of the hypotenuse to the adjacent side
- cosecant as the ratio of the hypotenuse to the opposite side
- cotangent as the ratio of the adjacent side to the opposite side

### Evaluating Trigonometric Functions of Angles Not in Standard Position

Using the triangle shown in (Figure), evaluate

## Show Solution

#### Analysis

Another approach would have been to find sine, cosine, and tangent first. Then find their reciprocals to determine the other functions.

### Try It

Using the triangle shown in (Figure),evaluate