**Trigonometry**

photo taken by Joe Sueyoshi

# Trigonometric Functions of Any Angle

## Learning objectives

A student will be able to:

- Identify the reference angles for angles in the unit circle.
- Identify the ordered pair on the unit circle for angles whose reference angle is , , and , or a quadrantal angle, including negative angles, and angles whose measure is greater than .
- Use these ordered pairs to determine values of trig functions of these angles.
- Use tables and calculators to find values of trig functions of any angle.

## Introduction

In the previous lesson we introduced the six trigonometric functions, and we worked with these functions in two ways: first, in right triangles, and second, for angles of rotation. In this lesson we will extend our work with trig functions of angles of rotation to any angle in the unit circle, including negative angles, and angles greater than . In the previous lesson, we worked with the quadrantal angles, and with the angles , , and . In this lesson we will work with angles related to these angles, as well as other angles in the unit circle. One of the key ideas of this lesson is that angles may share the same trig values. This idea will be developed throughout the lesson.

## Reference Angles and Angles in the Unit Circle

In the previous lesson, one of the review questions asked you to consider the angle . If we graph this angle in standard position, we see that the terminal side of this angle is a reflection of the terminal side of , across the axis.

Notice that makes a angle with the negative axis. Therefore we say that is the **reference angle** for . Formally, the **reference angle** of an angle in standard position is the angle formed with the closest portion of the axis. Notice that is the reference angle for many angles. For example, it is the reference angle for and for .

**Review Video: Determining trig function values using refence triangles**

**Review Video: Determining trig function values using the unit circle**

Unit Circle: You Try It Animation