Coterminal and Reference Angles Help

Introduction to Coterminal and Reference Angles

Two angles are coterminal if their terminal sides are the same. For example, the terminal sides of the angles 300° and −60° are the same. See Figure 13.3.

Fig. 13.3

Two angles are coterminal if their difference is a multiple of 360° or 2 π radians. In the example above, the difference of 300° and −60° is 300° − (−60°) = 360°.

Examples

Every angle, θ (the Greek letter theta ), has a reference angle , , associated with it. The reference angle is the smallest angle between the terminal side of the angle and the x -axis. A reference angle will be between 0 and π /2 radians, or 0° and 90°. The reference angle for all of the angles shown in Figures 13.4 through 13.7 is .

Fig. 13.4

Fig. 13.5

Fig. 13.6

Fig. 13.7

The xy plane is divided into four quadrants. The trigonometric functions of angles in the different quadrants will have different signs. It is important to be familiar with the signs of the trigonometric functions in the different quadrants. One reason is that formulas have ± signs in them, and the sign of + or − depends on the quadrant in which the angle lies. Before we find reference angles, we will become familiar with the quadrants in the xy plane. (see Figure 13.8.)

Fig. 13.8

Examples

Below is an outline for finding the reference angle.

1. If the angle is not between 0 radians and 2 π radians, find an angle between these two angles by adding or subtracting a multiple of 2 π . Call this angle θ .
2. If θ is Quadrant I, θ is its own reference angle.
3. If θ is in Quadrant II, the reference angle is π − θ .
4. If θ is in Quadrant III, the reference angle is θ − π .
5. If θ is in Quadrant IV, the reference angle is 2 π − θ .

Examples

Practice problems for this concept can be found at: Trigonometry Practice Test.