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.
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
Determine whether or not the angles are coterminal.
 18° and 738°

Is the difference between 18° and 738° a multiple of 360? 738° − 18° = 720°, 720° = 2.360°, so the angles are coterminal.
 −170° and 350°

350° − (−170°) = 350° + 170° = 520° and 520° is not a multiple of 360°, so the angles are not coterminal.
 π /8 radians and − 7 π /8 radians

Is the difference of π /8 and − 7 π /8 a multiple of 2 π ?
Because π radians is not a multiple of 2 π radians, the angles are not coterminal.
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 .
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.)
Examples
Determine the quadrant in which the point lies.
 (5, −3)

x = 5 is positive, and y = −3 is negative, the point is in Quadrant IV.
 (4, 7)

Both x = 4 and y = 7 are positive, the point is in Quadrant I.
 (−1, −6)

Both x = −1 and y = −6 are negative, the point is in Quadrant III.
 (−2, 10)

x = −2 is negative, y = 10 is positive, the point is in Quadrant II.
Below is an outline for finding the reference angle.
 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 .
 If θ is Quadrant I, θ is its own reference angle.
 If θ is in Quadrant II, the reference angle is π − θ .
 If θ is in Quadrant III, the reference angle is θ − π .
 If θ is in Quadrant IV, the reference angle is 2 π − θ .
Examples
Find the reference angle.

This angle is in Quadrant III (bigger than π but smaller than 3 π /2), so = 9 π /8 − π = π /8.

This angle is not between 0 and 2 π , so we need to add or subtract some multiple of 2 π so that we do have an angle between 0 and 2 π . The coterminal angle we need is 7 π /3 − 2 π = 7 π /3 − 6 π /3 = π /3, π /3 is its own reference angle because it is in Quadrant I, so = π /3.

This angle is in Quadrant II (between π /2 and π ), so = π − 5 π /7 = 7 π /7 − 5 π /7 = 2 π /7.

This angle is not between 0 and 2 π . It is coterminal with 2 π + (−2 π /3) = 6 π /3 − 2 π /3 = 4 π /3. The angles are in Quadrant III, so = 4 π /3 − π = 4 π /3 − 3 π /3 = π /3.
Practice problems for this concept can be found at: Trigonometry Practice Test.
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