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312 Chapter 4 Trigonometry
Reference Angles
The values of the trigonometric functions of angles greater than 90 (or less than 0) can
be determined from their values at corresponding acute angles called reference angles.
Definition of Reference Angle
Let be an angle in standard position. Its reference angle is the acute angle
formed by the terminal side of and the horizontal axis.
Figure 4.39 shows the reference angles for
in Quadrants II, III, and IV.
Quadrant II
θ
Reference
angle: ′ θ
θ
θ
Reference
angle: ′ θ
Reference
angle: θ ′
θ′ = π − θ (radians)
θ ′ = 180° − θ (degrees)
FIGURE 4.39
Quadrant III
θ ′ = θ − (radians)
′ = θ − 180° π
θ
(degrees)
Quadrant
IV
θ′ = 2π
− θ (radians)
θ′ = 360° − θ (degrees)
y
Example 4
Finding Reference Angles
θ = 300°
x
θ′ = 60°
Find the reference angle .
300
2.3
a. b. c.
135
FIGURE 4.40
θ′ = π − 2.3
FIGURE 4.41
θ′ = 45°
225°
FIGURE 4.42
y
y
θ = 2.3
x
225° and −135°
are coterminal.
x
θ = −135°
Solution
a. Because 300 lies in Quadrant IV, the angle it makes with the x-axis is
360 300
60.
Degrees
Figure 4.40 shows the angle
and its reference angle
b. Because 2.3 lies between 2 1.5708 and it follows that it is in
Quadrant II and its reference angle is
2.3
0.8416.
Radians
Figure 4.41 shows the angle
and its reference angle
c. First, determine that 135 is coterminal with 225, which lies in Quadrant III. So,
the reference angle is
225 180
45.
Degrees
Figure 4.42 shows the angle
2.3
Now try Exercise 45.
300
135
3.1416,
and its reference angle
60.
2.3.
45.