1. EXAMPLE 3
Find reference angles
Find the reference angle θ' for (a) θ = 5π 3
and (b) θ = – 130°.
SOLUTION a.
The terminal side of θ lies in Quadrant IV.
So, θ' = 2π – 5π 3 π = b.
Note that θ is coterminal with 230°, whose terminal side lies in Quadrant III. So, θ' = 230° – 180° + 50°.

2. EXAMPLE 4
Use reference angles to evaluate functions
Evaluate (a) tan ( – 240°) and (b) csc
17π 6
SOLUTION a.
The angle – 240° is coterminal with 120°. The reference angle is θ' = 180° – 120° = 60°. The tangent function is negative in Quadrant II, so you can write:
tan (–240°) = – tan 60° = – √ 3

3. EXAMPLE 4
Use reference angles to evaluate functions b.
The angle is coterminal
with The reference
angle is θ' = π – =
The cosecant function is positive in Quadrant II, so you can write:
17π 6 5π π
csc = csc = 2
17π 6 5π

4. GUIDED PRACTICE
for Examples 3 and 4
Sketch the angle. Then find its reference angle. °
The terminal side of θ lies in Quadrant III,
so θ' = 210° – 180° = 30°

5. GUIDED PRACTICE
for Examples 3 and 4
Sketch the angle. Then find its reference angle.
6. – 260°
– 260° is coterminal with 100°, whose terminal side of θ lies in Quadrant III, so θ' = 180° – 100° = 80°

6. GUIDED PRACTICE
for Examples 3 and 4
Sketch the angle. Then find its reference angle. 7. 7π 9 – 7π 9
The angle – is coterminal with The terminal side lies in Quadrant III,
so θ' = – π =
11π 2π

7. GUIDED PRACTICE
for Examples 3 and 4
Sketch the angle. Then find its reference angle.
15π 4 8.
The terminal side lies in Quadrant III,
so θ' = 2π – =
15π 4 π

8. GUIDED PRACTICE
for Examples 3 and 4 9.
Evaluate cos ( – 210°) without using a calculator.
– 210° is coterminal with 150°. The terminal side lies in Quadrant II, which means it will have a negative value.
So, cos (– 210°) = – 2 √ 3