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°.

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

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

GUIDED PRACTICE for Examples 3 and 4 Sketch the angle. Then find its reference angle. 5. 210° The terminal side of θ lies in Quadrant III, so θ' = 210° – 180° = 30°

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°

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

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

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