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EXAMPLE 3

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

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EXAMPLE 3

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

  2. 17π Evaluate (a) tan ( – 240°) and (b) csc . 6 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: 3 √ tan (–240°) = – tan 60° = – EXAMPLE 4 Use reference angles to evaluate functions SOLUTION

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

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

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

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

  8. √ 3 2 for Examples 3 and 4 GUIDED PRACTICE 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°) = –

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