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Finding Reference Angles and Evaluating Trigonometric Functions

This guide provides methods to find reference angles and evaluate trigonometric functions for various angles. It includes step-by-step solutions for identifying the terminal side of angles in different quadrants, determining coterminal angles, and calculating reference angles like θ' for given values. Learn how to apply these concepts to angles in Quadrants II, III, and IV, using examples such as θ = 210° and θ = -260°. Understanding reference angles is essential for solving trigonometric problems efficiently.

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Finding Reference Angles and Evaluating Trigonometric Functions

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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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