Reference Angle

1. Reference Angle • Reference angle: the positiveacute angle that lies between the terminal side of a given angle θ and the x-axis Note: the given angle θ MUST be in standard position

2. Reference Angle Examples – Quadrant I Note that both θ and the reference angle are 60°

3. Reference Angle Examples – Quadrant II

4. Reference Angle Examples – Quadrant III

5. Reference Angle Examples – Quadrant IV

6. Reference Angle Summary • Depending in which quadrant θ terminates, we can formulate a general rule for finding reference angles: • For any positive angle θ, 0° ≤ θ ≤ 360°: • If θЄ QI: Ref angle = θ • If θ Є QII: Ref angle = 180° – θ • If θ Є QIII: Ref angle = θ – 180° • If θЄ QIV: Ref angle = 360° – θ

7. Reference Angle Summary (Continued) • If θ > 360°: • Keep subtracting 360° from θ until 0° ≤ θ ≤ 360° • Go back to the first step on the previous slide • If θ < 0°: • Keep adding 360° to θ until 0° ≤ θ ≤ 360° • Go back to the first step on the previous slide

8. Reference Angle (Example) Ex 1: Draw each angle in standard position and then name the reference angle: a) 210° b) 101° c) 543° d) -342° e) -371°

9. Relationship Between Trigonometric Functions with Equivalent Values • Consider the value of cos 60° and the value of cos 120°: cos 60° = ½ (Should have this MEMORIZED!) cos 120° = -½ (From Definition I with and 30° – 60° – 90° triangle)

10. Relationship Between Trigonometric Functions with Equivalent Values (Continued) • What is the reference angle of 120°? 60° • Need to adjust the final answer depending on which quadrant θ terminates in: 120° terminates in QII AND cos θ is negative in QII • Therefore, cos 120° = -cos 60° = -½ • The VALUES are the same – just the signs are different!

11. Reference Angle Theorem • Reference Angle Theorem: the value of a trigonometric function of an angle θ is EQUIVALENT to the VALUE of the trigonometric function of its reference angle • The ONLY thing that may be different is the sign • Determine the sign based on the trigonometric function and which quadrant θ terminates in • The Reference Angle Theorem is the reason why we need to memorize the exact values of 30°, 45°, and 60° only in Quadrant I!

12. Reference Angle Summary • Recall: • For any positive angle θ, 0° ≤ θ ≤ 360° • If θЄ QI: Ref angle = θ • If θ Є QII: Ref angle = 180° – θ • If θ Є QIII: Ref angle = θ – 180° • If θЄ QIV: Ref angle = 360° – θ

13. Reference Angle Summary (Continued) • If θ > 360°: • Keep subtracting 360° from θ until 0° ≤ θ ≤ 360° • Go back to the first step • If θ < 0°: • Keep adding 360° to θ until 0° ≤ θ ≤ 360° • Go back to the the first step

14. Reference Angle Theorem (Example) Ex 2: Use reference angles to find the exact value of the following: a) cos 135° b) tan 315° c) sec(-60°) d) cot 390°

15. Approximating Angles (Continued) • To circumvent this problem, we can use reference angles: • Find the reference angle that corresponds to the given value of a trigonometric function: • Recall that a reference angle is a positive acute angle which terminates in QI • Because cos θ and sin θ are both positive in QI, always use the POSITIVE value of the trigonometric function • Apply the reference angle by utilizing the quadrant in which θ terminates

16. Approximating Angles (Example) Ex 3: Use a calculator to approximate θ if 0° < θ < 360° and: a) cos θ = 0.0644, θЄ QIV b) tan θ = 0.5890, θЄ QI c) sec θ = -3.4159, θЄ QII d) csc θ = -1.7876, θЄ QIII