1. Lesson 6: TRIGONOMETRIC IDENTITIES Math 12 Plane and Spherical Trigonometry

2. OBJECTIVES At the end of the lesson the students are expected to: Review basic identities. Simplify a trigonometric expression using identities. Verify a trigonometric identity. Apply the sum and difference identities. Apply the double-angle and half-angle identities. Apply the product-to-sum and sum-to-product identities.

3. TRIGONOMETRIC IDENTITIES A trigonometric identity is an equation involving trigonometric functions that hold for all values of the argument, typically chosen to be 𝜃.

4. BASIC TRIGONOMETRIC IDENTITIES Reciprocal Identities

5. Quotient (or Ratio) Identities

6. Pythagorean Identities Negative Arguments Identities

7. Guidelines for Verifying Trigonometric Identities The following suggestions help guide the way to verifying trigonometric identities: Start with the more complicated side of the equation. Combine all sums and differences of fractions (quotients) into a single fraction (quotient). Use basic trigonometric identities. Use algebraic techniques to manipulate one side of the other side of the equation is achieved. Sometimes it is helpful to convert all trigonometric functions into sines and cosines. Note: Trigonometric identities must be valid for all values of the independent variable for which the expressions in the equation are defined (domain of the equation).

8. Examples Verify the following identities: 2𝑠𝑒𝑐2𝜃=11−sin𝜃+11+sin𝜃 1+cos𝜃cos𝜃=sec𝜃+1 cos𝜃sec𝜃+tan𝜃=1−sin𝜃 1tan𝜃+cot𝜃=sin𝜃 cos𝜃 2𝑐𝑜𝑠2𝜃−1=𝑐𝑜𝑠4𝜃−𝑠𝑖𝑛4𝜃 tan𝑥+cot𝑥=csc𝑥sec𝑥

9. sin𝜃−tan𝜃2=𝑡𝑎𝑛2𝜃cos𝜃−12 cos𝜃1+sin𝜃+cos𝜃1−sin𝜃=2cos𝜃 sin𝜃+cos𝜃2tan𝜃=tan𝜃+2 𝑠𝑖𝑛2𝜃 1+sin𝜃+cos𝜃1−sin𝜃−cos𝜃=−2sin𝜃cos𝜃 𝑡𝑎𝑛2𝜃1−𝑐𝑜𝑠2𝜃+𝑠𝑖𝑛𝜃𝑠𝑒𝑐2𝜃−1=𝑐𝑜𝑠𝜃𝑠𝑒𝑐3𝜃+𝑐𝑜𝑡𝜃 𝑐𝑜𝑠2𝑥+1+sin𝑥𝑐𝑜𝑠2𝑥+3=1+sin𝑥2+sin𝑥

10. Sum and Difference Identities

11. Examples 1. Find the exact value for each trigonometric expression. a) sin𝜋12 b) sin105° c) tan165° 2. Write each expression as a single trigonometric function. a) sin2𝑥sin3𝑥+cos2𝑥cos3𝑥 b) cos𝜋−𝑥sin𝑥+sin𝜋−𝑥cos𝑥 c) tan49°+tan23°1−tan49°tan23° Find the exact value of a) sin𝛼−𝛽 and b) 𝑡𝑎𝑛𝛼+𝛽 if sin𝛼=−35 and sin𝛽=15; the terminal side of 𝛼 lies in Q3 and the terminal side of 𝛽 lies in Q1. Verify: sin𝑥−𝜋2=cos𝑥+𝜋2

12. Double-Angle Identities

13. Examples If cos𝑥=513 and sin𝑥<0, find a) tan2𝑥 b) cos2𝑥 If csc𝑥=−25 and 𝜋<𝑥<3𝜋2, find sin2𝑥. Simplify each expression and evaluate the resulting expression exactly, if possible. a) 2tan15°1−𝑡𝑎𝑛215° b) 𝑐𝑜𝑠2𝑥+2−𝑠𝑖𝑛2𝑥+2 Verify each identity. a) sin𝑥+cos𝑥2=1+sin2𝑥 b) sin3𝑥 =sin𝑥4𝑐𝑜𝑠2𝑥−1

14. Half-Angle Identities

15. Examples Use half-angle identities to find the exact values of the following: a) cos22.5° b) cot7𝜋8 c) sin75° 2. If csc𝑥=−3 and cos𝑥>0, find cos𝑥2. If cot𝑥=−245 and 𝜋2<𝑥<𝜋, find sin𝑥2. Verify the following: a) sin−𝑥=−2sin𝑥2cos𝑥2.

16. Product-to-Sum and Sum-to-Prroduct Identities

17. Product-to-Sum and Sum-to-Product Identities

18. Examples Write each expression as a sum or difference of sines and/or cosines. a) cos10𝑥sin5𝑥 c) sin3𝑥2sin5𝑥2 b) 4cos−𝑥cos2𝑥 d) sin−𝜋4𝑥cos−𝜋2𝑥 Write each expressions as a product of sines and/or cosines: a) cos2𝑥−cos4𝑥 c) sin0.4𝑥+sin0.6𝑥 b) sin𝑥2−sin5𝑥2 d) cos−𝜋4𝑥+cos𝜋6𝑥

19. Examples Simplify the following trigonometric expressions: a) cos3𝑥−cos𝑥sin3𝑥+sin𝑥 b) cos5𝑥+cos2𝑥sin5𝑥−sin2𝑥 Verify the following: a) 𝑠𝑖𝑛 𝐴+sin𝐵cos𝐴+cos𝐵=𝑡𝑎𝑛𝐴+𝐵2 b) sin𝐴−sin𝐵𝑐𝑜𝑠𝐴+𝑐𝑜𝑠𝐵=𝑡𝑎𝑛𝐴−𝐵2

20. References Algebra and Trigonometry by Cynthia Young Trigonometry by Jerome Hayden and Bettye Hall Trigonometry by Academe/Scott, Foresman Plane and Spherical Trigonometry by Paul Rider