1. Then/Now You used sum and difference identities. (Lesson 5-4) Use double-angle, power-reducing, and half-angle identities to evaluate trigonometric expressions and solve trigonometric equations. Use product-to-sum identities to evaluate trigonometric expressions and solve trigonometric equations.

2. Key Concept 1

3. Example 1 Evaluate Expressions Involving Double Angles If on the interval, find sin 2θ, cos 2θ, and tan 2θ. Since on the interval, one point on the terminal side of θ has y-coordinate 3 and a distance of 4 units from the origin, as shown.

4. Example 1 Evaluate Expressions Involving Double Angles Use these values and the double-angle identities for sine and cosine to find sin 2θ and cos 2θ. Then find tan 2θ using either the tangent double-angle identity or the definition of tangent. sin 2θ = 2sin θ cos θcos 2θ = 2cos 2 θ – 1 The x-coordinate of this point is therefore or. Using this point, we find that cos θ = and tan θ =.

5. Example 1 Evaluate Expressions Involving Double Angles Method 1

6. Example 1 Evaluate Expressions Involving Double Angles Method 2

7. Example 1 Evaluate Expressions Involving Double Angles Answer:

8. Example 1 A. B. C. D. If on the interval, find, sin2 , cos 2 , and tan 2 .

9. Example 2 Solve an Equation Using a Double-Angle Identity Solve cos 2θ – cos θ = 2 on the interval [0, 2 π ). cos 2θ – cos θ= 2Original equation 2 cos 2 θ – 1 – cos θ – 2= 0Cosine Double-Angle Identity 2 cos 2 θ – cos θ – 3= 0Simplify. (2 cos θ – 3)( cos θ + 1)= 0Factor. 2 cos θ – 3 = 0 or cos θ + 1= 0Zero Product Property cos θ = or cos θ= –1Solve for cos θ. θ= πSolve for 

10. Example 2 Solve an Equation Using a Double-Angle Identity Answer: π Since cos θ = has no solution, the solution on the interval [0, 2 π ) is θ = π.

11. Example 2 Solve tan2  + tan  = 0 on the interval [0, 2 π). A. B. C. D.

12. Key Concept 3

13. Example 3 Use an Identity to Reduce a Power Rewrite csc 4 θ in terms of cosines of multiple angles with no power greater than 1. csc 4 θ = (csc 2 θ) 2 (csc 2 θ) 2 = csc 4 θ Reciprocal Identity Pythagorean Identity Cosine Power- Reducing Identity

14. Example 3 Use an Identity to Reduce a Power Common denominator Simplify. Cosine Power- Reducing Identity Square the fraction.

15. Example 3 Use an Identity to Reduce a Power Simplify. Common denominator So, csc 4 θ =. Answer:

16. Example 3 Rewrite tan 4 x in terms of cosines of multiple angles with no power greater than 1. A. B. C. D.

17. Example 4 Solve an Equation Using a Power-Reducing Identity Solve sin 2 θ + cos 2θ – cos θ = 0. Solve Algebraically sin 2 θ + cos 2θ – cos θ= 0Original equation Sine Power-Reducing Identity Add like terms. Double-Angle Identity Simplify. Multiply each side by 2.

18. Example 4 Solve an Equation Using a Power-Reducing Identity Factor. 2cos θ= 0cos θ – 1= 0Zero Product Property cos  = 0cos  = 1Solve for cos .  =  = 0Solve for θ on [0, 2 π ). The graph of y = sin 2 θ + cos 2θ – cos θ has a period of 2 , so the solutions are

19. Example 4 Solve an Equation Using a Power-Reducing Identity Support Graphically The graph of y = sin 2 θ + cos 2θ – cos θ has zeros at on the interval [0, 2 π ). Answer:

20. Example 4 Solve cos 2x + 2cos 2 x = 0. A. B. C. D.

21. Key Concept 5

22. Example 5 Evaluate an Expression Involving a Half Angle Find the exact value of sin 22.5°. Notice that 22.5° is half of 45°. Therefore, apply the half-angle identity for sine, noting that since 22.5° lies in Quadrant I, its sine is positive. Sine Half-Angle Identity (Quadrant I angle)

23. Example 5 Evaluate an Expression Involving a Half Angle Quotient Property of Square Roots Subtract and then divide. Answer:

24. Example 5 Evaluate an Expression Involving a Half Angle CHECK Use a calculator to support your assertion that sin 22.5° =. sin 22.5° = 0.3826834324 and = 0.3826834324

25. Example 5 Find the exact value of. A. B. C. D.

26. Example 6 Solve an Equation Using a Half-Angle Identity Solve on the interval [0, 2π). Subtract 1 – cos x from each side. Square each side. Sine and Cosine Half- Angle Identities Original equation Multiply each side by 2.1 – cos x = 1 + cos x

27. Example 6 Solve an Equation Using a Half-Angle Identity Solve for cos x. Answer: Solve for x. The solutions on the interval [0, 2 π ) are.

28. Example 6 Solve on the interval [0, 2 π). A. B. C. D.

29. Key Concept 7

30. Example 7 Use an Identity to Write a Product as a Sum or Difference Rewrite cos 6x cos 3x as a sum or difference. Product-to-Sum Identity Simplify. Distributive Property Answer:

31. Example 7 Rewrite sin 4x cos 2x as a sum or difference. A. B. C. D.

32. Key Concept 8

33. Example 8 Find the exact value of cos 255° + cos 195°. Use a Product-to-Sum or Sum-to-Product Identity Simplify. Sum-to-Product Identity

34. Example 8 Use a Product-to-Sum or Sum-to-Product Identity Simplify. The exact value of cos 255° + cos 195° is. Answer:

35. Example 8 Find the exact value of sin 255° + sin 195°. A. B. C. D.

36. Example 9 Solve an Equation Using a Sum-to-Product Identity Solve sin 8x – sin 2x = 0. Solve Algebraically sin 8x – sin 2x= 0Original equation Sine Sum-to-Product Identity Simplify. Set each factor equal to zero and find solutions on the interval [0, 2 π).

37. Example 9 Solve an Equation Using a Sum-to-Product Identity Divide each side by 2. First factor set equal to 0 2cos 5x = 0 Second factor set equal to 0 sin 3x = 0 Divide each solution by 3. Divide each solution by 5. Multiple angle solutions in [0, 2 π).

38. Example 9 Solve an Equation Using a Sum-to-Product Identity The period of y = cos 5x is and the period of y = sin 3x is, so the solutions are where n is an integer.

39. Example 9 Solve an Equation Using a Sum-to-Product Identity Support Graphically The graph of y = sin 8x – sin 2x has zeros at on the interval.

40. Example 9 Solve an Equation Using a Sum-to-Product Identity Answer:

41. Example 9 Solve sin 6x + sin 2x = 0. A. B. C. D.

42. End of the Lesson