1. Splash Screen

2. Five-Minute Check (over Lesson 5-4) Then/Now
Key Concept: Double-Angle Identities and Proof: Double-Angle Identity for Sine
Example 1: Evaluate Expressions Involving Double Angles
Example 2: Solve an Equation Using a Double-Angle Identity
Key Concept: Power-Reducing Identities and Proof: Power-Reducing Identity for Sine
Example 3: Use an Identity to Reduce a Power
Example 4: Solve an Equation Using a Power-Reducing Identity
Key Concept: Half-Angle Identities and Proof: Half-Angle Identity for Cosine
Example 5: Evaluate an Expression Involving a Half Angle
Example 6: Solve an Equation Using a Half-Angle Identity
Key Concept: Product-to-Sum Identities and Proof: Product-to-Sum Identity for sin α cos β
Example 7: Use an Identity to Write a Product as a Sum or Difference
Key Concept: Sum-to-Product Identities and Proof: Sum-to-Product Identity for sin α + sin β
Example 8: Use a Product-to-Sum or Sum-to-Product Identity
Example 9: Solve an Equation Using a Sun-to-Product Identity
Lesson Menu

3. Find the exact value of sin 75°.B. C. D.
5–Minute Check 1

4. Find the exact value of A. B. C. D.
5–Minute Check 2

5. Find the exact value of A. B. C. D.
5–Minute Check 3

6. Simplify
A. tan( + 19°)
B. tan19
C. tan(19 )° D.
5–Minute Check 4

7. Find the solution to = 1 in the interval [0, 2).B. C. D.
5–Minute Check 5

8. 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.
Then/Now

9. Key Concept 1

10. If on the interval , find sin 2θ, cos 2θ, and tan 2θ.
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.
Example 1

11. sin 2θ = 2sin θ cos θ cos 2θ = 2cos2θ – 1
Evaluate Expressions Involving Double Angles
The x-coordinate of this point is therefore or Using this point, we find that cos θ = and tan θ =
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 θ cos 2θ = 2cos2θ – 1
Example 1

12. Evaluate Expressions Involving Double Angles
Method 1
Example 1

13. Evaluate Expressions Involving Double Angles
Method 2
Example 1

14. Evaluate Expressions Involving Double Angles
Answer:
Example 1

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

16. Solve cos 2θ – cos θ = 2 on the interval [0, 2π).
Solve an Equation Using a Double-Angle Identity
Solve cos 2θ – cos θ = 2 on the interval [0, 2π).
cos 2θ – cos θ = 2 Original equation
2 cos2 θ – 1 – cos θ – 2 = 0 Cosine Double-Angle Identity
2 cos2 θ – cos θ – 3 = 0 Simplify.
(2 cos θ – 3)( cos θ + 1) = 0 Factor.
2 cos θ – 3 = 0 or cos θ + 1 = 0 Zero Product Property
cos θ = or cos θ = –1 Solve for cos θ.
θ = π Solve for 
Example 2

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

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

19. Key Concept 3

20. Cosine Power-Reducing Identity
Use an Identity to Reduce a Power
Rewrite csc4 θ in terms of cosines of multiple angles with no power greater than 1.
csc4 θ = (csc2 θ)2
(csc2 θ)2 = csc4 θ
Reciprocal Identity
Pythagorean Identity
Cosine Power-Reducing Identity
Example 3

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

22. Common denominator Simplify. So, csc4 θ = . Answer:
Use an Identity to Reduce a Power
Common denominator
Simplify.
So, csc4 θ =
Answer:
Example 3

23. Rewrite tan4 x in terms of cosines of multiple angles with no power greater than 1.B. C. D.
Example 3

24. Solve sin2 θ + cos 2θ – cos θ = 0.
Solve an Equation Using a Power-Reducing Identity
Solve sin2 θ + cos 2θ – cos θ = 0.
Solve Algebraically
sin2 θ + cos 2θ – cos θ = 0 Original equation
Sine Power-Reducing Identity
Multiply each side by 2.
Add like terms.
Double-Angle Identity
Simplify.
Example 4

25. 2cos θ = 0 cos θ – 1 = 0 Zero Product Property
Solve an Equation Using a Power-Reducing Identity
Factor.
2cos θ = 0 cos θ – 1 = 0 Zero Product Property
cos  = 0 cos  = 1 Solve for cos .
 =  = 0 Solve for θ on [0, 2π).
The graph of y = sin2 θ + cos 2θ – cos θ has a period of 2, so the solutions are
Example 4

26. Solve an Equation Using a Power-Reducing Identity
Support Graphically
The graph of y = sin2 θ + cos 2θ – cos θ has zeros at on the interval [0, 2π). 
Answer:
Example 4

27. Solve cos 2x + 2cos2 x = 0. A. B. C. D.
Example 4

28. Key Concept 5

29. Find the exact value of sin 22.5°.
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)
Example 5

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

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

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

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

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

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

36. Key Concept 7

37. Rewrite cos 6x cos 3x as a sum or difference.
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:
Example 7

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

39. Key Concept 8

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

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

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

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

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

45. 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.
Example 9

46. The graph of y = sin 8x – sin 2x has zeros at on the interval . 
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 . 
Example 9

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

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

49. End of the Lesson