1. Splash Screen

2. Five-Minute Check (over Lesson 5-3) Then/Now New Vocabulary
Key Concept: Sum and Difference Identities
Example 1: Evaluate a Trigonometric Expression
Example 2: Real-World Example: Use a Sum or Difference Identity
Example 3: Rewrite as a Single Trigonometric Expression
Example 4: Write as an Algebraic Expression
Example 5: Verify Cofunction Identities
Example 6: Verify Reduction Identities
Example 7: Solve a Trigonometric Equation
Lesson Menu

3. Solve for all values of x.B. C. D.
5–Minute Check 1

4. Find all solutions of 2cos2 x + 3cos x + 1 = 0 in the interval [0, 2π).B. C. D.
5–Minute Check 2

5. Find all solutions of 4 cos2 x = 5 – 4 sin x in the interval [0, 2π).B. C. D.
5–Minute Check 3

6. Find all solutions of sin x + cos x = 1 in the interval [0, 2π).B. C. D.
5–Minute Check 4

7. Solve 4 sin θ – 1 = 2 sin θ for all values of θ.B. C. D.
5–Minute Check 5

8. Use sum and difference identities to evaluate trigonometric functions.
You found values of trigonometric functions using the unit circle. (Lesson 4-3)
Use sum and difference identities to evaluate trigonometric functions.
Use sum and difference identities to solve trigonometric equations.
Then/Now

9. reduction identity
Vocabulary

10. Key Concept 1

11. A. Find the exact value of cos 75°.
Evaluate a Trigonometric Expression
A. Find the exact value of cos 75°.
30° + 45° = 75°
Cosine Sum Identity
Example 1

12. Multiply. Combine the fractions. Answer:
Evaluate a Trigonometric Expression
Multiply.
Combine the fractions.
Answer:
Example 1

13. B. Find the exact value of tan .
Evaluate a Trigonometric Expression
B. Find the exact value of tan
Write as the sum or difference of angle measures with tangents that you know.
Example 1

14. Rationalize the denominator.
Evaluate a Trigonometric Expression
Tangent Sum Identity
Simplify.
Rationalize the denominator.
Example 1

15. Multiply. Simplify. Simplify. Answer:
Evaluate a Trigonometric Expression
Multiply.
Simplify.
Simplify.
Answer:
Example 1

16. Find the exact value of tan 15°.B. C. D.
Example 1

17. Rewrite the formula in terms of the sum of two angle measures.
Use a Sum or Difference Identity
A. ELECTRICITY An alternating current i in amperes in a certain circuit can be found after t seconds using i = 4 sin 255t, where 255 is a degree measure. Rewrite the formula in terms of the sum of two angle measures.
Rewrite the formula in terms of the sum of two angle measures.
i = 4 sin 255t Original equation
= 4 sin (210t + 45t) 255t = 210t + 45t
The formula is i = 4 sin (210t + 45t).
Answer: i = 4 sin (210t + 45t)
Example 2

18. Use a sum identity to find the exact current after 1 second.
Use a Sum or Difference Identity
B. ELECTRICITY An alternating current i in amperes in a certain circuit can be found after t seconds using i = 4 sin 255t. Use a sum identity to find the exact current after 1 second.
Use a sum identity to find the exact current after 1 second.
i = 4 sin (210t + 45t) Rewritten equation
= 4 sin ( ) t = 1
= 4[sin(210)cos(45) + cos(210)sin(45)] Sine Sum Identity
Example 2

19. The exact current after 1 second is amperes.
Use a Sum or Difference Identity
Substitute.
Multiply.
Simplify.
The exact current after 1 second is amperes.
Answer: amperes
Example 2

20. A. ELECTRICITY An alternating current i in amperes in a certain circuit can be found after t seconds using i = 4 sin 210t, where 210 is a degree measure. Rewrite the formula in terms of the sum of two angle measures.
A. i = 4 sin (240t – 30t)
B. i = 4 sin ( )
C. i = 4 sin [7(30t)]
D. i = 4 sin (150t + 60t)
Example 2

21. B. ELECTRICITY An alternating current i in amperes in a certain circuit can be found after t seconds using i = 4 sin 210t, where 210 is a degree measure. Use a sum identity to find the exact current after 1 second.
A. –1 ampere
B. –2 amperes
C. 1 ampere
D. 2 amperes
Example 2

22. A. Find the exact value of
Rewrite as a Single Trigonometric Expression
A. Find the exact value of
Tangent Difference Identity
Simplify.
Substitute.
Answer:
Example 3

23. Rewrite as fractions with a common denominator.
Rewrite as a Single Trigonometric Expression
B. Simplify
Sine Sum Identity
Rewrite as fractions with a common denominator.
Simplify.
Answer:
Example 3

24. Find the exact value of
A. 0 B. C.
D. 1
Example 3

25. Applying the Cosine Sum Identity, we find that
Write as an Algebraic Expression
Write as an algebraic expression of x that does not involve trigonometric functions.
Applying the Cosine Sum Identity, we find that
Example 4

26. Write as an Algebraic Expression
If we let α = and β = arccos x, then sin α = and cos β = x. Sketch one right triangle with an acute angle α and another with an acute angle β. Label the sides such that sin α = and cos β = x. Then use the Pythagorean Theorem to express the length of each third side.
Example 4

27. Write as an Algebraic Expression
Using these triangles, we find that = cos α or , cos (arccos x) = cos β or x, = sin α or , and sin (arccos x) = sin β or
Example 4

28. Now apply substitution and simplify.
Write as an Algebraic Expression
Now apply substitution and simplify.
Example 4

29. Write as an Algebraic Expression
Answer:
Example 4

30. Write sin(arccos 2x + arcsin x) as an algebraic expression of x does not involve trigonometric functions. A. B. C. D.
Example 4

31. cos (–θ) = cos (0 – θ) Rewrite as a difference.
Verify Cofunction Identities
Verify cos (–θ) = cos θ.
cos (–θ) = cos (0 – θ) Rewrite as a difference.
= cos 0 cos θ + sin 0 sin θ Cosine Difference Identity
= 1 cos θ + 0 sin θ cos 0 = 1 and sin 0 = 0
= cos θ  Multiply.
Answer: cos (–θ) = cos (0 – θ) = cos 0 cos θ + sin 0 sin θ = 1 cos θ + 0 sin θ = cos θ
Example 5

32. Verify tan (– ) = –tan . A. B. C. D.
Example 5

33. Cosine Difference Identity
Verify Reduction Identities
A. Verify
Cosine Difference Identity
Simplify. 
Example 6

34. Verify Reduction Identities
Answer:
Example 6

35. B. Verify tan (x – 360°) = tan x.
Verify Reduction Identities
B. Verify tan (x – 360°) = tan x.
Tangent Difference Identity
tan 360° = 0
Simplify. 
Answer:
Example 6

36. Verify the cofunction identity .A. B. C. D.
Example 6

37. Find the solutions of on the interval [ 0, 2).
Solve a Trigonometric Equation
Find the solutions of on the interval [ 0, 2).
Original equation
Sine Sum Identity and Sine Difference Identity
Example 7

38. Simplify. Divide each side by 2. Substitute. Solve for cos x.
Solve a Trigonometric Equation
Simplify.
Divide each side by 2.
Substitute.
Solve for cos x.
Example 7

39. On the interval [0, 2π), cos x = 0 when x =
Solve a Trigonometric Equation
On the interval [0, 2π), cos x = 0 when x =
Answer:
CHECK The graph of has zeros at on the interval [ 0, 2π). 
Example 7

40. Find the solutions of on the interval [0, 2π).B. C. D.
Example 7

41. End of the Lesson