Problem Set 52: Sum and Difference Identities

1. Explain the basis for the cofunction identities and when they apply.

2. Is there only one way to evaluate cos(5π4)? Explain how to set up the solution in two different ways, and then compute to make sure they give the same answer.

3. Explain to someone who has forgotten the even-odd properties of sinusoidal functions how the addition and subtraction formulas can determine this characteristic for f(x)=sin(x) and g(x)=cos(x). (Hint: 0x=x )

For the following exercises, find the exact value.

4. cos(7π12)

5. cos(π12)

6. sin(5π12)

7. sin(11π12)

8. tan(π12)

9. tan(19π12)

For the following exercises, rewrite in terms of sinx and cosx.

10. sin(x+11π6)

11. sin(x3π4)

12. cos(x5π6)

13. cos(x+2π3)

For the following exercises, simplify the given expression.

14. csc(π2t)

15. sec(π2θ)

16. cot(π2x)

17. tan(π2x)

18. sin(2x)cos(5x)sin(5x)cos(2x)

19. tan(32x)tan(75x)1+tan(32x)tan(75x)

For the following exercises, find the requested information.

20. Given that sina=23 and cosb=14, with a and b both in the interval [π2,π), find sin(a+b) and cos(ab).

21. Given that sina=45, and cosb=13, with a and b both in the interval [0,π2), find sin(ab) and cos(a+b).

For the following exercises, find the exact value of each expression.

22. sin(cos1(0)cos1(12))

23. cos(cos1(22)+sin1(32))

24. tan(sin1(12)cos1(12))

For the following exercises, simplify the expression, and then graph both expressions as functions to verify the graphs are identical.

25. cos(π2x)

26. sin(πx)

27. tan(π3+x)

28. sin(π3+x)

29. tan(π4x)

30. cos(7π6+x)

31. sin(π4+x)

32. cos(5π4+x)

For the following exercises, use a graph to determine whether the functions are the same or different. If they are the same, show why. If they are different, replace the second function with one that is identical to the first. (Hint: think 2x=x+x.
)

33. f(x)=sin(4x)sin(3x)cosx,g(x)=sinxcos(3x)

34. f(x)=cos(4x)+sinxsin(3x),g(x)=cosxcos(3x)

35. f(x)=sin(3x)cos(6x),g(x)=sin(3x)cos(6x)

36. f(x)=sin(4x),g(x)=sin(5x)cosxcos(5x)sinx

37. f(x)=sin(2x),g(x)=2sinxcosx

38. f(θ)=cos(2θ),g(θ)=cos2θsin2θ

39. f(θ)=tan(2θ),g(θ)=tanθ1+tan2θ

40. f(x)=sin(3x)sinx,g(x)=sin2(2x)cos2xcos2(2x)sin2x

41. f(x)=tan(x),g(x)=tanxtan(2x)1tanxtan(2x)

For the following exercises, find the exact value algebraically, and then confirm the answer with a calculator to the fourth decimal point.

42. sin(75)

43. sin(195)

44. cos(165)

45. cos(345)

46. tan(15)

For the following exercises, prove the identities provided.

47. tan(x+π4)=tanx+11tanx

48. tan(a+b)tan(ab)=sinacosa+sinbcosbsinacosasinbcosb

49. cos(a+b)cosacosb=1tanatanb

50. cos(x+y)cos(xy)=cos2xsin2y

51. cos(x+h)cosxh=cosxcosh1hsinxsinhh

For the following exercises, prove or disprove the statements.

52. tan(u+v)=tanu+tanv1tanutanv

53. tan(uv)=tanutanv1+tanutanv

54. tan(x+y)1+tanxtanx=tanx+tany1tan2xtan2y

55. If α,β, and γ are angles in the same triangle, then prove or disprove sin(α+β)=sinγ.

57. If α,β, and y are angles in the same triangle, then prove or disprove tanα+tanβ+tanγ=tanαtanβtanγ