1. Explain the basis for the cofunction identities and when they apply.
2. Is there only one way to evaluate cos(5π4)?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)f(x)=sin(x) and g(x)=cos(x)g(x)=cos(x). (Hint: 0−x=−x0−x=−x )
For the following exercises, find the exact value.
4. cos(7π12)cos(7π12)
5. cos(π12)cos(π12)
6. sin(5π12)sin(5π12)
7. sin(11π12)sin(11π12)
8. tan(−π12)tan(−π12)
9. tan(19π12)tan(19π12)
For the following exercises, rewrite in terms of sinxsinx and cosxcosx.
10. sin(x+11π6)sin(x+11π6)
11. sin(x−3π4)sin(x−3π4)
12. cos(x−5π6)cos(x−5π6)
13. cos(x+2π3)cos(x+2π3)
For the following exercises, simplify the given expression.
14. csc(π2−t)csc(π2−t)
15. sec(π2−θ)sec(π2−θ)
16. cot(π2−x)cot(π2−x)
17. tan(π2−x)tan(π2−x)
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(a−b).
21. Given that sina=45, and cosb=13, with a and b both in the interval [0,π2), find sin(a−b) and cos(a+b).
For the following exercises, find the exact value of each expression.
22. sin(cos−1(0)−cos−1(12))
23. cos(cos−1(√22)+sin−1(√32))
24. tan(sin−1(12)−cos−1(12))
For the following exercises, simplify the expression, and then graph both expressions as functions to verify the graphs are identical.
25. cos(π2−x)
26. sin(π−x)
27. tan(π3+x)
28. sin(π3+x)
29. tan(π4−x)
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)cosx−cos(5x)sinx
37. f(x)=sin(2x),g(x)=2sinxcosx
38. f(θ)=cos(2θ),g(θ)=cos2θ−sin2θ
39. f(θ)=tan(2θ),g(θ)=tanθ1+tan2θ
40. f(x)=sin(3x)sinx,g(x)=sin2(2x)cos2x−cos2(2x)sin2x
41. f(x)=tan(−x),g(x)=tanx−tan(2x)1−tanxtan(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+11−tanx
48. tan(a+b)tan(a−b)=sinacosa+sinbcosbsinacosa−sinbcosb
49. cos(a+b)cosacosb=1−tanatanb
50. cos(x+y)cos(x−y)=cos2x−sin2y
51. cos(x+h)−cosxh=cosxcosh−1h−sinxsinhh
For the following exercises, prove or disprove the statements.
52. tan(u+v)=tanu+tanv1−tanutanv
53. tan(u−v)=tanu−tanv1+tanutanv
54. tan(x+y)1+tanxtanx=tanx+tany1−tan2xtan2y
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
Candela Citations
- Precalculus. Authored by: Jay Abramson, et al.. Provided by: OpenStax. Located at: http://cnx.org/contents/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1/Preface. License: CC BY: Attribution. License Terms: Download for free at: http://cnx.org/contents/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1/Preface