Solutions to Odd-Numbered Exercises

1. Yes, when the reference angle is [latex]\frac{\pi }{4}[/latex] and the terminal side of the angle is in quadrants I and III. Thus, at [latex]x=\frac{\pi }{4},\frac{5\pi }{4}[/latex], the sine and cosine values are equal.

3. Substitute the sine of the angle in for [latex]y[/latex] in the Pythagorean Theorem [latex]{x}^{2}+{y}^{2}=1[/latex]. Solve for [latex]x[/latex] and take the negative solution.

5. The outputs of tangent and cotangent will repeat every [latex]\pi [/latex] units.

7. [latex]\frac{2\sqrt{3}}{3}[/latex]

9. [latex]\sqrt{3}[/latex]

11. [latex]\sqrt{2}[/latex]

13. 1

15. 2

17. [latex]\frac{\sqrt{3}}{3}[/latex]

19. [latex]-\frac{2\sqrt{3}}{3}[/latex]

21. [latex]\sqrt{3}[/latex]

23. [latex]-\sqrt{2}[/latex]

25. −1

27. −2

29. [latex]-\frac{\sqrt{3}}{3}[/latex]

31. 2

33. [latex]\frac{\sqrt{3}}{3}[/latex]

35. −2

37. −1

39. If [latex]\sin t=-\frac{2\sqrt{2}}{3},\sec t=-3,\csc t=-\frac{3\sqrt{2}}{4},\tan t=2\sqrt{2},\cot t=\frac{\sqrt{2}}{4}[/latex]

41. [latex]\sec t=2,\csc t=\frac{2\sqrt{3}}{3},\tan t=\sqrt{3},\cot t=\frac{\sqrt{3}}{3}[/latex]

43. [latex]-\frac{\sqrt{2}}{2}[/latex]

45. 3.1

47. 1.4

49. [latex]\sin t=\frac{\sqrt{2}}{2},\cos t=\frac{\sqrt{2}}{2},\tan t=1,\cot t=1,\sec t=\sqrt{2},\csc t=\sqrt{2}[/latex]

51. [latex]\sin t=-\frac{\sqrt{3}}{2},\cos t=-\frac{1}{2},\tan t=\sqrt{3},\cot t=\frac{\sqrt{3}}{3},\sec t=-2,\csc t=-\frac{2\sqrt{3}}{3}[/latex]

53. –0.228

55. –2.414

57. 1.414

59. 1.540

61. 1.556

63. [latex]\sin \left(t\right)\approx 0.79[/latex]

65. [latex]\csc t\approx 1.16[/latex]

67. even

69. even

71. [latex]\frac{\sin t}{\cos t}=\tan t[/latex]

73. 13.77 hours, period: [latex]1000\pi [/latex]

75. 7.73 inches