Solutions for the Other Trigonometric Functions

Solutions to Try Its

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

2. [latex]\begin{array}{l}\sin \frac{\pi }{3}=\frac{\sqrt{3}}{2}\\ \cos \frac{\pi }{3}=\frac{1}{2}\\ \tan \frac{\pi }{3}=\sqrt{3}\\ \sec \frac{\pi }{3}=2\\ \csc \frac{\pi }{3}=\frac{2\sqrt{3}}{3}\\ \cot \frac{\pi }{3}=\frac{\sqrt{3}}{3}\end{array}[/latex]

3. [latex]\sin \left(\frac{-7\pi }{4}\right)=\frac{\sqrt{2}}{2},\cos \left(\frac{-7\pi }{4}\right)=\frac{\sqrt{2}}{2},\tan \left(\frac{-7\pi }{4}\right)=1[/latex],

[latex]\sec \left(\frac{-7\pi }{4}\right)=\sqrt{2},\csc \left(\frac{-7\pi }{4}\right)=\sqrt{2},\cot \left(\frac{-7\pi }{4}\right)=1[/latex]

4. [latex]-\sqrt{3}[/latex]

5. [latex]-2[/latex]

6. [latex]\sin t[/latex]

7. [latex]\cos t=-\frac{8}{17},\sin t=\frac{15}{17},\tan t=-\frac{15}{8}[/latex]

[latex]\csc t=\frac{17}{15},\cot t=-\frac{8}{15}[/latex]

8. [latex]\begin{array}{l}\sin t=-1,\cos t=0,\tan t=\text{Undefined}\\ \sec t=\text{\hspace{0.17em}Undefined,}\csc t=-1,\cot t=0\end{array}[/latex]

9. [latex]\sec t=\sqrt{2},\csc t=\sqrt{2},\tan t=1,\cot t=1[/latex]

10. [latex]\approx -2.414[/latex]

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