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