Solutions to Try Its

1. $\sin t=-\frac{\sqrt{2}}{2},\cos t=\frac{\sqrt{2}}{2},\tan t=-1,\sec t=\sqrt{2},\csc t=-\sqrt{2},\cot t=-1$

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

3. $\sin \left(\frac{-7\pi }{4}\right)=\frac{\sqrt{2}}{2},\cos \left(\frac{-7\pi }{4}\right)=\frac{\sqrt{2}}{2},\tan \left(\frac{-7\pi }{4}\right)=1$,
$\sec \left(\frac{-7\pi }{4}\right)=\sqrt{2},\csc \left(\frac{-7\pi }{4}\right)=\sqrt{2},\cot \left(\frac{-7\pi }{4}\right)=1$

4. $-\sqrt{3}$

5. $-2$

6. $\sin t$

7. $\cos t=-\frac{8}{17},\sin t=\frac{15}{17},\tan t=-\frac{15}{8}$
$\csc t=\frac{17}{15},\cot t=-\frac{8}{15}$

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

9. $\sec t=\sqrt{2},\csc t=\sqrt{2},\tan t=1,\cot t=1$

10. $\approx -2.414$

Solutions to Odd-Numbered Exercises

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

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

5. The outputs of tangent and cotangent will repeat every $\pi$ units.

7. $\frac{2\sqrt{3}}{3}$

9. $\sqrt{3}$

11. $\sqrt{2}$

13. 1

15. 2

17. $\frac{\sqrt{3}}{3}$

19. $-\frac{2\sqrt{3}}{3}$

21. $\sqrt{3}$

23. $-\sqrt{2}$

25. −1

27. −2

29. $-\frac{\sqrt{3}}{3}$

31. 2

33. $\frac{\sqrt{3}}{3}$

35. −2

37. −1

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

41. $\sec t=2,\csc t=\frac{2\sqrt{3}}{3},\tan t=\sqrt{3},\cot t=\frac{\sqrt{3}}{3}$

43. $-\frac{\sqrt{2}}{2}$

45. 3.1

47. 1.4

49. $\sin t=\frac{\sqrt{2}}{2},\cos t=\frac{\sqrt{2}}{2},\tan t=1,\cot t=1,\sec t=\sqrt{2},\csc t=\sqrt{2}$

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

53. –0.228

55. –2.414

57. 1.414

59. 1.540

61. 1.556

63. $\sin \left(t\right)\approx 0.79$

65. $\csc t\approx 1.16$

67. even

69. even

71. $\frac{\sin t}{\cos t}=\tan t$

73. 13.77 hours, period: $1000\pi$

75. 7.73 inches