Solutions to Try Its

1. $\cos \left(t\right)=-\frac{\sqrt{2}}{2},\sin \left(t\right)=\frac{\sqrt{2}}{2}$

2. $\cos \left(\pi \right)=-1$, $\sin \left(\pi \right)=0$

3. $\sin \left(t\right)=-\frac{7}{25}$

4. approximately 0.866025403

5. $\frac{\pi }{3}$

6. a. $\text{cos}\left(315^\circ \right)=\frac{\sqrt{2}}{2},\text{sin}\left(315^\circ \right)=\frac{-\sqrt{2}}{2}$
b. $\cos \left(-\frac{\pi }{6}\right)=\frac{\sqrt{3}}{2},\sin \left(-\frac{\pi }{6}\right)=-\frac{1}{2}$

7. $\left(\frac{1}{2},-\frac{\sqrt{3}}{2}\right)$

Solutions to Odd-Numbered Exercises

1. The unit circle is a circle of radius 1 centered at the origin.

3. Coterminal angles are angles that share the same terminal side. A reference angle is the size of the smallest acute angle, $t$, formed by the terminal side of the angle $t$ and the horizontal axis.

5. The sine values are equal.

7. I

9. IV

11. $\frac{\sqrt{3}}{2}$

13. $\frac{1}{2}$

15. $\frac{\sqrt{2}}{2}$

17. 0

19. −1

21. $\frac{\sqrt{3}}{2}$

23. $60^\circ$

25. $80^\circ$

27. $45^\circ$

29. $\frac{\pi }{3}$

31. $\frac{\pi }{3}$

33. $\frac{\pi }{8}$

35. $60^\circ$, Quadrant IV, $\text{sin}\left(300^\circ \right)=-\frac{\sqrt{3}}{2},\cos \left(300^\circ \right)=\frac{1}{2}$

37. $45^\circ$, Quadrant II, $\text{sin}\left(135^\circ \right)=\frac{\sqrt{2}}{2}$, $\cos \left(135^\circ \right)=-\frac{\sqrt{2}}{2}$

39. $60^\circ$, Quadrant II, $\text{sin}\left(120^\circ \right)=\frac{\sqrt{3}}{2}$, $\cos \left(120^\circ \right)=-\frac{1}{2}$

41. $30^\circ$, Quadrant II, $\text{sin}\left(150^\circ \right)=\frac{1}{2}$, $\cos \left(150^\circ \right)=-\frac{\sqrt{3}}{2}$

43. $\frac{\pi }{6}$, Quadrant III, $\text{sin}\left(\frac{7\pi }{6}\right)=-\frac{1}{2}$, $\text{cos}\left(\frac{7\pi }{6}\right)=-\frac{\sqrt{3}}{2}$

45. $\frac{\pi }{4}$, Quadrant II, $\text{sin}\left(\frac{3\pi }{4}\right)=\frac{\sqrt{2}}{2}$, $\cos \left(\frac{4\pi }{3}\right)=-\frac{\sqrt[]{2}}{2}$

47. $\frac{\pi }{3}$, Quadrant II, $\text{sin}\left(\frac{2\pi }{3}\right)=\frac{\sqrt{3}}{2}$, $\cos \left(\frac{2\pi }{3}\right)=-\frac{1}{2}$

49. $\frac{\pi }{4}$, Quadrant IV, $\text{sin}\left(\frac{7\pi }{4}\right)=-\frac{\sqrt{2}}{2}$, $\text{cos}\left(\frac{7\pi }{4}\right)=\frac{\sqrt{2}}{2}$

51. $\frac{\sqrt{77}}{9}$

53. $-\frac{\sqrt{15}}{4}$

55. $\left(-10,10\sqrt{3}\right)$

57. $\left(-2.778,15.757\right)$

59. $\left[-1,1\right]$

61. $\sin t=\frac{1}{2},\cos t=-\frac{\sqrt{3}}{2}$

63. $\sin t=-\frac{\sqrt{2}}{2},\cos t=-\frac{\sqrt{2}}{2}$

65. $\sin t=\frac{\sqrt{3}}{2},\cos t=-\frac{1}{2}$

67. $\sin t=-\frac{\sqrt{2}}{2},\cos t=\frac{\sqrt{2}}{2}$

69. $\sin t=0,\cos t=-1$

71. $\sin t=-0.596,\cos t=0.803$

73. $\sin t=\frac{1}{2},\cos t=\frac{\sqrt{3}}{2}$

75. $\sin t=-\frac{1}{2},\cos t=\frac{\sqrt{3}}{2}$

77. $\sin t=0.761,\cos t=-0.649$

79. $\sin t=1,\cos t=0$

81. −0.1736

83. 0.9511

85. −0.7071

87. −0.1392

89. −0.7660

91. $\frac{\sqrt{2}}{4}$

93. $-\frac{\sqrt{6}}{4}$

95. $\frac{\sqrt{2}}{4}$

97. $\frac{\sqrt{2}}{4}$

99. 0

101. $\left(0,-1\right)$

103. 37.5 seconds, 97.5 seconds, 157.5 seconds, 217.5 seconds, 277.5 seconds, 337.5 seconds