Solutions to Try Its
1. [latex]\cos \left(t\right)=-\frac{\sqrt{2}}{2},\sin \left(t\right)=\frac{\sqrt{2}}{2}[/latex]
2. [latex]\cos \left(\pi \right)=-1[/latex], [latex]\sin \left(\pi \right)=0[/latex]
3. [latex]\sin \left(t\right)=-\frac{7}{25}[/latex]
4. approximately 0.866025403
5. [latex]\frac{\pi }{3}[/latex]
6. a. [latex]\text{cos}\left(315^\circ \right)=\frac{\sqrt{2}}{2},\text{sin}\left(315^\circ \right)=\frac{-\sqrt{2}}{2}[/latex]
b. [latex]\cos \left(-\frac{\pi }{6}\right)=\frac{\sqrt{3}}{2},\sin \left(-\frac{\pi }{6}\right)=-\frac{1}{2}[/latex]
7. [latex]\left(\frac{1}{2},-\frac{\sqrt{3}}{2}\right)[/latex]
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, [latex]t[/latex], formed by the terminal side of the angle [latex]t[/latex] and the horizontal axis.
5. The sine values are equal.
7. I
9. IV
11. [latex]\frac{\sqrt{3}}{2}[/latex]
13. [latex]\frac{1}{2}[/latex]
15. [latex]\frac{\sqrt{2}}{2}[/latex]
17. 0
19. −1
21. [latex]\frac{\sqrt{3}}{2}[/latex]
23. [latex]60^\circ [/latex]
25. [latex]80^\circ [/latex]
27. [latex]45^\circ [/latex]
29. [latex]\frac{\pi }{3}[/latex]
31. [latex]\frac{\pi }{3}[/latex]
33. [latex]\frac{\pi }{8}[/latex]
35. [latex]60^\circ [/latex], Quadrant IV, [latex]\text{sin}\left(300^\circ \right)=-\frac{\sqrt{3}}{2},\cos \left(300^\circ \right)=\frac{1}{2}[/latex]
37. [latex]45^\circ [/latex], Quadrant II, [latex]\text{sin}\left(135^\circ \right)=\frac{\sqrt{2}}{2}[/latex], [latex]\cos \left(135^\circ \right)=-\frac{\sqrt{2}}{2}[/latex]
39. [latex]60^\circ [/latex], Quadrant II, [latex]\text{sin}\left(120^\circ \right)=\frac{\sqrt{3}}{2}[/latex], [latex]\cos \left(120^\circ \right)=-\frac{1}{2}[/latex]
41. [latex]30^\circ [/latex], Quadrant II, [latex]\text{sin}\left(150^\circ \right)=\frac{1}{2}[/latex], [latex]\cos \left(150^\circ \right)=-\frac{\sqrt{3}}{2}[/latex]
43. [latex]\frac{\pi }{6}[/latex], Quadrant III, [latex]\text{sin}\left(\frac{7\pi }{6}\right)=-\frac{1}{2}[/latex], [latex]\text{cos}\left(\frac{7\pi }{6}\right)=-\frac{\sqrt{3}}{2}[/latex]
45. [latex]\frac{\pi }{4}[/latex], Quadrant II, [latex]\text{sin}\left(\frac{3\pi }{4}\right)=\frac{\sqrt{2}}{2}[/latex], [latex]\cos \left(\frac{4\pi }{3}\right)=-\frac{\sqrt[]{2}}{2}[/latex]
47. [latex]\frac{\pi }{3}[/latex], Quadrant II, [latex]\text{sin}\left(\frac{2\pi }{3}\right)=\frac{\sqrt{3}}{2}[/latex], [latex]\cos \left(\frac{2\pi }{3}\right)=-\frac{1}{2}[/latex]
49. [latex]\frac{\pi }{4}[/latex], Quadrant IV, [latex]\text{sin}\left(\frac{7\pi }{4}\right)=-\frac{\sqrt{2}}{2}[/latex], [latex]\text{cos}\left(\frac{7\pi }{4}\right)=\frac{\sqrt{2}}{2}[/latex]
51. [latex]\frac{\sqrt{77}}{9}[/latex]
53. [latex]-\frac{\sqrt{15}}{4}[/latex]
55. [latex]\left(-10,10\sqrt{3}\right)[/latex]
57. [latex]\left(-2.778,15.757\right)[/latex]
59. [latex]\left[-1,1\right][/latex]
61. [latex]\sin t=\frac{1}{2},\cos t=-\frac{\sqrt{3}}{2}[/latex]
63. [latex]\sin t=-\frac{\sqrt{2}}{2},\cos t=-\frac{\sqrt{2}}{2}[/latex]
65. [latex]\sin t=\frac{\sqrt{3}}{2},\cos t=-\frac{1}{2}[/latex]
67. [latex]\sin t=-\frac{\sqrt{2}}{2},\cos t=\frac{\sqrt{2}}{2}[/latex]
69. [latex]\sin t=0,\cos t=-1[/latex]
71. [latex]\sin t=-0.596,\cos t=0.803[/latex]
73. [latex]\sin t=\frac{1}{2},\cos t=\frac{\sqrt{3}}{2}[/latex]
75. [latex]\sin t=-\frac{1}{2},\cos t=\frac{\sqrt{3}}{2}[/latex]
77. [latex]\sin t=0.761,\cos t=-0.649[/latex]
79. [latex]\sin t=1,\cos t=0[/latex]
81. −0.1736
83. 0.9511
85. −0.7071
87. −0.1392
89. −0.7660
91. [latex]\frac{\sqrt{2}}{4}[/latex]
93. [latex]-\frac{\sqrt{6}}{4}[/latex]
95. [latex]\frac{\sqrt{2}}{4}[/latex]
97. [latex]\frac{\sqrt{2}}{4}[/latex]
99. 0
101. [latex]\left(0,-1\right)[/latex]
103. 37.5 seconds, 97.5 seconds, 157.5 seconds, 217.5 seconds, 277.5 seconds, 337.5 seconds