Solutions for Unit Circle: Sine and Cosine Angles

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