Angles

3. Whether the angle is positive or negative determines the direction. A positive angle is drawn in the counterclockwise direction, and a negative angle is drawn in the clockwise direction.

5. Linear speed is a measurement found by calculating distance of an arc compared to time. Angular speed is a measurement found by calculating the angle of an arc compared to time.

7.

9.

11.

13.

15.

17. 240°

19. [latex]\frac{4\pi }{3}[/latex]

21. [latex]\frac{2\pi }{3}[/latex]

22. [latex]s=\frac{7\pi}{3}\approx 7.33[/latex]

23. [latex]\frac{7\pi }{2}\approx 11.00{\text{ in}}^{2}[/latex]

24. [latex]s=\frac{9\pi}{5}\approx 5.65[/latex]

25. [latex]\frac{81\pi }{20}\approx 12.72{\text{ cm}}^{2}[/latex]

27. 20°

29. 60°

31. −75°

33. [latex]\frac{\pi }{2}[/latex] radians

35. [latex]-3\pi[/latex] radians

37. [latex]\pi[/latex] radians

39. [latex]\frac{5\pi }{6}[/latex] radians

41. [latex]\frac{5.02\pi }{3}\approx 5.26[/latex] miles

45. [latex]\frac{21\pi }{10}\approx 6.60[/latex] meters

47. 104.7198 cm2

49. 0.7697 in2

51. 250°

53. 320°

55. [latex]\frac{4\pi }{3}[/latex]

57. [latex]\frac{8\pi }{9}[/latex]

59. 1320 rad 210.085 RPM

61. 7 in./s, 4.77 RPM, 28.65 deg/s

65. [latex]5.76[/latex] miles

67. [latex]120^\circ[/latex]

73. 11.5 inches

74. [latex]52^\circ 22' 30''[/latex]

75. [latex]-73^\circ 15' 0''[/latex]

76. 7[latex]140^\circ 0' 28.8''[/latex]

77. [latex]38.2567^\circ[/latex]

78. [latex]72.0100^\circ[/latex]

79. [latex]-19.5000^\circ[/latex]

80. Complement: [latex]62^\circ[/latex], Supplement: [latex]152^\circ[/latex]

81. Complement: [latex]47^\circ 41' 24''[/latex], Supplement: [latex]137^\circ 41' 24''[/latex]

82. Complement: No complement exists, Supplement: [latex]68^\circ[/latex]

83. Complement: [latex]\frac{3\pi}{8}[/latex], Supplement: [latex]\frac{7\pi}{8}[/latex]

Unit Circle: Sine and Cosine Functions

2. On the unit circle, the coordinates of a point corresponding to an angle t are [latex](\cos t,\ \sin t)[/latex]. The x-coordinate represents [latex]\cos t[/latex] and the y-coordinate represents [latex]\sin t[/latex].

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

21. [latex]\frac{\sqrt{3}}{2}[/latex]

23. [latex]60^\circ[/latex]

27. [latex]45^\circ[/latex]

29. [latex]\frac{\pi }{3}[/latex]

31. [latex]\frac{\pi }{3}[/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]

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]

51. [latex]\frac{\sqrt{77}}{9}[/latex]

53. [latex]-\frac{\sqrt{15}}{4}[/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]

73. [latex]\sin t=\frac{1}{2},\cos t=\frac{\sqrt{3}}{2}[/latex]

81. −0.1736

83. 0.9511

87. −0.1392

89. −0.7660

Other Trigonometric Functions

7. [latex]\frac{2\sqrt{3}}{3}[/latex]

13. 1

19. [latex]-\frac{2\sqrt{3}}{3}[/latex]

21. [latex]\sqrt{3}[/latex]

33. [latex]\frac{\sqrt{3}}{3}[/latex]

35. −2

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

53. –0.228

55. –2.414

57. 1.414

61. 1.556

73. 13.77 hours, period: [latex]1000\pi[/latex]

75. 7.73 inches