{"id":1370,"date":"2023-06-05T14:51:03","date_gmt":"2023-06-05T14:51:03","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/tulsacc-math1613\/chapter\/solutions-for-unit-circle-sine-and-cosine-angles\/"},"modified":"2023-06-05T14:51:03","modified_gmt":"2023-06-05T14:51:03","slug":"solutions-for-unit-circle-sine-and-cosine-angles","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/tulsacc-math1613\/chapter\/solutions-for-unit-circle-sine-and-cosine-angles\/","title":{"raw":"Solutions 45: Unit Circle: Sine and Cosine Angles","rendered":"Solutions 45: Unit Circle: Sine and Cosine Angles"},"content":{"raw":"\n

Solutions to Odd-Numbered Exercises<\/h2>\n1. The unit circle is a circle of radius 1 centered at the origin.\n\n3. 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.\n\n5. The sine values are equal.\n\n7. I\n\n9. IV\n\n11. [latex]\\frac{\\sqrt{3}}{2}[\/latex]\n\n13. [latex]\\frac{1}{2}[\/latex]\n\n15. [latex]\\frac{\\sqrt{2}}{2}[\/latex]\n\n17. 0\n\n19. \u22121\n\n21. [latex]\\frac{\\sqrt{3}}{2}[\/latex]\n\n23. [latex]60^\\circ [\/latex]\n\n25. [latex]80^\\circ [\/latex]\n\n27. [latex]45^\\circ [\/latex]\n\n29. [latex]\\frac{\\pi }{3}[\/latex]\n\n31. [latex]\\frac{\\pi }{3}[\/latex]\n\n33. [latex]\\frac{\\pi }{8}[\/latex]\n\n35. [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]\n\n37. [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]\n\n39. [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]\n\n41. [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]\n\n43. [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]\n\n45. [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]\n\n47. [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]\n\n49. [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]\n\n51. [latex]\\frac{\\sqrt{77}}{9}[\/latex]\n\n53. [latex]-\\frac{\\sqrt{15}}{4}[\/latex]\n\n55. [latex]\\left(-10,10\\sqrt{3}\\right)[\/latex]\n\n57. [latex]\\left(-2.778,15.757\\right)[\/latex]\n\n59. [latex]\\left[-1,1\\right][\/latex]\n\n61. [latex]\\sin t=\\frac{1}{2},\\cos t=-\\frac{\\sqrt{3}}{2}[\/latex]\n\n63. [latex]\\sin t=-\\frac{\\sqrt{2}}{2},\\cos t=-\\frac{\\sqrt{2}}{2}[\/latex]\n\n65. [latex]\\sin t=\\frac{\\sqrt{3}}{2},\\cos t=-\\frac{1}{2}[\/latex]\n\n67. [latex]\\sin t=-\\frac{\\sqrt{2}}{2},\\cos t=\\frac{\\sqrt{2}}{2}[\/latex]\n\n69. [latex]\\sin t=0,\\cos t=-1[\/latex]\n\n71. [latex]\\sin t=-0.596,\\cos t=0.803[\/latex]\n\n73. [latex]\\sin t=\\frac{1}{2},\\cos t=\\frac{\\sqrt{3}}{2}[\/latex]\n\n75. [latex]\\sin t=-\\frac{1}{2},\\cos t=\\frac{\\sqrt{3}}{2}[\/latex]\n\n77. [latex]\\sin t=0.761,\\cos t=-0.649[\/latex]\n\n79. [latex]\\sin t=1,\\cos t=0[\/latex]\n\n81. \u22120.1736\n\n83. 0.9511\n\n85. \u22120.7071\n\n87. \u22120.1392\n\n89. \u22120.7660\n\n91. [latex]\\frac{\\sqrt{2}}{4}[\/latex]\n\n93. [latex]-\\frac{\\sqrt{6}}{4}[\/latex]\n\n95. [latex]\\frac{\\sqrt{2}}{4}[\/latex]\n\n97. [latex]\\frac{\\sqrt{2}}{4}[\/latex]\n\n99. 0\n\n101. [latex]\\left(0,-1\\right)[\/latex]\n\n103. 37.5 seconds, 97.5 seconds, 157.5 seconds, 217.5 seconds, 277.5 seconds, 337.5 seconds\n","rendered":"

Solutions to Odd-Numbered Exercises<\/h2>\n

1. The unit circle is a circle of radius 1 centered at the origin.<\/p>\n

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.<\/p>\n

5. The sine values are equal.<\/p>\n

7. I<\/p>\n

9. IV<\/p>\n

11. [latex]\\frac{\\sqrt{3}}{2}[\/latex]<\/p>\n

13. [latex]\\frac{1}{2}[\/latex]<\/p>\n

15. [latex]\\frac{\\sqrt{2}}{2}[\/latex]<\/p>\n

17. 0<\/p>\n

19. \u22121<\/p>\n

21. [latex]\\frac{\\sqrt{3}}{2}[\/latex]<\/p>\n

23. [latex]60^\\circ[\/latex]<\/p>\n

25. [latex]80^\\circ[\/latex]<\/p>\n

27. [latex]45^\\circ[\/latex]<\/p>\n

29. [latex]\\frac{\\pi }{3}[\/latex]<\/p>\n

31. [latex]\\frac{\\pi }{3}[\/latex]<\/p>\n

33. [latex]\\frac{\\pi }{8}[\/latex]<\/p>\n

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]<\/p>\n

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]<\/p>\n

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]<\/p>\n

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]<\/p>\n

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]<\/p>\n

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]<\/p>\n

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]<\/p>\n

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]<\/p>\n

51. [latex]\\frac{\\sqrt{77}}{9}[\/latex]<\/p>\n

53. [latex]-\\frac{\\sqrt{15}}{4}[\/latex]<\/p>\n

55. [latex]\\left(-10,10\\sqrt{3}\\right)[\/latex]<\/p>\n

57. [latex]\\left(-2.778,15.757\\right)[\/latex]<\/p>\n

59. [latex]\\left[-1,1\\right][\/latex]<\/p>\n

61. [latex]\\sin t=\\frac{1}{2},\\cos t=-\\frac{\\sqrt{3}}{2}[\/latex]<\/p>\n

63. [latex]\\sin t=-\\frac{\\sqrt{2}}{2},\\cos t=-\\frac{\\sqrt{2}}{2}[\/latex]<\/p>\n

65. [latex]\\sin t=\\frac{\\sqrt{3}}{2},\\cos t=-\\frac{1}{2}[\/latex]<\/p>\n

67. [latex]\\sin t=-\\frac{\\sqrt{2}}{2},\\cos t=\\frac{\\sqrt{2}}{2}[\/latex]<\/p>\n

69. [latex]\\sin t=0,\\cos t=-1[\/latex]<\/p>\n

71. [latex]\\sin t=-0.596,\\cos t=0.803[\/latex]<\/p>\n

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

75. [latex]\\sin t=-\\frac{1}{2},\\cos t=\\frac{\\sqrt{3}}{2}[\/latex]<\/p>\n

77. [latex]\\sin t=0.761,\\cos t=-0.649[\/latex]<\/p>\n

79. [latex]\\sin t=1,\\cos t=0[\/latex]<\/p>\n

81. \u22120.1736<\/p>\n

83. 0.9511<\/p>\n

85. \u22120.7071<\/p>\n

87. \u22120.1392<\/p>\n

89. \u22120.7660<\/p>\n

91. [latex]\\frac{\\sqrt{2}}{4}[\/latex]<\/p>\n

93. [latex]-\\frac{\\sqrt{6}}{4}[\/latex]<\/p>\n

95. [latex]\\frac{\\sqrt{2}}{4}[\/latex]<\/p>\n

97. [latex]\\frac{\\sqrt{2}}{4}[\/latex]<\/p>\n

99. 0<\/p>\n

101. [latex]\\left(0,-1\\right)[\/latex]<\/p>\n

103. 37.5 seconds, 97.5 seconds, 157.5 seconds, 217.5 seconds, 277.5 seconds, 337.5 seconds<\/p>\n\n\t\t\t

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Candela Citations<\/h3>\n\t\t\t\t\t
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CC licensed content, Specific attribution<\/div>