{"id":1372,"date":"2023-06-05T14:51:04","date_gmt":"2023-06-05T14:51:04","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/tulsacc-math1613\/chapter\/solutions-for-the-other-trigonometric-functions\/"},"modified":"2023-06-05T14:51:04","modified_gmt":"2023-06-05T14:51:04","slug":"solutions-for-the-other-trigonometric-functions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/tulsacc-math1613\/chapter\/solutions-for-the-other-trigonometric-functions\/","title":{"raw":"Solutions 46: Other Trigonometric Functions","rendered":"Solutions 46: Other Trigonometric Functions"},"content":{"raw":"\n

Solutions to Odd-Numbered Exercises<\/h2>\n1. Yes, when the reference angle is [latex]\\frac{\\pi }{4}[\/latex] and the terminal side of the angle is in quadrants I and III. Thus, at [latex]x=\\frac{\\pi }{4},\\frac{5\\pi }{4}[\/latex], the sine and cosine values are equal.\n\n3. Substitute the sine of the angle in for [latex]y[\/latex] in the Pythagorean Theorem [latex]{x}^{2}+{y}^{2}=1[\/latex]. Solve for [latex]x[\/latex] and take the negative solution.\n\n5. The outputs of tangent and cotangent will repeat every [latex]\\pi [\/latex] units.\n\n7. [latex]\\frac{2\\sqrt{3}}{3}[\/latex]\n\n9. [latex]\\sqrt{3}[\/latex]\n\n11. [latex]\\sqrt{2}[\/latex]\n\n13. 1\n\n15. 2\n\n17. [latex]\\frac{\\sqrt{3}}{3}[\/latex]\n\n19. [latex]-\\frac{2\\sqrt{3}}{3}[\/latex]\n\n21. [latex]\\sqrt{3}[\/latex]\n\n23. [latex]-\\sqrt{2}[\/latex]\n\n25. \u22121\n\n27. \u22122\n\n29. [latex]-\\frac{\\sqrt{3}}{3}[\/latex]\n\n31. 2\n\n33. [latex]\\frac{\\sqrt{3}}{3}[\/latex]\n\n35. \u22122\n\n37. \u22121\n\n39. 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]\n\n41. [latex]\\sec t=2,\\csc t=\\frac{2\\sqrt{3}}{3},\\tan t=\\sqrt{3},\\cot t=\\frac{\\sqrt{3}}{3}[\/latex]\n\n43. [latex]-\\frac{\\sqrt{2}}{2}[\/latex]\n\n45. 3.1\n\n47. 1.4\n\n49. [latex]\\sin t=\\frac{\\sqrt{2}}{2},\\cos t=\\frac{\\sqrt{2}}{2},\\tan t=1,\\cot t=1,\\sec t=\\sqrt{2},\\csc t=\\sqrt{2}[\/latex]\n\n51. [latex]\\sin t=-\\frac{\\sqrt{3}}{2},\\cos t=-\\frac{1}{2},\\tan t=\\sqrt{3},\\cot t=\\frac{\\sqrt{3}}{3},\\sec t=-2,\\csc t=-\\frac{2\\sqrt{3}}{3}[\/latex]\n\n53. \u20130.228\n\n55. \u20132.414\n\n57. 1.414\n\n59. 1.540\n\n61. 1.556\n\n63. [latex]\\sin \\left(t\\right)\\approx 0.79[\/latex]\n\n65. [latex]\\csc t\\approx 1.16[\/latex]\n\n67. even\n\n69. even\n\n71. [latex]\\frac{\\sin t}{\\cos t}=\\tan t[\/latex]\n\n73. 13.77 hours, period: [latex]1000\\pi [\/latex]\n\n75. 7.73 inches\n","rendered":"

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

1. Yes, when the reference angle is [latex]\\frac{\\pi }{4}[\/latex] and the terminal side of the angle is in quadrants I and III. Thus, at [latex]x=\\frac{\\pi }{4},\\frac{5\\pi }{4}[\/latex], the sine and cosine values are equal.<\/p>\n

3. Substitute the sine of the angle in for [latex]y[\/latex] in the Pythagorean Theorem [latex]{x}^{2}+{y}^{2}=1[\/latex]. Solve for [latex]x[\/latex] and take the negative solution.<\/p>\n

5. The outputs of tangent and cotangent will repeat every [latex]\\pi[\/latex] units.<\/p>\n

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

9. [latex]\\sqrt{3}[\/latex]<\/p>\n

11. [latex]\\sqrt{2}[\/latex]<\/p>\n

13. 1<\/p>\n

15. 2<\/p>\n

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

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

21. [latex]\\sqrt{3}[\/latex]<\/p>\n

23. [latex]-\\sqrt{2}[\/latex]<\/p>\n

25. \u22121<\/p>\n

27. \u22122<\/p>\n

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

31. 2<\/p>\n

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

35. \u22122<\/p>\n

37. \u22121<\/p>\n

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]<\/p>\n

41. [latex]\\sec t=2,\\csc t=\\frac{2\\sqrt{3}}{3},\\tan t=\\sqrt{3},\\cot t=\\frac{\\sqrt{3}}{3}[\/latex]<\/p>\n

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

45. 3.1<\/p>\n

47. 1.4<\/p>\n

49. [latex]\\sin t=\\frac{\\sqrt{2}}{2},\\cos t=\\frac{\\sqrt{2}}{2},\\tan t=1,\\cot t=1,\\sec t=\\sqrt{2},\\csc t=\\sqrt{2}[\/latex]<\/p>\n

51. [latex]\\sin t=-\\frac{\\sqrt{3}}{2},\\cos t=-\\frac{1}{2},\\tan t=\\sqrt{3},\\cot t=\\frac{\\sqrt{3}}{3},\\sec t=-2,\\csc t=-\\frac{2\\sqrt{3}}{3}[\/latex]<\/p>\n

53. \u20130.228<\/p>\n

55. \u20132.414<\/p>\n

57. 1.414<\/p>\n

59. 1.540<\/p>\n

61. 1.556<\/p>\n

63. [latex]\\sin \\left(t\\right)\\approx 0.79[\/latex]<\/p>\n

65. [latex]\\csc t\\approx 1.16[\/latex]<\/p>\n

67. even<\/p>\n

69. even<\/p>\n

71. [latex]\\frac{\\sin t}{\\cos t}=\\tan t[\/latex]<\/p>\n

73. 13.77 hours, period: [latex]1000\\pi[\/latex]<\/p>\n

75. 7.73 inches<\/p>\n\n\t\t\t

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Candela Citations<\/h3>\n\t\t\t\t\t
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