{"id":1384,"date":"2023-06-05T14:51:11","date_gmt":"2023-06-05T14:51:11","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/tulsacc-math1613\/chapter\/solutions-for-inverse-trigonometric-functions\/"},"modified":"2023-06-05T14:51:11","modified_gmt":"2023-06-05T14:51:11","slug":"solutions-for-inverse-trigonometric-functions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/tulsacc-math1613\/chapter\/solutions-for-inverse-trigonometric-functions\/","title":{"raw":"Solutions 50: Inverse Trigonometric Functions","rendered":"Solutions 50: Inverse Trigonometric Functions"},"content":{"raw":"\n

Solutions to Odd-Numbered Exercises<\/h2>\n1. The function [latex]y=\\sin x[\/latex] is one-to-one on [latex]\\left[\u2212\\frac{\\pi}{2}\\text{, }\\frac{\\pi}{2}\\right][\/latex]; thus, this interval is the range of the inverse function of [latex]y=\\sin x\\text{, }f\\left(x\\right)=\\sin^{\u22121}x[\/latex]. The function [latex]y=\\cos x[\/latex] is one-to-one on [0,\u03c0]; thus, this interval is the range of the inverse function of [latex]y=\\cos x\\text{, }f(x)=\\cos^{\u22121}x[\/latex].\n\n3. [latex]\\frac{\\pi}{6}[\/latex] is the radian measure of an angle between [latex]\u2212\\frac{\\pi}{2}[\/latex] and [latex]\\frac{\\pi}{2}[\/latex] whose sine is 0.5.\n\n5. In order for any function to have an inverse, the function must be one-to-one and must pass the horizontal line test. The regular sine function is not one-to-one unless its domain is restricted in some way. Mathematicians have agreed to restrict the sine function to the interval [latex]\\left[\u2212\\frac{\\pi}{2}\\text{, }\\frac{\\pi}{2}\\right][\/latex] so that it is one-to-one and possesses an inverse.\n\n7. True. The angle, [latex]\\theta_{1}[\/latex] that equals [latex]\\arccos(\u2212x)\\text{, }x\\text{>}0[\/latex], will be a second quadrant angle with reference angle, [latex]\\theta_{2}[\/latex], where [latex]\\theta_{2}[\/latex] equals [latex]\\arccos x\\text{, }x\\text{>}0[\/latex]. Since [latex]\\theta_{2}[\/latex] is the reference angle for [latex]\\theta_{1}[\/latex], [latex]\\theta_{2}=\\pi(\u2212x)=\\pi\u2212\\arccos x[\/latex]\n\n9. [latex]\u2212\\frac{\\pi}{6}[\/latex]\n\n11. [latex]\\frac{3\\pi}{4}[\/latex]\n\n13. [latex]\u2212\\frac{\\pi}{3}[\/latex]\n\n15. [latex]\\frac{\\pi}{3}[\/latex]\n\n17. 1.98\n\n19. 0.93\n\n21. 1.41\n\n23. 0.56 radians\n\n25. 0\n\n27. 0.71\n\n29. \u22120.71\n\n31. [latex]\u2212\\frac{\\pi}{4}[\/latex]\n\n33. 0.8\n\n35. [latex]\\frac{5}{13}[\/latex]\n\n37. [latex]\\frac{x\u22121}{\\sqrt{\u2212x^{2}+2x}}[\/latex]\n\n39. [latex]\\frac{\\sqrt{x^{2}\u22121}}{x}[\/latex]\n\n41. [latex]\\frac{x+0.5}{\\sqrt{\u2212x^{2}\u2212x+\\frac{3}{4}}}[\/latex]\n\n43. [latex]\\frac{\\sqrt{2x+1}}{x+1}[\/latex]\n\n45. [latex]\\frac{\\sqrt{2x+1}}{x+1}[\/latex]\n\n47. t\n\n49. domain [\u22121,1]; range [0,\u03c0]\n\"A\n\n51. approximately [latex]x=0.00[\/latex]\n\n53. 0.395 radians\n\n55. 1.11 radians\n\n57. 1.25 radians\n\n59. 0.405 radians\n\n61. No. The angle the ladder makes with the horizontal is 60 degrees.\n","rendered":"

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

1. The function [latex]y=\\sin x[\/latex] is one-to-one on [latex]\\left[\u2212\\frac{\\pi}{2}\\text{, }\\frac{\\pi}{2}\\right][\/latex]; thus, this interval is the range of the inverse function of [latex]y=\\sin x\\text{, }f\\left(x\\right)=\\sin^{\u22121}x[\/latex]. The function [latex]y=\\cos x[\/latex] is one-to-one on [0,\u03c0]; thus, this interval is the range of the inverse function of [latex]y=\\cos x\\text{, }f(x)=\\cos^{\u22121}x[\/latex].<\/p>\n

3. [latex]\\frac{\\pi}{6}[\/latex] is the radian measure of an angle between [latex]\u2212\\frac{\\pi}{2}[\/latex] and [latex]\\frac{\\pi}{2}[\/latex] whose sine is 0.5.<\/p>\n

5. In order for any function to have an inverse, the function must be one-to-one and must pass the horizontal line test. The regular sine function is not one-to-one unless its domain is restricted in some way. Mathematicians have agreed to restrict the sine function to the interval [latex]\\left[\u2212\\frac{\\pi}{2}\\text{, }\\frac{\\pi}{2}\\right][\/latex] so that it is one-to-one and possesses an inverse.<\/p>\n

7. True. The angle, [latex]\\theta_{1}[\/latex] that equals [latex]\\arccos(\u2212x)\\text{, }x\\text{>}0[\/latex], will be a second quadrant angle with reference angle, [latex]\\theta_{2}[\/latex], where [latex]\\theta_{2}[\/latex] equals [latex]\\arccos x\\text{, }x\\text{>}0[\/latex]. Since [latex]\\theta_{2}[\/latex] is the reference angle for [latex]\\theta_{1}[\/latex], [latex]\\theta_{2}=\\pi(\u2212x)=\\pi\u2212\\arccos x[\/latex]<\/p>\n

9. [latex]\u2212\\frac{\\pi}{6}[\/latex]<\/p>\n

11. [latex]\\frac{3\\pi}{4}[\/latex]<\/p>\n

13. [latex]\u2212\\frac{\\pi}{3}[\/latex]<\/p>\n

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

17. 1.98<\/p>\n

19. 0.93<\/p>\n

21. 1.41<\/p>\n

23. 0.56 radians<\/p>\n

25. 0<\/p>\n

27. 0.71<\/p>\n

29. \u22120.71<\/p>\n

31. [latex]\u2212\\frac{\\pi}{4}[\/latex]<\/p>\n

33. 0.8<\/p>\n

35. [latex]\\frac{5}{13}[\/latex]<\/p>\n

37. [latex]\\frac{x\u22121}{\\sqrt{\u2212x^{2}+2x}}[\/latex]<\/p>\n

39. [latex]\\frac{\\sqrt{x^{2}\u22121}}{x}[\/latex]<\/p>\n

41. [latex]\\frac{x+0.5}{\\sqrt{\u2212x^{2}\u2212x+\\frac{3}{4}}}[\/latex]<\/p>\n

43. [latex]\\frac{\\sqrt{2x+1}}{x+1}[\/latex]<\/p>\n

45. [latex]\\frac{\\sqrt{2x+1}}{x+1}[\/latex]<\/p>\n

47. t<\/p>\n

49. domain [\u22121,1]; range [0,\u03c0]
\n\"A<\/p>\n

51. approximately [latex]x=0.00[\/latex]<\/p>\n

53. 0.395 radians<\/p>\n

55. 1.11 radians<\/p>\n

57. 1.25 radians<\/p>\n

59. 0.405 radians<\/p>\n

61. No. The angle the ladder makes with the horizontal is 60 degrees.<\/p>\n\n\t\t\t

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