{"id":1382,"date":"2023-06-05T14:51:10","date_gmt":"2023-06-05T14:51:10","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/tulsacc-math1613\/chapter\/solutions-for-the-other-trigonometric-functions-2\/"},"modified":"2023-06-05T14:51:10","modified_gmt":"2023-06-05T14:51:10","slug":"solutions-for-the-other-trigonometric-functions-2","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/tulsacc-math1613\/chapter\/solutions-for-the-other-trigonometric-functions-2\/","title":{"raw":"Solutions 49: Graphs of Other Trigonometric Functions","rendered":"Solutions 49: Graphs of Other Trigonometric Functions"},"content":{"raw":"\n

Solutions to Odd-Numbered Exercises<\/h2>\n1.  Since [latex]y=\\csc x[\/latex] is the reciprocal function of [latex]y=\\sin x[\/latex], you can plot the reciprocal of the coordinates on the graph of [latex]y=\\sin x[\/latex] to obtain the y<\/em>-coordinates of [latex]y=\\csc x[\/latex]. The x<\/em>-intercepts of the graph [latex]y=\\sin x[\/latex] are the vertical asymptotes for the graph of [latex]y=\\csc x[\/latex].\n\n3. Answers will vary. Using the unit circle, one can show that [latex]\\tan(x+\\pi)=\\tan x[\/latex].\n\n5. The period is the same: 2\u03c0.\n\n7. IV\n\n9. III\n\n11. period: 8; horizontal shift: 1 unit to left\n\n13. 1.5\n\n15. 5\n\n17. [latex]\u2212\\cot x\\cos x\u2212\\sin x[\/latex]\n\n19. stretching factor: 2; period: [latex]\\frac{\\pi}{4}[\/latex]; asymptotes: [latex]x=\\frac{1}{4}\\left(\\frac{\\pi}{2}+\\pi k\\right)+8[\/latex], where k<\/em> is an integer\n\"A\n\n21. stretching factor: 6; period: 6; asymptotes: [latex]x=3k[\/latex], where k<\/em> is an integer\n\"A\n\n23. stretching factor: 1; period: \u03c0; asymptotes: [latex]x=\u03c0k[\/latex], where k<\/em> is an integer\n\"A\n\n25. Stretching factor: 1; period: \u03c0; asymptotes: [latex]x=\\frac{\\pi}{4}+{\\pi}k[\/latex], where k<\/em> is an integer\n\"A\n\n27. stretching factor: 2; period: 2\u03c0; asymptotes: [latex]x=\u03c0k[\/latex], where k<\/em> is an integer\n\"A\n\n29. stretching factor: 4; period: [latex]\\frac{2\\pi}{3}[\/latex]; asymptotes: [latex]x=\\frac{\\pi}{6}k[\/latex], where k<\/em> is an odd integer\n\"A\n\n31. stretching factor: 7; period: [latex]\\frac{2\\pi}{5}[\/latex]; asymptotes: [latex]x=\\frac{\\pi}{10}k[\/latex], where k<\/em> is an odd integer\n\"A\n\n33. stretching factor: 2; period: 2\u03c0; asymptotes: [latex]x=\u2212\\frac{\\pi}{4}+\\pi k[\/latex], where k<\/em> is an integer\n\"A\n\n35. stretching factor: [latex]\\frac{7}{5}[\/latex]; period: 2\u03c0; asymptotes: [latex]x=\\frac{\\pi}{4}+\\pi[\/latex]k<\/em>, where k<\/em> is an integer\n\"A\n\n37. [latex]y=\\tan\\left(3\\left(x\u2212\\frac{\\pi}{4}\\right)\\right)+2[\/latex]\n\"A\n\n39. [latex]f(x)=\\csc(2x)[\/latex]\n\n41. [latex]f(x)=\\csc(4x)[\/latex]\n\n43. [latex]f(x)=2\\csc x[\/latex]\n\n45. [latex]f(x)=\\frac{1}{2}\\tan(100\\pi x)[\/latex]\n\nFor the following exercises, use a graphing calculator to graph two periods of the given function. Note: most graphing calculators do not have a cosecant button; therefore, you will need to input [latex]\\csc x[\/latex] as [latex]\\frac{1}{\\sin x}[\/latex].\n\n46. [latex]f(x)=|\\csc(x)|[\/latex]\n\n47. [latex]f(x)=|\\cot(x)|[\/latex]\n\n48. [latex]f(x)=2^{\\csc(x)}[\/latex]\n\n49. [latex]f(x)=\\frac{\\csc(x)}{\\sec(x)}[\/latex]\n\n51.\n\"A\n\n53.\n\"A\n\n55. a. [latex](\u2212\\frac{\\pi}{2}\\text{,}\\frac{\\pi}{2})[\/latex];\nb.\n\"A\nc. [latex]x=\u2212\\frac{\\pi}{2}[\/latex] and [latex]x=\\frac{\\pi}{2}[\/latex]; the distance grows without bound as |x<\/em>| approaches [latex]\\frac{\\pi}{2}[\/latex]\u2014i.e., at right angles to the line representing due north, the boat would be so far away, the fisherman could not see it;\nd. 3; when [latex]x=\u2212\\frac{\\pi}{3}[\/latex], the boat is 3 km away;\ne. 1.73; when [latex]x=\\frac{\\pi}{6}[\/latex], the boat is about 1.73 km away;\nf. 1.5 km; when [latex]x=0[\/latex].\n\n57. a. [latex]h(x)=2\\tan\\left(\\frac{\\pi}{120}x\\right)[\/latex];\nb.\n\"An\nc. [latex]h(0)=0:[\/latex] after 0 seconds, the rocket is 0 mi above the ground; [latex]h(30)=2:[\/latex] after 30 seconds, the rockets is 2 mi high;\nd. As x<\/em> approaches 60 seconds, the values of [latex]h(x)[\/latex] grow increasingly large. The distance to the rocket is growing so large that the camera can no longer track it.\n","rendered":"

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

1.  Since [latex]y=\\csc x[\/latex] is the reciprocal function of [latex]y=\\sin x[\/latex], you can plot the reciprocal of the coordinates on the graph of [latex]y=\\sin x[\/latex] to obtain the y<\/em>-coordinates of [latex]y=\\csc x[\/latex]. The x<\/em>-intercepts of the graph [latex]y=\\sin x[\/latex] are the vertical asymptotes for the graph of [latex]y=\\csc x[\/latex].<\/p>\n

3. Answers will vary. Using the unit circle, one can show that [latex]\\tan(x+\\pi)=\\tan x[\/latex].<\/p>\n

5. The period is the same: 2\u03c0.<\/p>\n

7. IV<\/p>\n

9. III<\/p>\n

11. period: 8; horizontal shift: 1 unit to left<\/p>\n

13. 1.5<\/p>\n

15. 5<\/p>\n

17. [latex]\u2212\\cot x\\cos x\u2212\\sin x[\/latex]<\/p>\n

19. stretching factor: 2; period: [latex]\\frac{\\pi}{4}[\/latex]; asymptotes: [latex]x=\\frac{1}{4}\\left(\\frac{\\pi}{2}+\\pi k\\right)+8[\/latex], where k<\/em> is an integer
\n\"A<\/p>\n

21. stretching factor: 6; period: 6; asymptotes: [latex]x=3k[\/latex], where k<\/em> is an integer
\n\"A<\/p>\n

23. stretching factor: 1; period: \u03c0; asymptotes: [latex]x=\u03c0k[\/latex], where k<\/em> is an integer
\n\"A<\/p>\n

25. Stretching factor: 1; period: \u03c0; asymptotes: [latex]x=\\frac{\\pi}{4}+{\\pi}k[\/latex], where k<\/em> is an integer
\n\"A<\/p>\n

27. stretching factor: 2; period: 2\u03c0; asymptotes: [latex]x=\u03c0k[\/latex], where k<\/em> is an integer
\n\"A<\/p>\n

29. stretching factor: 4; period: [latex]\\frac{2\\pi}{3}[\/latex]; asymptotes: [latex]x=\\frac{\\pi}{6}k[\/latex], where k<\/em> is an odd integer
\n\"A<\/p>\n

31. stretching factor: 7; period: [latex]\\frac{2\\pi}{5}[\/latex]; asymptotes: [latex]x=\\frac{\\pi}{10}k[\/latex], where k<\/em> is an odd integer
\n\"A<\/p>\n

33. stretching factor: 2; period: 2\u03c0; asymptotes: [latex]x=\u2212\\frac{\\pi}{4}+\\pi k[\/latex], where k<\/em> is an integer
\n\"A<\/p>\n

35. stretching factor: [latex]\\frac{7}{5}[\/latex]; period: 2\u03c0; asymptotes: [latex]x=\\frac{\\pi}{4}+\\pi[\/latex]k<\/em>, where k<\/em> is an integer
\n\"A<\/p>\n

37. [latex]y=\\tan\\left(3\\left(x\u2212\\frac{\\pi}{4}\\right)\\right)+2[\/latex]
\n\"A<\/p>\n

39. [latex]f(x)=\\csc(2x)[\/latex]<\/p>\n

41. [latex]f(x)=\\csc(4x)[\/latex]<\/p>\n

43. [latex]f(x)=2\\csc x[\/latex]<\/p>\n

45. [latex]f(x)=\\frac{1}{2}\\tan(100\\pi x)[\/latex]<\/p>\n

For the following exercises, use a graphing calculator to graph two periods of the given function. Note: most graphing calculators do not have a cosecant button; therefore, you will need to input [latex]\\csc x[\/latex] as [latex]\\frac{1}{\\sin x}[\/latex].<\/p>\n

46. [latex]f(x)=|\\csc(x)|[\/latex]<\/p>\n

47. [latex]f(x)=|\\cot(x)|[\/latex]<\/p>\n

48. [latex]f(x)=2^{\\csc(x)}[\/latex]<\/p>\n

49. [latex]f(x)=\\frac{\\csc(x)}{\\sec(x)}[\/latex]<\/p>\n

51.
\n\"A<\/p>\n

53.
\n\"A<\/p>\n

55. a. [latex](\u2212\\frac{\\pi}{2}\\text{,}\\frac{\\pi}{2})[\/latex];
\nb.
\n\"A
\nc. [latex]x=\u2212\\frac{\\pi}{2}[\/latex] and [latex]x=\\frac{\\pi}{2}[\/latex]; the distance grows without bound as |x<\/em>| approaches [latex]\\frac{\\pi}{2}[\/latex]\u2014i.e., at right angles to the line representing due north, the boat would be so far away, the fisherman could not see it;
\nd. 3; when [latex]x=\u2212\\frac{\\pi}{3}[\/latex], the boat is 3 km away;
\ne. 1.73; when [latex]x=\\frac{\\pi}{6}[\/latex], the boat is about 1.73 km away;
\nf. 1.5 km; when [latex]x=0[\/latex].<\/p>\n

57. a. [latex]h(x)=2\\tan\\left(\\frac{\\pi}{120}x\\right)[\/latex];
\nb.
\n\"An
\nc. [latex]h(0)=0:[\/latex] after 0 seconds, the rocket is 0 mi above the ground; [latex]h(30)=2:[\/latex] after 30 seconds, the rockets is 2 mi high;
\nd. As x<\/em> approaches 60 seconds, the values of [latex]h(x)[\/latex] grow increasingly large. The distance to the rocket is growing so large that the camera can no longer track it.<\/p>\n\n\t\t\t

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