{"id":15674,"date":"2019-09-05T17:41:33","date_gmt":"2019-09-05T17:41:33","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/precalculus\/?post_type=chapter&p=15674"},"modified":"2025-02-05T05:22:56","modified_gmt":"2025-02-05T05:22:56","slug":"problem-set-48-graphs-of-the-sine-and-cosine-function","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/precalculus\/chapter\/problem-set-48-graphs-of-the-sine-and-cosine-function\/","title":{"raw":"Problem Set 48: Graphs of the Sine and Cosine Function","rendered":"Problem Set 48: Graphs of the Sine and Cosine Function"},"content":{"raw":"1. Why are the sine and cosine functions called periodic functions?\r\n\r\n2. How does the graph of [latex]y=\\sin x[\/latex] compare with the graph of [latex]y=\\cos x[\/latex]? Explain how you could horizontally translate the graph of [latex]y=\\sin x[\/latex] to obtain [latex]y=\\cos x[\/latex].\r\n\r\n3. For the equation [latex]A\\cos(Bx+C)+D[\/latex], what constants affect the range of the function and how do they affect the range?\r\n\r\n4. How does the range of a translated sine function relate to the equation [latex]y=A\\sin(Bx+C)+D[\/latex]?\r\n\r\n5. How can the unit circle be used to construct the graph of [latex]f(t)=\\sin t[\/latex]?\r\n\r\n6.\u00a0[latex]f(x)=2\\sin x[\/latex]\r\n\r\n7.\u00a0[latex]f(x)=\\frac{2}{3}\\cos x[\/latex]\r\n\r\n8.\u00a0[latex]f(x)=\u22123\\sin x[\/latex]\r\n\r\n9.\u00a0[latex]f(x)=4\\sin x[\/latex]\r\n\r\n10.\u00a0[latex]f(x)=2\\cos x[\/latex]\r\n\r\n11.\u00a0[latex]f(x)=\\cos(2x)[\/latex]\r\n\r\n12.\u00a0[latex]f(x)=2\\sin\\left(\\frac{1}{2}x\\right)[\/latex]\r\n\r\n13.\u00a0[latex]f(x)=4\\cos(\\pi x)[\/latex]\r\n\r\n14.\u00a0[latex]f(x)=3\\cos\\left(\\frac{6}{5}x\\right)[\/latex]\r\n\r\n15.\u00a0[latex]y=3\\sin(8(x+4))+5[\/latex]\r\n\r\n16.\u00a0[latex]y=2\\sin(3x\u221221)+4[\/latex]\r\n\r\n17.\u00a0[latex]y=5\\sin(5x+20)\u22122[\/latex]\r\n\r\nFor the following exercises, graph one full period of each function, starting at [latex]x=0[\/latex]. For each function, state the amplitude, period, and midline. State the maximum and minimum y<\/em>-values and their corresponding x<\/em>-values on one period for [latex]x>0[\/latex]. State the phase shift and vertical translation, if applicable. Round answers to two decimal places if necessary.\r\n\r\n18.\u00a0[latex]f(t)=2\\sin\\left(t\u2212\\frac{5\\pi}{6}\\right)[\/latex]\r\n\r\n19.\u00a0[latex]f(t)=\u2212\\cos\\left(t+\\frac{\\pi}{3}\\right)+1[\/latex]\r\n\r\n20.\u00a0[latex]f(t)=4\\cos\\left(2\\left(t+\\frac{\\pi}{4}\\right)\\right)\u22123[\/latex]\r\n\r\n21.\u00a0[latex]f(t)=\u2212\\sin\\left(12t+\\frac{5\\pi}{3}\\right)[\/latex]\r\n\r\n22.\u00a0[latex]f(x)=4\\sin\\left(\\frac{\\pi}{2}(x\u22123)\\right)+7[\/latex]\r\n\r\n23. Determine the amplitude, midline, period, and an equation involving the sine function for the graph shown in Figure 26.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"371\"]\"A Figure 26<\/b>[\/caption]24. Determine the amplitude, period, midline, and an equation involving cosine for the graph shown in Figure 27.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"308\"]\"A Figure 27<\/b>[\/caption]25. Determine the amplitude, period, midline, and an equation involving cosine for the graph shown in Figure 28.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"432\"]\"A Figure 28<\/b>[\/caption]26. Determine the amplitude, period, midline, and an equation involving sine for the graph shown in Figure 29.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"400\"]\"A Figure 29<\/b>[\/caption]27. Determine the amplitude, period, midline, and an equation involving cosine for the graph shown in Figure 30.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"401\"]\"A Figure 30<\/b>[\/caption]28. Determine the amplitude, period, midline, and an equation involving sine for the graph shown in Figure 31.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"307\"]\"A Figure 31<\/b>[\/caption]\r\n\r\n29. Determine the amplitude, period, midline, and an equation involving cosine for the graph shown in Figure 32.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"308\"]\"A Figure 32<\/b>[\/caption]\r\n\r\n30. Determine the amplitude, period, midline, and an equation involving sine for the graph shown in Figure 33.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"306\"]\"A Figure 33<\/b>[\/caption]\r\n\r\nFor the following exercises, let [latex]f(x)=\\sin x[\/latex].\r\n\r\n31.\u00a0On [0,2\u03c0), solve [latex]f(x)=\\frac{1}{2}[\/latex].\r\n\r\n32. Evaluate [latex]f\\left(\\frac{\\pi}{2}\\right)[\/latex].\r\n\r\n33. On [0,2\u03c0), [latex]f(x)=\\frac{\\sqrt{2}}{2}[\/latex]. Find all values of x.\r\n\r\n34.\u00a0On [0,2\u03c0), the maximum value(s) of the function occur(s) at what x-value(s)?\r\n\r\n35. On [0,2\u03c0), the minimum value(s) of the function occur(s) at what x-value(s)?\r\n\r\n36. Show that [latex]f(\u2212x)=\u2212f(x)[\/latex].This means that [latex]f(x)=\\sin x[\/latex] is an odd function and possesses symmetry with respect to ________________.\r\n\r\nFor the following exercises, let [latex]f(x)=\\cos x[\/latex].\r\n\r\n37. On [0,2\u03c0), solve the equation [latex]f(x)=\\cos x=0[\/latex].\r\n\r\n38. On[0,2\u03c0), solve [latex]f(x)=\\frac{1}{2}[\/latex].\r\n\r\n39. On [0,2\u03c0), find the x<\/em>-intercepts of [latex]f(x)=\\cos x[\/latex].\r\n\r\n40. On [0,2\u03c0), find the x<\/em>-values at which the function has a maximum or minimum value.\r\n\r\n41. On [0,2\u03c0), solve the equation [latex]f(x)=\\frac{\\sqrt{3}}{2}[\/latex].\r\n\r\n42. Graph [latex]h(x)=x+\\sin x \\text{ on}[0,2\\pi][\/latex]. Explain why the graph appears as it does.\r\n\r\n43. Graph [latex]h(x)=x+\\sin x[\/latex] on[\u2212100,100]. Did the graph appear as predicted in the previous exercise?\r\n\r\n44. Graph [latex]f(x)=x\\sin x[\/latex] on [0,2\u03c0] and verbalize how the graph varies from the graph of [latex]f(x)=\\sin x[\/latex].\r\n\r\n45. Graph [latex]f(x)=x\\sin x[\/latex] on the window [\u221210,10] and explain what the graph shows.\r\n\r\n46. Graph [latex]f(x)=\\frac{\\sin x}{x}[\/latex] on the window [\u22125\u03c0,5\u03c0] and explain what the graph shows.\r\n\r\n47. A Ferris wheel is 25 meters in diameter and boarded from a platform that is 1 meter above the ground. The six o\u2019clock position on the Ferris wheel is level with the loading platform. The wheel completes 1 full revolution in 10 minutes. The function h<\/em>(t<\/em>) gives a person\u2019s height in meters above the ground t<\/em> minutes after the wheel begins to turn.\r\na. Find the amplitude, midline, and period of\u00a0h(t<\/em>).\r\nb. Find a formula for the height function\u00a0h(t<\/em>).\r\nc. How high off the ground is a person after 5 minutes?","rendered":"

1. Why are the sine and cosine functions called periodic functions?<\/p>\n

2. How does the graph of [latex]y=\\sin x[\/latex] compare with the graph of [latex]y=\\cos x[\/latex]? Explain how you could horizontally translate the graph of [latex]y=\\sin x[\/latex] to obtain [latex]y=\\cos x[\/latex].<\/p>\n

3. For the equation [latex]A\\cos(Bx+C)+D[\/latex], what constants affect the range of the function and how do they affect the range?<\/p>\n

4. How does the range of a translated sine function relate to the equation [latex]y=A\\sin(Bx+C)+D[\/latex]?<\/p>\n

5. How can the unit circle be used to construct the graph of [latex]f(t)=\\sin t[\/latex]?<\/p>\n

6.\u00a0[latex]f(x)=2\\sin x[\/latex]<\/p>\n

7.\u00a0[latex]f(x)=\\frac{2}{3}\\cos x[\/latex]<\/p>\n

8.\u00a0[latex]f(x)=\u22123\\sin x[\/latex]<\/p>\n

9.\u00a0[latex]f(x)=4\\sin x[\/latex]<\/p>\n

10.\u00a0[latex]f(x)=2\\cos x[\/latex]<\/p>\n

11.\u00a0[latex]f(x)=\\cos(2x)[\/latex]<\/p>\n

12.\u00a0[latex]f(x)=2\\sin\\left(\\frac{1}{2}x\\right)[\/latex]<\/p>\n

13.\u00a0[latex]f(x)=4\\cos(\\pi x)[\/latex]<\/p>\n

14.\u00a0[latex]f(x)=3\\cos\\left(\\frac{6}{5}x\\right)[\/latex]<\/p>\n

15.\u00a0[latex]y=3\\sin(8(x+4))+5[\/latex]<\/p>\n

16.\u00a0[latex]y=2\\sin(3x\u221221)+4[\/latex]<\/p>\n

17.\u00a0[latex]y=5\\sin(5x+20)\u22122[\/latex]<\/p>\n

For the following exercises, graph one full period of each function, starting at [latex]x=0[\/latex]. For each function, state the amplitude, period, and midline. State the maximum and minimum y<\/em>-values and their corresponding x<\/em>-values on one period for [latex]x>0[\/latex]. State the phase shift and vertical translation, if applicable. Round answers to two decimal places if necessary.<\/p>\n

18.\u00a0[latex]f(t)=2\\sin\\left(t\u2212\\frac{5\\pi}{6}\\right)[\/latex]<\/p>\n

19.\u00a0[latex]f(t)=\u2212\\cos\\left(t+\\frac{\\pi}{3}\\right)+1[\/latex]<\/p>\n

20.\u00a0[latex]f(t)=4\\cos\\left(2\\left(t+\\frac{\\pi}{4}\\right)\\right)\u22123[\/latex]<\/p>\n

21.\u00a0[latex]f(t)=\u2212\\sin\\left(12t+\\frac{5\\pi}{3}\\right)[\/latex]<\/p>\n

22.\u00a0[latex]f(x)=4\\sin\\left(\\frac{\\pi}{2}(x\u22123)\\right)+7[\/latex]<\/p>\n

23. Determine the amplitude, midline, period, and an equation involving the sine function for the graph shown in Figure 26.<\/p>\n

\"A<\/p>\n

Figure 26<\/b><\/p>\n<\/div>\n

24. Determine the amplitude, period, midline, and an equation involving cosine for the graph shown in Figure 27.<\/p>\n

\"A<\/p>\n

Figure 27<\/b><\/p>\n<\/div>\n

25. Determine the amplitude, period, midline, and an equation involving cosine for the graph shown in Figure 28.<\/p>\n

\"A<\/p>\n

Figure 28<\/b><\/p>\n<\/div>\n

26. Determine the amplitude, period, midline, and an equation involving sine for the graph shown in Figure 29.<\/p>\n

\"A<\/p>\n

Figure 29<\/b><\/p>\n<\/div>\n

27. Determine the amplitude, period, midline, and an equation involving cosine for the graph shown in Figure 30.<\/p>\n

\"A<\/p>\n

Figure 30<\/b><\/p>\n<\/div>\n

28. Determine the amplitude, period, midline, and an equation involving sine for the graph shown in Figure 31.<\/p>\n

\"A<\/p>\n

Figure 31<\/b><\/p>\n<\/div>\n

29. Determine the amplitude, period, midline, and an equation involving cosine for the graph shown in Figure 32.<\/p>\n

\"A<\/p>\n

Figure 32<\/b><\/p>\n<\/div>\n

30. Determine the amplitude, period, midline, and an equation involving sine for the graph shown in Figure 33.<\/p>\n

\"A<\/p>\n

Figure 33<\/b><\/p>\n<\/div>\n

For the following exercises, let [latex]f(x)=\\sin x[\/latex].<\/p>\n

31.\u00a0On [0,2\u03c0), solve [latex]f(x)=\\frac{1}{2}[\/latex].<\/p>\n

32. Evaluate [latex]f\\left(\\frac{\\pi}{2}\\right)[\/latex].<\/p>\n

33. On [0,2\u03c0), [latex]f(x)=\\frac{\\sqrt{2}}{2}[\/latex]. Find all values of x.<\/p>\n

34.\u00a0On [0,2\u03c0), the maximum value(s) of the function occur(s) at what x-value(s)?<\/p>\n

35. On [0,2\u03c0), the minimum value(s) of the function occur(s) at what x-value(s)?<\/p>\n

36. Show that [latex]f(\u2212x)=\u2212f(x)[\/latex].This means that [latex]f(x)=\\sin x[\/latex] is an odd function and possesses symmetry with respect to ________________.<\/p>\n

For the following exercises, let [latex]f(x)=\\cos x[\/latex].<\/p>\n

37. On [0,2\u03c0), solve the equation [latex]f(x)=\\cos x=0[\/latex].<\/p>\n

38. On[0,2\u03c0), solve [latex]f(x)=\\frac{1}{2}[\/latex].<\/p>\n

39. On [0,2\u03c0), find the x<\/em>-intercepts of [latex]f(x)=\\cos x[\/latex].<\/p>\n

40. On [0,2\u03c0), find the x<\/em>-values at which the function has a maximum or minimum value.<\/p>\n

41. On [0,2\u03c0), solve the equation [latex]f(x)=\\frac{\\sqrt{3}}{2}[\/latex].<\/p>\n

42. Graph [latex]h(x)=x+\\sin x \\text{ on}[0,2\\pi][\/latex]. Explain why the graph appears as it does.<\/p>\n

43. Graph [latex]h(x)=x+\\sin x[\/latex] on[\u2212100,100]. Did the graph appear as predicted in the previous exercise?<\/p>\n

44. Graph [latex]f(x)=x\\sin x[\/latex] on [0,2\u03c0] and verbalize how the graph varies from the graph of [latex]f(x)=\\sin x[\/latex].<\/p>\n

45. Graph [latex]f(x)=x\\sin x[\/latex] on the window [\u221210,10] and explain what the graph shows.<\/p>\n

46. Graph [latex]f(x)=\\frac{\\sin x}{x}[\/latex] on the window [\u22125\u03c0,5\u03c0] and explain what the graph shows.<\/p>\n

47. A Ferris wheel is 25 meters in diameter and boarded from a platform that is 1 meter above the ground. The six o\u2019clock position on the Ferris wheel is level with the loading platform. The wheel completes 1 full revolution in 10 minutes. The function h<\/em>(t<\/em>) gives a person\u2019s height in meters above the ground t<\/em> minutes after the wheel begins to turn.
\na. Find the amplitude, midline, and period of\u00a0h(t<\/em>).
\nb. Find a formula for the height function\u00a0h(t<\/em>).
\nc. How high off the ground is a person after 5 minutes?<\/p>\n\n\t\t\t

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