{"id":2343,"date":"2025-08-13T00:21:11","date_gmt":"2025-08-13T00:21:11","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&p=2343"},"modified":"2026-02-17T19:58:13","modified_gmt":"2026-02-17T19:58:13","slug":"periodic-functions-get-stronger","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/periodic-functions-get-stronger\/","title":{"raw":"Periodic Functions: Get Stronger","rendered":"Periodic Functions: Get Stronger"},"content":{"raw":"

Graphs of the Sine and Cosine Function<\/h1>\r\n1. Why are the sine and cosine functions called periodic functions?\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\nFor each function, state the amplitude, period, and midline. State the maximum and minimum y-values and their corresponding x-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\n7.\u00a0[latex]f(x)=\\frac{2}{3}\\cos x[\/latex]\r\n\r\n9.\u00a0[latex]f(x)=4\\sin x[\/latex]\r\n\r\n11.\u00a0[latex]f(x)=\\cos(2x)[\/latex]\r\n\r\n13.\u00a0[latex]f(x)=4\\cos(\\pi x)[\/latex]\r\n\r\n15.\u00a0[latex]y=3\\sin(8(x+4))+5[\/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\n19.\u00a0[latex]f(t)=\u2212\\cos\\left(t+\\frac{\\pi}{3}\\right)+1[\/latex]\r\n\r\n21.\u00a0[latex]f(t)=\u2212\\sin\\left(12t+\\frac{5\\pi}{3}\\right)[\/latex]\r\n\r\nDetermine the amplitude, midline, period, and an equation involving the sine function for the graph shown.\r\n\r\n23.\r\n\r\n\"A\r\n\r\n25.\r\n\r\n\"A\r\n\r\n27.\r\n\r\n\"A\r\n\r\n29.\r\n\r\n\"A\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\n33. On [0,2\u03c0), [latex]f(x)=\\frac{\\sqrt{2}}{2}[\/latex]. Find all values of x.\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\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\n39. On [0,2\u03c0), find the x<\/em>-intercepts of [latex]f(x)=\\cos x[\/latex].\r\n\r\n41. On [0,2\u03c0), solve the equation [latex]f(x)=\\frac{\\sqrt{3}}{2}[\/latex].\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?\r\n

Graphs of the Other Trigonometric Functions<\/h1>\r\n1. Explain how the graph of the sine function can be used to graph [latex]y=\\csc x[\/latex].\r\n\r\n3.\u00a0Explain why the period of [latex]\\tan x[\/latex] is equal to \u03c0.\r\n\r\nFor the following exercises, match each trigonometric function with one of the following graphs.\r\n\r\n\"Trigonometric<\/span>\"Trigonometric<\/span>\"Trigonometric<\/span>\"Trigonometric<\/span>\r\n\r\n6. [latex]f(x)=\\tan x[\/latex]\r\n\r\n7. [latex]f(x)=\\sec x[\/latex]\r\n\r\n8. [latex]f(x)=\\csc x[\/latex]\r\n\r\n9. [latex]f(x)=\\cot x[\/latex]\r\n\r\nFor the following exercises, find the period and horizontal shift of each of the functions.\r\n\r\n11. [latex]h(x)=2\\sec\\left(\\frac{\\pi}{4}(x+1)\\right)[\/latex]\r\n\r\n13. If tan\u00a0x<\/em> =\u00a0\u22121.5, find tan(\u2212x).\r\n\r\n15. If csc\u00a0x<\/em> =\u00a0\u22125, find csc(\u2212x).\r\n\r\nFor the following exercises, rewrite each expression such that the argument\u00a0x<\/em> is positive.\r\n\r\n17. [latex]\\cot(\u2212x)\\cos(\u2212x)+\\sin(\u2212x)[\/latex]\r\n\r\nFor the following exercises, sketch two periods of the graph for each of the following functions. Identify the stretching factor, period, and asymptotes.\r\n\r\n19. [latex]f(x)=2\\tan(4x\u221232)[\/latex]\r\n\r\n21. [latex]m(x)=6\\csc\\left(\\frac{\\pi}{3}x+\\pi\\right)[\/latex]\r\n\r\n23. [latex]p(x)=\\tan\\left(x\u2212\\frac{\\pi}{2}\\right)[\/latex]\r\n\r\n25. [latex]f(x)=\\tan\\left(x+\\frac{\\pi}{4}\\right)[\/latex]\r\n\r\n27. [latex]f(x)=2\\csc(x)[\/latex]\r\n\r\n29. [latex]f(x)=4\\sec(3x)[\/latex]\r\n\r\n31. [latex]f(x)=7\\sec(5x)[\/latex]\r\n\r\n33. [latex]f(x)=2\\csc \\left(x+\\frac{\\pi}{4}\\right)\u22121[\/latex]\r\n\r\n35. [latex]f(x)=\\frac{7}{5}\\csc \\left(x\u2212\\frac{\\pi}{4}\\right)[\/latex]\r\n\r\nFor the following exercises, find and graph two periods of the periodic function with the given stretching factor, |A<\/em>|, period, and phase shift.\r\n\r\n37. A tangent curve, [latex]A=1[\/latex], period of [latex]\\frac{\\pi}{3}[\/latex]; and phase shift [latex](h\\text{,}k)=\\left(\\frac{\\pi}{4}\\text{,}2\\right)[\/latex]\r\n\r\nFor the following exercises, find an equation for the graph of each function.\r\n\r\n39.\r\n\"A\r\n\r\n41.\r\n\"A\r\n\r\n43.\r\n\"A\r\n\r\n45.\r\n\"graph\r\n\r\n49.\r\n\"A\r\n\r\n55.\u00a0Standing on the shore of a lake, a fisherman sights a boat far in the distance to his left. Let x<\/em>, measured in radians, be the angle formed by the line of sight to the ship and a line due north from his position. Assume due north is 0 and x<\/em> is measured negative to the left and positive to the right. The boat travels from due west to due east and, ignoring the curvature of the Earth, the distance [latex]d(x)[\/latex], in kilometers, from the fisherman to the boat is given by the function [latex]d(x)=1.5\\sec(x)[\/latex].\r\n
a. What is a reasonable domain for [latex]d(x)[\/latex]?\r\nb. Graph d(x) on this domain.\r\nc. Find and discuss the meaning of any vertical asymptotes on the graph of [latex]d(x)[\/latex].\r\nd. Calculate and interpret [latex]d(\u2212\\frac{\\pi}{3})[\/latex]. Round to the second decimal place.\r\ne. Calculate and interpret [latex]d(\\frac{\\pi}{6})[\/latex]. Round to the second decimal place.\r\nf. What is the minimum distance between the fisherman and the boat? When does this occur?<\/div>\r\n\"An\r\n\r\n57.\u00a0A video camera is focused on a rocket on a launching pad 2 miles from the camera. The angle of elevation from the ground to the rocket after x<\/em> seconds is [latex]\\frac{\\pi}{120}x[\/latex].\r\n
a. Write a function expressing the altitude [latex]h(x)[\/latex], in miles, of the rocket above the ground after x<\/em> seconds. Ignore the curvature of the Earth.\r\nb. Graph [latex]h(x)[\/latex] on the interval (0,60).\r\nc. Evaluate and interpret the values [latex]h(0)[\/latex] and [latex]h(30)[\/latex].\r\nd. What happens to the values of [latex]h(x)[\/latex] as x<\/em> approaches 60 seconds? Interpret the meaning of this in terms of the problem.<\/div>\r\n ","rendered":"

Graphs of the Sine and Cosine Function<\/h1>\n

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

For each function, state the amplitude, period, and midline. State the maximum and minimum y-values and their corresponding x-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

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

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

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

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

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

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

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

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

23.<\/p>\n

\"A<\/p>\n

25.<\/p>\n

\"A<\/p>\n

27.<\/p>\n

\"A<\/p>\n

29.<\/p>\n

\"A<\/p>\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

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

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

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

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

Graphs of the Other Trigonometric Functions<\/h1>\n

1. Explain how the graph of the sine function can be used to graph [latex]y=\\csc x[\/latex].<\/p>\n

3.\u00a0Explain why the period of [latex]\\tan x[\/latex] is equal to \u03c0.<\/p>\n

For the following exercises, match each trigonometric function with one of the following graphs.<\/p>\n

\"Trigonometric<\/span>\"Trigonometric<\/span>\"Trigonometric<\/span>\"Trigonometric<\/span><\/p>\n

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

7. [latex]f(x)=\\sec x[\/latex]<\/p>\n

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

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

For the following exercises, find the period and horizontal shift of each of the functions.<\/p>\n

11. [latex]h(x)=2\\sec\\left(\\frac{\\pi}{4}(x+1)\\right)[\/latex]<\/p>\n

13. If tan\u00a0x<\/em> =\u00a0\u22121.5, find tan(\u2212x).<\/p>\n

15. If csc\u00a0x<\/em> =\u00a0\u22125, find csc(\u2212x).<\/p>\n

For the following exercises, rewrite each expression such that the argument\u00a0x<\/em> is positive.<\/p>\n

17. [latex]\\cot(\u2212x)\\cos(\u2212x)+\\sin(\u2212x)[\/latex]<\/p>\n

For the following exercises, sketch two periods of the graph for each of the following functions. Identify the stretching factor, period, and asymptotes.<\/p>\n

19. [latex]f(x)=2\\tan(4x\u221232)[\/latex]<\/p>\n

21. [latex]m(x)=6\\csc\\left(\\frac{\\pi}{3}x+\\pi\\right)[\/latex]<\/p>\n

23. [latex]p(x)=\\tan\\left(x\u2212\\frac{\\pi}{2}\\right)[\/latex]<\/p>\n

25. [latex]f(x)=\\tan\\left(x+\\frac{\\pi}{4}\\right)[\/latex]<\/p>\n

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

29. [latex]f(x)=4\\sec(3x)[\/latex]<\/p>\n

31. [latex]f(x)=7\\sec(5x)[\/latex]<\/p>\n

33. [latex]f(x)=2\\csc \\left(x+\\frac{\\pi}{4}\\right)\u22121[\/latex]<\/p>\n

35. [latex]f(x)=\\frac{7}{5}\\csc \\left(x\u2212\\frac{\\pi}{4}\\right)[\/latex]<\/p>\n

For the following exercises, find and graph two periods of the periodic function with the given stretching factor, |A<\/em>|, period, and phase shift.<\/p>\n

37. A tangent curve, [latex]A=1[\/latex], period of [latex]\\frac{\\pi}{3}[\/latex]; and phase shift [latex](h\\text{,}k)=\\left(\\frac{\\pi}{4}\\text{,}2\\right)[\/latex]<\/p>\n

For the following exercises, find an equation for the graph of each function.<\/p>\n

39.
\n\"A<\/p>\n

41.
\n\"A<\/p>\n

43.
\n\"A<\/p>\n

45.
\n\"graph<\/p>\n

49.
\n\"A<\/p>\n

55.\u00a0Standing on the shore of a lake, a fisherman sights a boat far in the distance to his left. Let x<\/em>, measured in radians, be the angle formed by the line of sight to the ship and a line due north from his position. Assume due north is 0 and x<\/em> is measured negative to the left and positive to the right. The boat travels from due west to due east and, ignoring the curvature of the Earth, the distance [latex]d(x)[\/latex], in kilometers, from the fisherman to the boat is given by the function [latex]d(x)=1.5\\sec(x)[\/latex].<\/p>\n

a. What is a reasonable domain for [latex]d(x)[\/latex]?
\nb. Graph d(x) on this domain.
\nc. Find and discuss the meaning of any vertical asymptotes on the graph of [latex]d(x)[\/latex].
\nd. Calculate and interpret [latex]d(\u2212\\frac{\\pi}{3})[\/latex]. Round to the second decimal place.
\ne. Calculate and interpret [latex]d(\\frac{\\pi}{6})[\/latex]. Round to the second decimal place.
\nf. What is the minimum distance between the fisherman and the boat? When does this occur?<\/div>\n

\"An<\/p>\n

57.\u00a0A video camera is focused on a rocket on a launching pad 2 miles from the camera. The angle of elevation from the ground to the rocket after x<\/em> seconds is [latex]\\frac{\\pi}{120}x[\/latex].<\/p>\n

a. Write a function expressing the altitude [latex]h(x)[\/latex], in miles, of the rocket above the ground after x<\/em> seconds. Ignore the curvature of the Earth.
\nb. Graph [latex]h(x)[\/latex] on the interval (0,60).
\nc. Evaluate and interpret the values [latex]h(0)[\/latex] and [latex]h(30)[\/latex].
\nd. What happens to the values of [latex]h(x)[\/latex] as x<\/em> approaches 60 seconds? Interpret the meaning of this in terms of the problem.<\/div>\n

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