Graphs of the Sine and Cosine Function

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

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?

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.

7. [latex]f(x)=\frac{2}{3}\cos x[/latex]

9. [latex]f(x)=4\sin x[/latex]

11. [latex]f(x)=\cos(2x)[/latex]

13. [latex]f(x)=4\cos(\pi x)[/latex]

15. [latex]y=3\sin(8(x+4))+5[/latex]

17. [latex]y=5\sin(5x+20)−2[/latex]

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-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.

19. [latex]f(t)=−\cos\left(t+\frac{\pi}{3}\right)+1[/latex]

21. [latex]f(t)=−\sin\left(12t+\frac{5\pi}{3}\right)[/latex]

Determine the amplitude, midline, period, and an equation involving the sine function for the graph shown.

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

31. On [0,2π), solve [latex]f(x)=\frac{1}{2}[/latex].

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

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

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

37. On [0,2π), solve the equation [latex]f(x)=\cos x=0[/latex].

39. On [0,2π), find the x-intercepts of [latex]f(x)=\cos x[/latex].

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

47. A Ferris wheel is 25 meters in diameter and boarded from a platform that is 1 meter above the ground. The six o’clock position on the Ferris wheel is level with the loading platform. The wheel completes 1 full revolution in 10 minutes. The function h(t) gives a person’s height in meters above the ground t minutes after the wheel begins to turn.
a. Find the amplitude, midline, and period of h(t).
b. Find a formula for the height function h(t).
c. How high off the ground is a person after 5 minutes?

Graphs of the Other Trigonometric Functions

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

3. Explain why the period of [latex]\tan x[/latex] is equal to π.

For the following exercises, match each trigonometric function with one of the following graphs.

6. [latex]f(x)=\tan x[/latex]

7. [latex]f(x)=\sec x[/latex]

8. [latex]f(x)=\csc x[/latex]

9. [latex]f(x)=\cot x[/latex]

For the following exercises, find the period and horizontal shift of each of the functions.

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

13. If tan x = −1.5, find tan(−x).

15. If csc x = −5, find csc(−x).

For the following exercises, rewrite each expression such that the argument x is positive.

17. [latex]\cot(−x)\cos(−x)+\sin(−x)[/latex]

For the following exercises, sketch two periods of the graph for each of the following functions. Identify the stretching factor, period, and asymptotes.

19. [latex]f(x)=2\tan(4x−32)[/latex]

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

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

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

27. [latex]f(x)=2\csc(x)[/latex]

29. [latex]f(x)=4\sec(3x)[/latex]

31. [latex]f(x)=7\sec(5x)[/latex]

33. [latex]f(x)=2\csc \left(x+\frac{\pi}{4}\right)−1[/latex]

35. [latex]f(x)=\frac{7}{5}\csc \left(x−\frac{\pi}{4}\right)[/latex]

For the following exercises, find and graph two periods of the periodic function with the given stretching factor, |A|, period, and phase shift.

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]

For the following exercises, find an equation for the graph of each function.

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55. Standing on the shore of a lake, a fisherman sights a boat far in the distance to his left. Let x, 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 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].

57. A 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 seconds is [latex]\frac{\pi}{120}x[/latex].