Our Discord hit 10K members! 🎉 Meet students and ask top educators your questions.Join Here!

Precalculus 7th

David Cohen, Theodore B. Lee, David Sklar

Chapter 8

Graphs of the Trigonometric Functions

Educators


Problem 1

Graph the functions for one period. In each case, specify the amplitude, period, $x$ -intercepts, and interval(s) on which the function is increasing.
(a) $y=2 \sin x$
(b) $y=-\sin 2 x$

Adnan G.
Numerade Educator

Problem 2

Graph the functions for one period. In each case, specify the amplitude, period, $x$ -intercepts, and interval(s) on which the function is increasing.
(a) $y=3 \sin x$
(b) $y=\sin 3 x$

Adnan G.
Numerade Educator

Problem 3

Graph the functions for one period. In each case, specify the amplitude, period, $x$ -intercepts, and interval(s) on which the function is increasing.
(a) $y=\cos 2 x$
(b) $y=2 \cos 2 x$

Adnan G.
Numerade Educator

Problem 4

Graph the functions for one period. In each case, specify the amplitude, period, $x$ -intercepts, and interval(s) on which the function is increasing.
(a) $y=\cos (x / 2)$
(b) $y=-\frac{1}{2} \cos (x / 2)$

Adnan G.
Numerade Educator

Problem 5

Graph the functions for one period. In each case, specify the amplitude, period, $x$ -intercepts, and interval(s) on which the function is increasing.
(a) $y=3 \sin (\pi x / 2)$
(b) $y=-3 \sin (\pi x / 2)$

Adnan G.
Numerade Educator

Problem 6

Graph the functions for one period. In each case, specify the amplitude, period, $x$ -intercepts, and interval(s) on which the function is increasing.
(a) $y=2 \sin \pi x$
(b) $y=-2 \sin \pi x$

Adnan G.
Numerade Educator

Problem 7

Graph the functions for one period. In each case, specify the amplitude, period, $x$ -intercepts, and interval(s) on which the function is increasing.
(a) $y=\cos 2 \pi x$
(b) $y=-4 \cos 2 \pi x$

Adnan G.
Numerade Educator

Problem 8

Graph the functions for one period. In each case, specify the amplitude, period, $x$ -intercepts, and interval(s) on which the function is increasing.
(a) $y=-2 \cos (x / 4)$
(b) $y=-2 \cos (\pi x / 4)$

Adnan G.
Numerade Educator

Problem 9

et the viewing rectangle so that it extends from 0 to $2 \pi$ in the $x$ -direction and from $-4$ to 4 in the $y$ -direction. On the same set of axes, graph the four functions $y=\sin x$ $y=2 \sin x, y=3 \sin x,$ and $y=4 \sin x .$ What is the amplitude in each case? What is the period?

Adnan G.
Numerade Educator

Problem 10

(a)Without using a graphing utility, specify the amplitude and the period for each of the following four functions:
$y=\cos x, y=2 \cos x, y=3 \cos x,$ and $y=4 \cos x$
(b) Check your answers in part (a) by graphing the four functions. (Use the viewing rectangle specified in Exercise 1.)

Adnan G.
Numerade Educator

Problem 11

(a)Without using a graphing utility, specify the amplitude and the period for $y=2 \sin \pi x$ and for $y=\sin 2 \pi x$
(b) Check your answers in part (a) by graphing the two functions. (Use a viewing rectangle that extends from 0 to 2 in the $x$ -direction and from $-2$ to 2 in the $y$ -direction.

Adnan G.
Numerade Educator

Problem 12

Graph the function for one period. Specify the amplitude, period, $x$ -intercepts, and interval(s) on which the function is increasing.
$$y=1+\sin 2 x$$

Adnan G.
Numerade Educator

Problem 13

Graph the function for one period. Specify the amplitude, period, $x$ -intercepts, and interval(s) on which the function is increasing.
$$y=\sin (x / 2)-2$$

Adnan G.
Numerade Educator

Problem 14

Graph the function for one period. Specify the amplitude, period, $x$ -intercepts, and interval(s) on which the function is increasing.
$$y=1-\cos (\pi x / 3)$$

Adnan G.
Numerade Educator

Problem 15

Graph the function for one period. Specify the amplitude, period, $x$ -intercepts, and interval(s) on which the function is increasing.
$$y=-2-2 \cos 3 \pi x$$

Adnan G.
Numerade Educator

Problem 16

Determine the amplitude, period, and phase shift for the given function. Graph the function over one period. Indicate the $x$ -intercepts and the coordinates of the highest and Iowest points on the graph.
$$f(x)=\sin \left(x-\frac{\pi}{6}\right)$$

Adnan G.
Numerade Educator

Problem 17

Determine the amplitude, period, and phase shift for the given function. Graph the function over one period. Indicate the $x$ -intercepts and the coordinates of the highest and Iowest points on the graph.
$$g(x)=\cos \left(x+\frac{\pi}{3}\right)$$

Adnan G.
Numerade Educator

Problem 18

Determine the amplitude, period, and phase shift for the given function. Graph the function over one period. Indicate the $x$ -intercepts and the coordinates of the highest and Iowest points on the graph.
$$F(x)=-\cos \left(x+\frac{\pi}{4}\right)$$

Adnan G.
Numerade Educator

Problem 19

Determine the amplitude, period, and phase shift for the given function. Graph the function over one period. Indicate the $x$ -intercepts and the coordinates of the highest and Iowest points on the graph.
$$G(x)=-\sin (x+2)$$

Adnan G.
Numerade Educator

Problem 20

Determine the amplitude, period, and phase shift for the given function. Graph the function over one period. Indicate the $x$ -intercepts and the coordinates of the highest and Iowest points on the graph.
$$y=\sin \left(2 x-\frac{\pi}{2}\right)$$

Adnan G.
Numerade Educator

Problem 21

Determine the amplitude, period, and phase shift for the given function. Graph the function over one period. Indicate the $x$ -intercepts and the coordinates of the highest and Iowest points on the graph.
$$y=\sin \left(3 x+\frac{\pi}{2}\right)$$

Adnan G.
Numerade Educator

Problem 22

Determine the amplitude, period, and phase shift for the given function. Graph the function over one period. Indicate the $x$ -intercepts and the coordinates of the highest and Iowest points on the graph.
$$y=\cos (2 x-\pi)$$

Adnan G.
Numerade Educator

Problem 23

Determine the amplitude, period, and phase shift for the given function. Graph the function over one period. Indicate the $x$ -intercepts and the coordinates of the highest and Iowest points on the graph.
$$y=\cos \left(x-\frac{\pi}{2}\right)$$

Adnan G.
Numerade Educator

Problem 24

Determine the amplitude, period, and phase shift for the given function. Graph the function over one period. Indicate the $x$ -intercepts and the coordinates of the highest and Iowest points on the graph.
$$y=3 \sin \left(\frac{1}{2} x+\frac{\pi}{6}\right)$$

Adnan G.
Numerade Educator

Problem 25

Determine the amplitude, period, and phase shift for the given function. Graph the function over one period. Indicate the $x$ -intercepts and the coordinates of the highest and Iowest points on the graph.
$$y=-2 \sin (\pi x+\pi)$$

Adnan G.
Numerade Educator

Problem 26

Determine the amplitude, period, and phase shift for the given function. Graph the function over one period. Indicate the $x$ -intercepts and the coordinates of the highest and Iowest points on the graph.
$$y=4 \cos \left(3 x-\frac{\pi}{4}\right)$$

Adnan G.
Numerade Educator

Problem 27

Determine the amplitude, period, and phase shift for the given function. Graph the function over one period. Indicate the $x$ -intercepts and the coordinates of the highest and Iowest points on the graph.
$$y=\cos (x+1)$$

Adnan G.
Numerade Educator

Problem 28

Determine the amplitude, period, and phase shift for the given function. Graph the function over one period. Indicate the $x$ -intercepts and the coordinates of the highest and Iowest points on the graph.
$$y=\frac{1}{2} \sin \left(\frac{\pi x}{2}-\pi^{2}\right)$$

Adnan G.
Numerade Educator

Problem 29

Determine the amplitude, period, and phase shift for the given function. Graph the function over one period. Indicate the $x$ -intercepts and the coordinates of the highest and Iowest points on the graph.
$$y=\cos \left(2 x-\frac{\pi}{3}\right)+1$$

Adnan G.
Numerade Educator

Problem 30

Determine the amplitude, period, and phase shift for the given function. Graph the function over one period. Indicate the $x$ -intercepts and the coordinates of the highest and Iowest points on the graph.
$$y=1-\cos \left(2 x-\frac{\pi}{3}\right)$$

Adnan G.
Numerade Educator

Problem 31

Determine the amplitude, period, and phase shift for the given function. Graph the function over one period. Indicate the $x$ -intercepts and the coordinates of the highest and Iowest points on the graph.
$$y=3 \cos \left(\frac{2 x}{3}+\frac{\pi}{6}\right)$$

Adnan G.
Numerade Educator

Problem 32

(a) Using pencil and paper, not a graphing utility, determine the amplitude, period, and (where appropriate) phase shift for each function.
(b) Use a graphing utility to graph each function for two complete cycles. [In choosing an appropriate viewing rectangle, you will need to use the information obtained in part (a).]
(c) Use the graphing utility to estimate the coordinates of the highest and the lowest points on the graph.
(d) Use the information obtained in part (a) to specify the exact values for the coordinates that you estimated in part (c).
$$y=-2.5 \cos (3 x+4)$$

Adnan G.
Numerade Educator

Problem 33

(a) Using pencil and paper, not a graphing utility, determine the amplitude, period, and (where appropriate) phase shift for each function.
(b) Use a graphing utility to graph each function for two complete cycles. [In choosing an appropriate viewing rectangle, you will need to use the information obtained in part (a).]
(c) Use the graphing utility to estimate the coordinates of the highest and the lowest points on the graph.
(d) Use the information obtained in part (a) to specify the exact values for the coordinates that you estimated in part (c).
$$y=-2.5 \cos (3 \pi x+4)$$

Adnan G.
Numerade Educator

Problem 34

(a) Using pencil and paper, not a graphing utility, determine the amplitude, period, and (where appropriate) phase shift for each function.
(b) Use a graphing utility to graph each function for two complete cycles. [In choosing an appropriate viewing rectangle, you will need to use the information obtained in part (a).]
(c) Use the graphing utility to estimate the coordinates of the highest and the lowest points on the graph.
(d) Use the information obtained in part (a) to specify the exact values for the coordinates that you estimated in part (c).
$$y=-2.5 \cos \left(\frac{1}{3} x+4\right)$$

Adnan G.
Numerade Educator

Problem 35

(a) Using pencil and paper, not a graphing utility, determine the amplitude, period, and (where appropriate) phase shift for each function.
(b) Use a graphing utility to graph each function for two complete cycles. [In choosing an appropriate viewing rectangle, you will need to use the information obtained in part (a).]
(c) Use the graphing utility to estimate the coordinates of the highest and the lowest points on the graph.
(d) Use the information obtained in part (a) to specify the exact values for the coordinates that you estimated in part (c).
$$y=-2.5 \cos \left(\frac{1}{3} \pi x+4\right)$$

Adnan G.
Numerade Educator

Problem 36

(a) Using pencil and paper, not a graphing utility, determine the amplitude, period, and (where appropriate) phase shift for each function.
(b) Use a graphing utility to graph each function for two complete cycles. [In choosing an appropriate viewing rectangle, you will need to use the information obtained in part (a).]
(c) Use the graphing utility to estimate the coordinates of the highest and the lowest points on the graph.
(d) Use the information obtained in part (a) to specify the exact values for the coordinates that you estimated in part (c).
$$y=\sin (0.5 x-0.75)$$

Adnan G.
Numerade Educator

Problem 37

(a) Using pencil and paper, not a graphing utility, determine the amplitude, period, and (where appropriate) phase shift for each function.
(b) Use a graphing utility to graph each function for two complete cycles. [In choosing an appropriate viewing rectangle, you will need to use the information obtained in part (a).]
(c) Use the graphing utility to estimate the coordinates of the highest and the lowest points on the graph.
(d) Use the information obtained in part (a) to specify the exact values for the coordinates that you estimated in part (c).
$$y=\sin (0.5 x+0.75)$$

Adnan G.
Numerade Educator

Problem 38

(a) Using pencil and paper, not a graphing utility, determine the amplitude, period, and (where appropriate) phase shift for each function.
(b) Use a graphing utility to graph each function for two complete cycles. [In choosing an appropriate viewing rectangle, you will need to use the information obtained in part (a).]
(c) Use the graphing utility to estimate the coordinates of the highest and the lowest points on the graph.
(d) Use the information obtained in part (a) to specify the exact values for the coordinates that you estimated in part (c).
$$y=0.02 \cos (100 \pi x-4 \pi)$$

Adnan G.
Numerade Educator

Problem 39

(a) Using pencil and paper, not a graphing utility, determine the amplitude, period, and (where appropriate) phase shift for each function.
(b) Use a graphing utility to graph each function for two complete cycles. [In choosing an appropriate viewing rectangle, you will need to use the information obtained in part (a).]
(c) Use the graphing utility to estimate the coordinates of the highest and the lowest points on the graph.
(d) Use the information obtained in part (a) to specify the exact values for the coordinates that you estimated in part (c).
$$y=0.02 \cos (0.01 \pi x-4 \pi)$$

Adnan G.
Numerade Educator

Problem 40

Determine whether the equation for the graph has the form $y=A \sin B x$ or $y=A \cos B x(\text { with } B>0)$ and then find the values of $A$ and $B$.
(GRAPH CANT COPY)

Adnan G.
Numerade Educator

Problem 41

Determine whether the equation for the graph has the form $y=A \sin B x$ or $y=A \cos B x(\text { with } B>0)$ and then find the values of $A$ and $B$.
(GRAPH CANT COPY)

Adnan G.
Numerade Educator

Problem 42

Determine whether the equation for the graph has the form $y=A \sin B x$ or $y=A \cos B x(\text { with } B>0)$ and then find the values of $A$ and $B$.
(GRAPH CANT COPY)

Adnan G.
Numerade Educator

Problem 43

Determine whether the equation for the graph has the form $y=A \sin B x$ or $y=A \cos B x(\text { with } B>0)$ and then find the values of $A$ and $B$.
(GRAPH CANT COPY)

Adnan G.
Numerade Educator

Problem 44

Determine whether the equation for the graph has the form $y=A \sin B x$ or $y=A \cos B x(\text { with } B>0)$ and then find the values of $A$ and $B$.
(GRAPH CANT COPY)

Adnan G.
Numerade Educator

Problem 45

Determine whether the equation for the graph has the form $y=A \sin B x$ or $y=A \cos B x(\text { with } B>0)$ and then find the values of $A$ and $B$.
(GRAPH CANT COPY)

Adnan G.
Numerade Educator

Problem 46

You are given a table of average monthly temperatures and a scatter plot based on the data. Use the methods of Example 8 to find a periodic function of the form $y=A \sin (B t-C)+D$ whose values approximate the monthly temperatures. The data in these exercises, as well as in Exercise $51,$ are from the Global Historical Climatology Network, and can be accessed on the Internet through the following website created by Robert Hoare.
(GRAPH CANT COPY)
Average Monthly Temperatures for Phoenix, Arizona (based on daily maximums)
$$\begin{array}{lc}\hline \text { Month } & \text { Temperature ('F) } \\\hline \text { Jan. } & 65.5 \\
\text { Feb. } & 70.2 \\\text { Mar. } & 75.2 \\\text { Apr. } & 84.6 \\\text { May } & 93.2 \\\text { June } & 102.7 \\\text { July } & 105.1 \\\text { Aug. } & 103.1 \\\text { Sept. } & 98.6 \\\text { Oct. } & 88.2 \\
\text { Nov. } & 74.7 \\\text { Dec. } & 66.4 \\\hline\end{array}$$

Adnan G.
Numerade Educator

Problem 47

You are given a table of average monthly temperatures and a scatter plot based on the data. Use the methods of Example 8 to find a periodic function of the form $y=A \sin (B t-C)+D$ whose values approximate the monthly temperatures. The data in these exercises, as well as in Exercise $51,$ are from the Global Historical Climatology Network, and can be accessed on the Internet through the following website created by Robert Hoare.
(GRAPH CANT COPY)
Average Monthly Temperatures for Bangor, Maine
$$\begin{array}{lc}\hline \text { Month } & \text { Temperature ('F) } \\\hline \text { Jan. } & 18.0 \\
\text { Feb. } & 20.8 \\\text { Mar. } & 30.7 \\\text { Apr. } & 42.6 \\\text { May } & 54.1 \\
\text { June } & 62.8 \\\text { July } & 68.5 \\\text { Aug. } & 67.3 \\\text { Sept. } & 58.1 \\
\text { Oct. } & 47.1 \\\text { Nov. } & 36.9 \\\text { Dec. } & 23.7 \\\hline\end{array}$$

Adnan G.
Numerade Educator

Problem 48

You are given a table of average monthly temperatures and a scatter plot based on the data. Use the methods of Example 8 to find a periodic function of the form $y=A \sin (B t-C)+D$ whose values approximate the monthly temperatures. The data in these exercises, as well as in Exercise $51,$ are from the Global Historical Climatology Network, and can be accessed on the Internet through the following website created by Robert Hoare.
(GRAPH CANT COPY)
Average Monthly Temperatures for Cape Town, South Africa
$$\begin{array}{lc}\hline \text { Month } & \text { Temperature }\left(^{\circ} \mathrm{F}\right) \\
\hline \text { Jan. } & 69.8 \\\text { Feb. } & 70.0 \\\text { Mar. } & 67.8 \\\text { Apr. } & 63.1 \\
\text { May } & 58.8 \\\text { June } & 55.6 \\\text { July } & 54.3 \\\text { Aug. } & 55.4 \\
\text { Sept. } & 57.7 \\\text { Oct. } & 61.2 \\\text { Nov. } & 64.8 \\\text { Dec. } & 67.8 \\
\hline\end{array}$$

Adnan G.
Numerade Educator

Problem 49

You are given a table of average monthly temperatures and a scatter plot based on the data. Use the methods of Example 8 to find a periodic function of the form $y=A \sin (B t-C)+D$ whose values approximate the monthly temperatures. The data in these exercises, as well as in Exercise $51,$ are from the Global Historical Climatology Network, and can be accessed on the Internet through the following website created by Robert Hoare.
(GRAPH CANT COPY)
Average Monthly Temperatures for Beira, Mozambique
$$\begin{array}{lc}\hline \text { Month } & \text { Temperature }\left(^{\circ} \mathrm{F}\right) \\
\hline \text { Jan. } & 81.3 \\\text { Feb. } & 81.3 \\\text { Mar. } & 80.1 \\\text { Apr. } & 77.5 \\
\text { May } & 73.0 \\\text { June } & 69.4 \\\text { July } & 68.7 \\\text { Aug. } & 70.2 \\
\text { Sept. } & 73.4 \\\text { Oct. } & 76.6 \\\text { Nov. } & 79.0 \\\text { Dec. } & 80.4 \\\hline\end{array}$$

Adnan G.
Numerade Educator

Problem 50

You are given a table of average monthly temperatures and a scatter plot based on the data. Use the methods of Example 8 to find a periodic function of the form $y=A \sin (B t-C)+D$ whose values approximate the monthly temperatures. The data in these exercises, as well as in Exercise $51,$ are from the Global Historical Climatology Network, and can be accessed on the Internet through the following website created by Robert Hoare.
(GRAPH CANT COPY)
The following two scatter plots and table display average temperature data for two locations with very different climates: Dar es Salaam is south of the Equator, in Tanzania, on the Indian Ocean; Tiksi is in Russia, far north of the Arctic Circle, on the Arctic Ocean.
Average Monthly Temperatures for Dar es Salaam, Tanzania, and Tiksi, Russia
(TABLE CANT COPY)
(a) Use the methods of Example 8 to find a periodic function of the form $y=A \sin (B t-C)+D$ whose values approximate the monthly temperatures for Dar es Salaam.
(b) Follow part (a) for Tiksi.
(c) By looking at the scatter plot for temperatures in Dar es Salaam, say if the average rate of change of temperature over the interval from January through July is positive or negative.
(d) Use the table of values to compute the average rate of change of temperature in Dar es Salaam over the interval from January through July. Be sure to include units as part of your answer, and check that the sign is consistent with your response in part (c).
(e) By looking at the scatter plot for temperatures in Tiksi, say if the average rate of change of temperature over the interval from January through July is positive or negative.
(f) Use the table of values to compute the average rate of change of temperature in Tiksi over the interval from January through July. As before, be sure to include units as part of your answer, and check that the sign is consistent with your response in part (e).

Adnan G.
Numerade Educator

Problem 51

The following scatter plot displays average monthly temperatures for Death Valley, California. (The averages were obtained using daily maximums.) Use the methods of Example 8 to find a periodic function of the form $y=A \sin (B t-C)+D$ whose values approximate the monthly temperatures. (No table is given; you will need to rely on the scatter plot and make estimates.)
(TABLE CANT COPY)

Adnan G.
Numerade Educator

Problem 52

In Section 9.2 you'll see the identity $\sin ^{2} x=\frac{1}{2}-\frac{1}{2} \cos 2 x$. Use this identity to graph the function $y=\sin ^{2} x$ for one period.

Adnan G.
Numerade Educator

Problem 53

In Section 9.2 you'll see the identity $\cos ^{2} x=\frac{1}{2}+\frac{1}{2} \cos 2 x$. Use this identity to graph the function $y=\cos ^{2} x$ for one period.

Adnan G.
Numerade Educator

Problem 54

In Section 9.2 we derive the identity $\sin 2 x=2 \sin x \cos x$. Use this to graph $y=\sin x \cos x$ for one period.

Adnan G.
Numerade Educator

Problem 55

In Section 9.2 we derive the identity $\cos 2 x=\cos ^{2} x-\sin ^{2} x .$ Use this to graph $y=\cos ^{2} x-\sin ^{2} x$ for one period.

Adnan G.
Numerade Educator

Problem 56

In Example 2 we showed that the period of $y=\cos 3 x$ is $2 \pi / 3 .$ Use the same method to show that the period of $y=A \cos B x,$ with $B>0,$ is $2 \pi / B$

Adnan G.
Numerade Educator

Problem 57

Adapt the method of Example 6 to determine the amplitude, period, and phase shift for the given function. Graph the function over one period indicating the $x$ -intercepts and the coordinates of the highest and lowest points on the graph.
(a) $y=3 \cos (\pi-2 x)$
(b) $y=\frac{5}{2} \sin (-4 \pi x-\pi)$

Adnan G.
Numerade Educator

Problem 58

( Let $F(x)=\sin x, G(x)=x^{2},$ and $H(x)=x^{3} .$ Which, if any, of the following four composite functions have graphs that do not go below the $x$ -axis? First, try to answer without using a graphing utility, then use the graphing utility to check yourself. (You will learn more this way than if you were to draw the graphs immediately.)
$$\begin{array}{ll}y=G(F(x)) & y=F(G(x)) \\y=F(H(x)) & y=H(F(x))\end{array}$$

Adnan G.
Numerade Educator

Problem 59

(a) Graph the two functions $y=\sin x$ and $y=\sin (\sin x)$ in the standard viewing rectangle. Then for a closer look, switch to a viewing rectangle extending from 0 to $2 \pi$ in the $x$ -direction and from $-1$ to 1 in the $y$ -direction. Compare the two graphs; write out your observations in complete sentences.
(b) Use the graphing utility to estimate the amplitude of the function $y=\sin (\sin x)$
(c) Using your knowledge of the sine function, explain why the amplitude of the function $y=\sin (\sin x)$ is the number sin $1 .$ Then evaluate $\sin 1$ and use the result to check your approximation in part (b).

Adnan G.
Numerade Educator

Problem 60

Let $f(x)=e^{x / 20}(\sin x)$
(a) Graph the function $f$ using a viewing rectangle that extends from $-5$ to 5 in both the $x$ - and the $y$ -directions. Note that the resulting graph resembles a sine curve.
(b) Change the viewing rectangle so that $x$ extends from 0 to 50 and $y$ extends from $-10$ to $10 .$ Describe what you see. Is the function periodic?
(c) Add the graphs of the two functions $y=e^{x / 20}$ and $y=-e^{x / 20}$ to your picture in part (b). Describe what you see.

Adnan G.
Numerade Educator

Problem 61

For this exercise, use the standard viewing rectangle.
(a) Graph the function $y=\ln \left(\sin ^{2} x\right)$
(b) Graph the function $y=\ln (1-\cos x)+\ln (1+\cos x)$
(c) Explain why the two graphs are identical.

Adnan G.
Numerade Educator