Basic Technical Mathematics with Calculus

Allyn J. Washington, Richard S. Evans

Chapter 10

Graphs of the Trigonometric Functions - all with Video Answers

Educators

Section 1

Graphs of y=a sin x and y=a cos x

Problem 1

Graph the function if the given changes are made in the indicated examples of this section.
In Example $2,$ if the sign of the coefficient of $\cos x$ is changed, plot the graph of the resulting function.

Nick Johnson

Nick Johnson

Numerade Educator

Problem 2

Graph the function if the given changes are made in the indicated examples of this section.
In Example $4,$ if the sign of the coefficient of $\sin x$ is changed, display the graph of the resulting function.

Nick Johnson

Numerade Educator

Problem 3

Complete the following table for the given functions and then plot the resulting graphs.
$$\begin{array}{c|c|c|c|c|c|c|c|c|c} x & -\pi & -\frac{3 \pi}{4} & -\frac{\pi}{2} & -\frac{\pi}{4} & 0 & \frac{\pi}{4} & \frac{\pi}{2} & \frac{3 \pi}{4} & \pi \\ \hline y & & & & & & & & & \end{array}$$
$$\begin{array}{c|c|c|c|c|c|c|c|c} x & \frac{5 \pi}{4} & \frac{3 \pi}{2} & \frac{7 \pi}{4} & 2 \pi & \frac{9 \pi}{4} & \frac{5 \pi}{2} & \frac{11 \pi}{4} & 3 \pi \\ \hline y & & & & & & & & \end{array}$$
$$y=\sin x$$

Nick Johnson

Numerade Educator

Problem 4

Complete the following table for the given functions and then plot the resulting graphs.
$$\begin{array}{c|c|c|c|c|c|c|c|c|c} x & -\pi & -\frac{3 \pi}{4} & -\frac{\pi}{2} & -\frac{\pi}{4} & 0 & \frac{\pi}{4} & \frac{\pi}{2} & \frac{3 \pi}{4} & \pi \\ \hline y & & & & & & & & & \end{array}$$
$$\begin{array}{c|c|c|c|c|c|c|c|c} x & \frac{5 \pi}{4} & \frac{3 \pi}{2} & \frac{7 \pi}{4} & 2 \pi & \frac{9 \pi}{4} & \frac{5 \pi}{2} & \frac{11 \pi}{4} & 3 \pi \\ \hline y & & & & & & & & \end{array}$$
$$y=-30 \sin x$$

Carson Merrill

Numerade Educator

Problem 5

Complete the following table for the given functions and then plot the resulting graphs.
$$\begin{array}{c|c|c|c|c|c|c|c|c|c} x & -\pi & -\frac{3 \pi}{4} & -\frac{\pi}{2} & -\frac{\pi}{4} & 0 & \frac{\pi}{4} & \frac{\pi}{2} & \frac{3 \pi}{4} & \pi \\ \hline y & & & & & & & & & \end{array}$$
$$\begin{array}{c|c|c|c|c|c|c|c|c} x & \frac{5 \pi}{4} & \frac{3 \pi}{2} & \frac{7 \pi}{4} & 2 \pi & \frac{9 \pi}{4} & \frac{5 \pi}{2} & \frac{11 \pi}{4} & 3 \pi \\ \hline y & & & & & & & & \end{array}$$
$$y=-3 \cos x$$

Carson Merrill

Numerade Educator

Problem 6

Complete the following table for the given functions and then plot the resulting graphs.
$$\begin{array}{c|c|c|c|c|c|c|c|c|c} x & -\pi & -\frac{3 \pi}{4} & -\frac{\pi}{2} & -\frac{\pi}{4} & 0 & \frac{\pi}{4} & \frac{\pi}{2} & \frac{3 \pi}{4} & \pi \\ \hline y & & & & & & & & & \end{array}$$
$$\begin{array}{c|c|c|c|c|c|c|c|c} x & \frac{5 \pi}{4} & \frac{3 \pi}{2} & \frac{7 \pi}{4} & 2 \pi & \frac{9 \pi}{4} & \frac{5 \pi}{2} & \frac{11 \pi}{4} & 3 \pi \\ \hline y & & & & & & & & \end{array}$$
$$y=-4 \sin x$$

Nick Johnson

Numerade Educator

Problem 7

Give the amplitude and sketch the graphs of the given functions. Check each using a calculator.
$$y=3 \sin x$$

Nick Johnson

Numerade Educator

Problem 8

Give the amplitude and sketch the graphs of the given functions. Check each using a calculator.
$$y=15 \sin x$$

Nick Johnson

Numerade Educator

Problem 9

Give the amplitude and sketch the graphs of the given functions. Check each using a calculator.
$$y=\frac{5}{2} \sin x$$

Nick Johnson

Numerade Educator

Problem 10

Give the amplitude and sketch the graphs of the given functions. Check each using a calculator.
$$y=35 \sin x$$

Nick Johnson

Numerade Educator

Problem 11

Give the amplitude and sketch the graphs of the given functions. Check each using a calculator.
$$y=200 \cos x$$

Nick Johnson

Numerade Educator

Problem 12

Give the amplitude and sketch the graphs of the given functions. Check each using a calculator.
$$y=0.25 \cos x$$

Nick Johnson

Numerade Educator

Problem 13

Give the amplitude and sketch the graphs of the given functions. Check each using a calculator.
$$y=0.8 \cos x$$

Nick Johnson

Numerade Educator

Problem 14

Give the amplitude and sketch the graphs of the given functions. Check each using a calculator.
$$y=\frac{3}{2} \cos x$$

Nick Johnson

Numerade Educator

Problem 15

Give the amplitude and sketch the graphs of the given functions. Check each using a calculator.
$$y=-\sin x$$

Nick Johnson

Numerade Educator

Problem 16

Give the amplitude and sketch the graphs of the given functions. Check each using a calculator.
$$y=-30 \sin x$$

Nick Johnson

Numerade Educator

Problem 17

Give the amplitude and sketch the graphs of the given functions. Check each using a calculator.
$$y=-1500 \sin x$$

Nick Johnson

Numerade Educator

Problem 18

Give the amplitude and sketch the graphs of the given functions. Check each using a calculator.
$$y=-0.2 \sin x$$

Nick Johnson

Numerade Educator

Problem 19

Give the amplitude and sketch the graphs of the given functions. Check each using a calculator.
$$y=-4 \cos x$$

Nick Johnson

Numerade Educator

Problem 20

Give the amplitude and sketch the graphs of the given functions. Check each using a calculator.
$$y=-8 \cos x$$

Nick Johnson

Numerade Educator

Problem 21

Give the amplitude and sketch the graphs of the given functions. Check each using a calculator.
$$y=-50 \cos x$$

Nick Johnson

Numerade Educator

Problem 22

Give the amplitude and sketch the graphs of the given functions. Check each using a calculator.
$$y=-0.4 \cos x$$

Nick Johnson

Numerade Educator

Problem 23

Although units of $\pi$ are convenient, we must remember that $\pi$ is only a number. Numbers that are not multiples of $\pi$ may be used. In Exercises $23-26,$ plot the indicated graphs by finding the values of $y$ that correspond to values of $x$ of $0,1,2,3,4,5,6,$ and 7 on a calculator. (Remember, the numbers $0,1,2,$ and so on represent radian measure.)
$$y=\sin x$$

Nick Johnson

Numerade Educator

Problem 24

Although units of $\pi$ are convenient, we must remember that $\pi$ is only a number. Numbers that are not multiples of $\pi$ may be used. In Exercises $23-26,$ plot the indicated graphs by finding the values of $y$ that correspond to values of $x$ of $0,1,2,3,4,5,6,$ and 7 on a calculator. (Remember, the numbers $0,1,2,$ and so on represent radian measure.)
$$y=-30 \sin x$$

Nick Johnson

Numerade Educator

Problem 25

Although units of $\pi$ are convenient, we must remember that $\pi$ is only a number. Numbers that are not multiples of $\pi$ may be used. In Exercises $23-26,$ plot the indicated graphs by finding the values of $y$ that correspond to values of $x$ of $0,1,2,3,4,5,6,$ and 7 on a calculator. (Remember, the numbers $0,1,2,$ and so on represent radian measure.)
$$y=-12 \cos x$$

Carson Merrill

Numerade Educator

Problem 26

Although units of $\pi$ are convenient, we must remember that $\pi$ is only a number. Numbers that are not multiples of $\pi$ may be used. In Exercises $23-26,$ plot the indicated graphs by finding the values of $y$ that correspond to values of $x$ of $0,1,2,3,4,5,6,$ and 7 on a calculator. (Remember, the numbers $0,1,2,$ and so on represent radian measure.)
$$y=2 \cos x$$

Nick Johnson

Numerade Educator

Problem 27

Solve the given problems.
Find the function and graph it for a function of the form $y=a \sin x$ that passes through $(\pi / 2,-2)$.

Nick Johnson

Numerade Educator

Problem 28

Solve the given problems.
Find the function and graph it for a function of the form $y=a \sin x$ that passes through $(3 \pi / 2,-2)$.

Nick Johnson

Numerade Educator

Problem 29

Solve the given problems.
The height (in $\mathrm{ft}$ ) of a person on a ferris wheel above its center is given by $h=-32 \cos t$ where $t$ is the time (in $\mathrm{s}$ ). Graph one complete cycle of this function.

Carson Merrill

Numerade Educator

Problem 30

Solve the given problems.
The horizontal displacement $d$ (in $\mathrm{m}$ ) of the bob on a large pendulum is $d=5 \sin t,$ where $t$ is the time (in $\mathrm{s}$ ). Graph two cycles of this function.

Carson Merrill

Numerade Educator

Problem 31

Solve the given problems.
The graph displayed on an oscilloscope can be represented by $y=-0.05 \sin x .$ Display this curve on a calculator.

Carson Merrill

Numerade Educator

Problem 32

Solve the given problems.
The displacement $y$ (in $\mathrm{cm}$ ) of the end of a robot arm for welding is $y=4.75 \cos t,$ where $t$ is the time (in $\mathrm{s}$ ). Display this curve on a calculator.

Carson Merrill

Numerade Educator

Problem 33

The graph of a function of the form $y=a \sin x$ or $y=a \cos x$ is shown. Determine the specific function of each. (GRAPH CAN'T COPY)

Carson Merrill

Numerade Educator

Problem 34

The graph of a function of the form $y=a \sin x$ or $y=a \cos x$ is shown. Determine the specific function of each. (GRAPH CAN'T COPY)

Carson Merrill

Numerade Educator

Problem 35

The graph of a function of the form $y=a \sin x$ or $y=a \cos x$ is shown. Determine the specific function of each. (GRAPH CAN'T COPY)

Carson Merrill

Numerade Educator

Problem 36

The graph of a function of the form $y=a \sin x$ or $y=a \cos x$ is shown. Determine the specific function of each. (GRAPH CAN'T COPY)

Carson Merrill

Numerade Educator

Problem 37

Each ordered pair given is located on the graph of either $y=a \sin x$ or $y=a \cos x$ where the amplitude is $|a|=2.50$ Use the trace feature on a calculator to determine the equation of the function that contains the given point. (All points are located such that the $x$ value is between $-\pi$ and $\pi .$ )
$$(0.67,-1.55)$$

Carson Merrill

Numerade Educator

Problem 38

Each ordered pair given is located on the graph of either $y=a \sin x$ or $y=a \cos x$ where the amplitude is $|a|=2.50$ Use the trace feature on a calculator to determine the equation of the function that contains the given point. (All points are located such that the $x$ value is between $-\pi$ and $\pi .$ )
$$(-1.20,0.90)$$

Carson Merrill

Numerade Educator

Problem 39

Each ordered pair given is located on the graph of either $y=a \sin x$ or $y=a \cos x$ where the amplitude is $|a|=2.50$ Use the trace feature on a calculator to determine the equation of the function that contains the given point. (All points are located such that the $x$ value is between $-\pi$ and $\pi .$ )
$$(2.07,1.20)$$

Carson Merrill

Numerade Educator

Problem 40

Each ordered pair given is located on the graph of either $y=a \sin x$ or $y=a \cos x$ where the amplitude is $|a|=2.50$ Use the trace feature on a calculator to determine the equation of the function that contains the given point. (All points are located such that the $x$ value is between $-\pi$ and $\pi .$ )
$$(-2.47,-1.55)$$

Carson Merrill

Numerade Educator