Chapter 0

Preparation for Calculus

Educators


Problem 1

Coterminal Angles Explain how to find coterminal angles in degrees.

Amy J.
Numerade Educator

Problem 2

Degrees to Radians Explain how to convert from degrees to radians.

Noah C.
Numerade Educator

Problem 3

Trigonometric Functions
Find $\sin \theta, \cos \theta,$ and $\tan \theta$

Amy J.
Numerade Educator

Problem 4

Characteristics of a Graph In your own words, describe the meaning of amplitude and period.

Noah C.
Numerade Educator

Problem 5

Coterminal Angles in Degrees In Exercises 5 and 6 , determine two coterminal angles in degree measure (one positive and one negative) for each angle.

Amy J.
Numerade Educator

Problem 6

Coterminal Angles in Degrees In Exercises 5 and 6 , determine two coterminal angles in degree measure (one positive and one negative) for each angle.

Noah C.
Numerade Educator

Problem 7

Coterminal Angles in Radians In Exercises 7 and 8 , determine two coterminal angles in radian measure (one positive and one negative) for each angle.

Amy J.
Numerade Educator

Problem 8

Coterminal Angles in Radians In Exercises 7 and 8 , determine two coterminal angles in radian measure (one positive and one negative) for each angle.

Noah C.
Numerade Educator

Problem 9

Degrees to Radians In Exercises 9 and 10, convert the degree measure to radian measure as a multiple of $\pi$ and as a decimal accurate to three decimal places.
$\begin{array}{lll}{\text { (a) }-20^{\circ}} & {\text { (b) }-240^{\circ}} & {\text { (c) }-270^{\circ}} & {\text { (d) } 144^{\circ}}\end{array}$

Amy J.
Numerade Educator

Problem 10

Degrees to Radians In Exercises 9 and 10, convert the degree measure to radian measure as a multiple of $\pi$ and as a decimal accurate to three decimal places.
$\begin{array}{lll}{\text { (a) }-20^{\circ}} & {\text { (b) }-240^{\circ}} & {\text { (c) }-270^{\circ}} & {\text { (d) } 144^{\circ}}\end{array}$

Noah C.
Numerade Educator

Problem 11

Radians to Degrees In Exercises 11 and $12,$ convert the radian measure to degree measure.
$\begin{array}{lll}{\text { (a) } \frac{3 \pi}{2}} & {\text { (b) } \frac{7 \pi}{6}} & {\text { (c) }-\frac{7 \pi}{12}}\end{array} \quad$ (d) $-2.367$

Amy J.
Numerade Educator

Problem 12

Radians to Degrees In Exercises 11 and $12$ convert the radian measure to degree measure.
(a) $\frac{7 \pi}{3} \quad$ (b) $-\frac{11 \pi}{30}$ (c) $\frac{11 \pi}{6} \quad$ (d) 0.438

Noah C.
Numerade Educator

Problem 13

Completing a Table Let $r$ represent the radius of a circle, $\theta$ the central angle (measured in radians), and s the length of the arc subtended by the angle. Use the relationship $s=r \theta$ to complete the table.

Amy J.
Numerade Educator

Problem 14

Angular Speed A car is moving at the rate of 50 miles per hour, and the diameter of its wheels is 2.5 feet.
(a) Find the number of revolutions per minute that the wheels are rotating.
(b) Find the angular speed of the wheels in radians per minute.

Noah C.
Numerade Educator

Problem 15

Evaluating Trigonometric Functions In Exercises 15 and $16,$ evaluate the six trigonometric functions of the angle $\theta$ .

Amy J.
Numerade Educator

Problem 16

Evaluating Trigonometric Functions In Exercises 15 and $16,$ evaluate the six trigonometric functions of the angle $\theta$ .

Noah C.
Numerade Educator

Problem 17

Evaluating Trigonometric Functions In Exercises $17-20$ , sketch a right triangle corresponding to the trigonometric function of the acute angle $\theta$ . Then evaluate the other five trigonometric functions of $\theta$ .
$$\sin \theta=\frac{1}{2}$$

Amy J.
Numerade Educator

Problem 18

Evaluating Trigonometric Functions In Exercises $17-20$ , sketch a right triangle corresponding to the trigonometric function of the acute angle $\theta$ . Then evaluate the other five trigonometric functions of $\theta$.
$$\sin \theta=\frac{1}{3}$$

Noah C.
Numerade Educator

Problem 19

Evaluating Trigonometric Functions In Exercises $17-20$ , sketch a right triangle corresponding to the trigonometric function of the acute angle $\theta$ . Then evaluate the other five trigonometric functions of $\theta$.
$$\cos \theta=\frac{4}{5}$$

Amy J.
Numerade Educator

Problem 20

Evaluating Trigonometric Functions In Exercises $17-20$ , sketch a right triangle corresponding to the trigonometric function of the acute angle $\theta$ . Then evaluate the other five trigonometric functions of $\theta$.
$$\sec \theta=\frac{13}{5}$$

Noah C.
Numerade Educator

Problem 21

Evaluating Trigonometric Functions In Exerrises $21-24$ , evaluate the sine, cosine, and tangent of each angle. Do not use a calculator.
$$\begin{array}{lll}{\text { (a) } 60^{\circ}} & {\text { (b) } 120^{\circ}} & {\text { (c) } \frac{\pi}{4}} & {\text { (d) } \frac{5 \pi}{4}}\end{array}$$

Amy J.
Numerade Educator

Problem 22

Evaluating Trigonometric Functions In Exerrises $21-24$ , evaluate the sine, cosine, and tangent of each angle. Do not use a calculator.
$$\begin{array}{lll}{(a)-30^{\circ}} & {\text { (b) } 150^{\circ}} & {\text { (c) }-\frac{\pi}{6}} & {\text { (d) } \frac{\pi}{2}}\end{array}$$

Noah C.
Numerade Educator

Problem 23

Evaluating Trigonometric Functions In Exerrises $21-24$ , evaluate the sine, cosine, and tangent of each angle. Do not use a calculator.
$$\begin{array}{ll}{\text { (a) } 225^{\circ}} & {\text { (b) }-225^{\circ}} & {\text { (c) } \frac{5 \pi}{3}} & {\text { (d) } \frac{11 \pi}{6}}\end{array}$$

Amy J.
Numerade Educator

Problem 24

Evaluating Trigonometric Functions In Exerrises $21-24$ , evaluate the sine, cosine, and tangent of each angle. Do not use a calculator.
$$\begin{array}{ll}{\text { (a) } 750^{\circ}} & {\text { (b) } 510^{\circ}} & {\text { (c) } \frac{10 \pi}{3}} & {\text { (d) } \frac{17 \pi}{3}}\end{array}$$

Noah C.
Numerade Educator

Problem 25

Evaluating Trigonometric Functions Using Technology In Exercises $25-28$ , use a calculator to evaluate each trigonometric function. Round your answers to four decimal places.
$$\begin{array}{l}{\text { (a) } \sin 10^{\circ}} \\ {\text { (b) } \csc 10^{\circ}}\end{array}$$

Amy J.
Numerade Educator

Problem 26

Evaluating Trigonometric Functions Using Technology
In Exercises $25-28$ , use a calculator to evaluate each trigonometric function. Round your answers to four decimal places.
$$\begin{array}{l}{\text { (a) } \sec 225^{\circ}} \\ {\text { (b) } \sec 135^{\circ}}\end{array}$$

Noah C.
Numerade Educator

Problem 27

Evaluating Trigonometric Functions Using Technology
In Exercises $25-28$ , use a calculator to evaluate each trigonometric function. Round your answers to four decimal places.
$$\begin{array}{l}{\text { (a) } \tan \frac{\pi}{9}} \\ {\text { (b) } \tan \frac{10 \pi}{9}}\end{array}$$

Amy J.
Numerade Educator

Problem 28

Evaluating Trigonometric Functions Using Technology
In Exercises $25-28$ , use a calculator to evaluate each trigonometric function. Round your answers to four decimal places.
$$\begin{array}{l}{\text { (a) } \cot (1.35)} \\ {\text { (b) } \tan (1.35)}\end{array}$$

Noah C.
Numerade Educator

Problem 29

Determining a Quadrant In Exercises 29 and 30 , determine the quadrant in which $\theta$ lies.
$$\begin{array}{l}{\text { (a) } \sin \theta<0 \text { and } \cos \theta<0} \\ {\text { (b) } \sec \theta>0 \text { and } \cot \theta<0}\end{array}$$

Amy J.
Numerade Educator

Problem 30

Determining a Quadrant In Exercises 29 and 30 , determine the quadrant in which $\theta$ lies.
$$\begin{array}{l}{\text { (a) } \sin \theta>0 \text { and } \cos \theta<0} \\ {\text { (b) } \csc \theta<0 \text { and } \tan \theta>0}\end{array}$$

Noah C.
Numerade Educator

Problem 31

Solving a Trigonometric Equation In Exercises $31-34$ , find two solutions of each equation. Give your answers in radians $(0 \leq \theta \leq 2 \pi) .$ Do not use a calculator.
$$\begin{array}{l}{\text { (a) } \cos \theta=\frac{\sqrt{2}}{2}} \\ {\text { (b) } \cos \theta=-\frac{\sqrt{2}}{2}}\end{array}$$

Amy J.
Numerade Educator

Problem 32

Solving a Trigonometric Equation In Exercises $31-34$ , find two solutions of each equation. Give your answers in radians $(0 \leq \theta \leq 2 \pi) .$ Do not use a calculator.
$$\begin{array}{l}{\text { (a) } \sec \theta=2} \\ {\text { (b) } \sec \theta=-2}\end{array}$$

Noah C.
Numerade Educator

Problem 33

Solving a Trigonometric Equation In Exercises $31-34$ , find two solutions of each equation. Give your answers in radians $(0 \leq \theta \leq 2 \pi) .$ Do not use a calculator.
$$\begin{array}{l}{\text { (a) } \tan \theta=1} \\ {\text { (b) } \cot \theta=-\sqrt{3}}\end{array}$$

Amy J.
Numerade Educator

Problem 34

Solving a Trigonometric Equation In Exercises $31-34$ , find two solutions of each equation. Give your answers in radians $(0 \leq \theta \leq 2 \pi) .$ Do not use a calculator.
$$\begin{array}{l}{\text { (a) } \sin \theta=\frac{\sqrt{3}}{2}} \\ {\text { (b) } \sin \theta=-\frac{\sqrt{3}}{2}}\end{array}$$

Noah C.
Numerade Educator

Problem 35

Solving a Trigonometric Equation In Exercises $35-42,$ solve the equation for $\theta,$ where $0 \leq \theta \leq 2 \pi .$
$$2 \sin ^{2} \theta=1$$

Amy J.
Numerade Educator

Problem 36

Solving a Trigonometric
Equation In Exercises $35-42,$ solve the equation for $\theta,$ where $0 \leq \theta \leq 2 \pi .$
$$\tan ^{2} \theta=3$$

Noah C.
Numerade Educator

Problem 37

Solving a Trigonometric
Equation In Exercises $35-42,$ solve the equation for $\theta,$ where $0 \leq \theta \leq 2 \pi .$
$$\tan ^{2} \theta-\tan \theta=0$$

Amy J.
Numerade Educator

Problem 38

Solving a Trigonometric
Equation In Exercises $35-42,$ solve the equation for $\theta,$ where $0 \leq \theta \leq 2 \pi .$
$$2 \cos ^{2} \theta-\cos \theta=1$$

Noah C.
Numerade Educator

Problem 39

Solving a Trigonometric Equation
In Exercises $35-42$ , solve the equation for $\theta,$ where $0 \leq \theta \leq 2 \pi$
$$\sec \theta \csc \theta=2 \csc \theta$$

Amy J.
Numerade Educator

Problem 40

Solving a Trigonometric Equation
In Exercises $35-42$ , solve the equation for $\theta,$ where $0 \leq \theta \leq 2 \pi$
$$\sin \theta=\cos \theta$$

Noah C.
Numerade Educator

Problem 41

Solving a Trigonometric Equation
In Exercises $35-42$ , solve the equation for $\theta,$ where $0 \leq \theta \leq 2 \pi$
$$\cos ^{2} \theta+\sin \theta=1$$

Amy J.
Numerade Educator

Problem 42

Solving a Trigonometric Equation
In Exercises $35-42$ , solve the equation for $\theta,$ where $0 \leq \theta \leq 2 \pi$
$$\cos \frac{\theta}{2}-\cos \theta=1$$

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Problem 43

Airplane Ascent An airplane leaves the runway climbing at an angle of $18^{\circ}$ with a speed of 275 feet per second (see figure). Find the altitude $a$ of the plane after 1 minute.

Amy J.
Numerade Educator

Problem 44

Height of a Mountain While traveling across flat land, you notice a mountain directly in front of you. Its angle of elevation (to the peak) is $3.5^{\circ}$ . After you drive 13 miles closer to the mountain, the angle of elevation is $9^{\circ}$ . Approximate the height of the mountain.

Noah C.
Numerade Educator

Problem 45

Period and Amplitude In Exercises $45-48,$ determine the period and amplitude of each function.
$$y=2 \sin 2 x$$

Amy J.
Numerade Educator

Problem 46

Period and Amplitude In Exercises $45-48,$ determine the period and amplitude of each function.
$$y=\frac{3}{2} \cos \frac{x}{2}$$

Noah C.
Numerade Educator

Problem 47

Period and Amplitude In Exercises $45-48,$ determine the period and amplitude of each function.
$$y=-3 \sin 4 \pi x$$

Amy J.
Numerade Educator

Problem 48

Period and Amplitude In Exercises $45-48,$ determine the period and amplitude of each function.
$$y=\frac{2}{3} \cos \frac{\pi x}{10}$$

Noah C.
Numerade Educator

Problem 49

Period In Exercises $49-52,$ find the period of the function.
$$y=7 \tan 2 \pi x$$

Amy J.
Numerade Educator

Problem 50

Period In Exercises $49-52,$ find the period of the function.
$$y=7 \tan 2 \pi x$$

Noah C.
Numerade Educator

Problem 51

Period In Exercises $49-52,$ find the period of the function.
$$y=\sec 5 x$$

Amy J.
Numerade Educator

Problem 52

Period In Exercises $49-52,$ find the period of the function.
$$y=\csc 4 x$$

Noah C.
Numerade Educator

Problem 53

Writing In Exercises 53 and $54,$ use a graphing utility to graph each function $f$ in the same viewing window for $c=-2$ , $c=-1, c=1,$ and $c=2 .$ Give a written description of the change in the graph caused by changing $c .$
$$\begin{array}{l}{\text { (a) } f(x)=c \sin x} \\ {\text { (b) } f(x)=\cos (c x)} \\ {\text { (c) } f(x)=\cos (\pi x-c)}\end{array}$$

Amy J.
Numerade Educator

Problem 54

Writing In Exercises 53 and $54,$ use a graphing utility to graph each function $f$ in the same viewing window for $c=-2,$ $c=-1, c=1,$ and $c=2 .$ Give a written description of the change in the graph caused by changing $c .$
$$\begin{array}{l}{\text { (a) } f(x)=\sin x+c} \\ {\text { (b) } f(x)=-\sin (2 \pi x-c)} \\ {\text { (c) } f(x)=c \cos x}\end{array}$$

Noah C.
Numerade Educator

Problem 55

Sketching the Graph of a Trigonometric Function In Exercises $55-66,$ sketch the graph of the function.
$$y=\sin \frac{x}{2}$$

Amy J.
Numerade Educator

Problem 56

Sketching the Graph of a Trigonometric Function In Exercises $55-66,$ sketch the graph of the function.
$$y=2 \cos 2 x$$

Noah C.
Numerade Educator

Problem 57

Sketching the Graph of a Trigonometric Function In Exercises $55-66,$ sketch the graph of the function.
$$y=-\sin \frac{2 \pi x}{3}$$

Amy J.
Numerade Educator

Problem 58

Sketching the Graph of a Trigonometric Function In Exercises $55-66,$ sketch the graph of the function.
$$y=2 \tan x$$

Noah C.
Numerade Educator

Problem 59

Sketching the Graph of a Trigonometric Function In Exercises $55-66,$ sketch the graph of the function.
$$y=\csc \frac{x}{2}$$

Amy J.
Numerade Educator

Problem 60

Sketching the Graph of a Trigonometric Function In Exercises $55-66,$ sketch the graph of the function.
$$y=\tan 2 x$$

Noah C.
Numerade Educator

Problem 61

Sketching the Graph of a Trigonometric Function In Exercises $55-66,$ sketch the graph of the function.
$$y=2 \sec 2 x$$

Amy J.
Numerade Educator

Problem 62

Sketching the Graph of a Trigonometric Function In Exercises $55-66,$ sketch the graph of the function.
$$y=\csc 2 \pi x$$

Noah C.
Numerade Educator

Problem 63

Sketching the Graph of a Trigonometric Function In Exercises $55-66,$ sketch the graph of the function.
$$y=\sin (x+\pi)$$

Amy J.
Numerade Educator

Problem 64

Sketching the Graph of a Trigonometric Function In Exercises $55-66,$ sketch the graph of the function.
$$y=\cos \left(x-\frac{\pi}{3}\right)$$

Noah C.
Numerade Educator

Problem 65

Sketching the Graph of a Trigonometric Function In Exercises $55-66,$ sketch the graph of the function.
$$y=1+\cos \left(x-\frac{\pi}{2}\right)$$

Amy J.
Numerade Educator

Problem 66

Sketching the Graph of a Trigonometric Function In Exercises $55-66,$ sketch the graph of the function.
$$y=1+\sin \left(x+\frac{\pi}{2}\right)$$

Noah C.
Numerade Educator

Problem 67

Graphical Reasoning In Exercises 67 and $68,$ find $a, b,$ and $c$ such that the graph of the function matches the graph in the figure.
$$y=a \cos (b x-c)$$

Amy J.
Numerade Educator

Problem 68

Graphical Reasoning In Exercises 67 and $68,$ find $a, b,$ and $c$ such that the graph of the function matches the graph in the figure.
$$y=a \sin (b x-c)$$

Noah C.
Numerade Educator

Problem 69

Think About It You are given the value of tan $\theta$ . Is it possible to find the value of sec $\theta$ without finding the measure of $\theta ?$ Explain.

Amy J.
Numerade Educator

Problem 70

Restricted Domain Explain how to restrict the domain of the sine function so that it becomes a one-to-one function.

Noah C.
Numerade Educator

Problem 71

Think About It How do the ranges of the cosine function and the secant function compare?

Amy J.
Numerade Educator

Problem 72

HOW DO YOU SEE IT? Consider an angle in standard position with $r=12$ centimeters, as shown in the figure. Describe the changes in the values of $x, y, \sin \theta, \cos \theta,$ and $\tan \theta$ as $\theta$ increases continually from $0^{\circ}$ to $90^{\circ} .$

Noah C.
Numerade Educator

Problem 73

Think About It Sketch the graphs of
$$f(x)=\sin x, \quad g(x)=|\sin x|, \quad \text { and } \quad h(x)=\sin (|x|)$$
In general, how are the graphs of $|f(x)|$ and $f(|x|)$ related to the graph of $f ?$

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Problem 74

Ferris wheel. The model for the height $h$ of a Ferris wheel car is $h=51+50 \sin 8 \pi t$
where $t$ is measured in minutes. (The Ferris wheel has a radius of 50 feet.) This model yields a height of 51 feet when $t=0$ . Alter the model so that the height of the car is 1 foot when $t=0$ .

Noah C.
Numerade Educator

Problem 75

Sales The monthly sales $S$ (in thousands of units) of a seasonal product are modeled by
$$S=58.3+32.5 \cos \frac{\pi t}{6}$$
where $t$ is the time (in months), with $t=1$ corresponding to January. Use a graphing utility to graph the model for $S$ and determine the months when sales exceed $75,000$ units.

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Problem 76

Pattern Recognition Use a graphing utility to compare the graph of
$$f(x)=\frac{4}{\pi}\left(\sin \pi x+\frac{1}{3} \sin 3 \pi x\right)$$
with the given graph. Try to improve the approximation by
adding a term to $f(x) .$ Use a graphing utility to verify that
your new approximation is better than the original. Can you
find other terms to add to make the approximation even better?
What is the pattern? (Hint: Use sine terms.)

Noah C.
Numerade Educator

Problem 77

True or False? In Exercises $77-80$ , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.
$$\begin{array}{l}{\text { A measurement of } 4 \text { radians corresponds to two complete }} \\ {\text { revolutions from the initial side to the terminal side of an angle. }}\end{array}$$

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Problem 78

True or False? In Exercises $77-80$ , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.
Amplitude is always positive

Noah C.
Numerade Educator

Problem 79

True or False? In Exercises $77-80$ , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.
$$\begin{array}{l}{\text { The function } y=\frac{1}{2} \sin 2 x \text { has an amplitude that is twice that of }} \\ {\text { the function } y=\sin x .}\end{array}$$

Amy J.
Numerade Educator

Problem 80

True or False? In Exercises $77-80$ , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.
$$\begin{array}{l}{\text { The function } y=3 \cos (x / 3) \text { has a period that is three times }} \\ {\text { that of the function } y=\cos x .}\end{array}$$

Noah C.
Numerade Educator