Trigonometry A GRAPHING APPROACH WIT H TECHNOLOGY UPDATES

Roland E. Larson Robert P. Hostetler THE PENNSYLVANIA STATE UNIVERSITY

THE BEHREND COLLEGE

Bruce H. Edwards UNIVERSITY OF FLORIDA

WITH THE ASSISTANCE OF

David E. Heyd

THE PENNSYLVANIA STATE UNIVERSITY

THE BEHREND COLLEGE

Houghton Mifflin Company Boston New York

Sponsoring Editor: Christine B. Hoag

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Technical Art: Folium

Trademark Acknowledgments: TI is a registered trademark of Texas Instruments Incorporated. Casio is a registered trademark of Casio, Inc. Sharp is a registered trademark of Sharp Electronics Corporation. Hewlett Packard is a registered Trademark.

Art and Photo Credits: p. 327, from The Fractal Geometry ofNature by Benoit B. Mandelbrot, by permission; p. 328, from Chaos and Fractals: The Mathematics Behind the Computer Graphics, edited by Robert L. Devaney and Linda Keen. Copyright 1989 by the American Mathematical Society. All rights reserved. Used by permission of the publisher.

We have included examples and exercises that use real-life data. This would not have been possible without the help of many people and organizations. Our wholehearted thanks goes to all for their time and effort.

Copyright 1997 by Houghton Mifflin Company. All rights reserved.

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Printed in the U.S.A.

ISBN: 0-669-41760-2

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CONTENTS

INTRODUCTION TO CALCULATORS XXVI

READING A MATHEMATICS TEXT XXIX

WRITI NG XXXI

Chapter I PREREQUISITES FOR TRIGONOMETRY

2 3 Quarter 1.7 GRAPHS OF FUNCTIONS 63

1.1 THE REAL NUMBER SYSTEM

1.2 SOLVING EQUATIONS 8

1.3 THE CARTESIAN PLANE 18

1.4 GRAPHS AND GRAPHING UTILITIES 25

1.5 LINES IN THE PLANE AND SLOPE 36

1.6 FUNCTIONS 50

1.8 SHIFTING, REFLECTING, AND STRETCHING

GRAPHS 74

1.9 COMBINATIONS OF FUNCTIONS 83

1.10 INVERSE FUNCTIONS 92

REVIEW EXERCISES 101

I

XXI

XXII CON TEN T 5

Chapter 2 TRIGONOMETRY 105

2.1 RADIAN AND DEGREE MEASURE 105

2.2 THE TRIGONOMETRIC FUNCTIONS AND THE UNIT

CIRCLE 118

2.3 TRIGONOMETRIC FUNCTIONS AND RIGHT

TRIANGLES 127

2.4 TRIGONOMETRIC FUNCTIONS OF ANY

ANGLE 139

2.5 GRAPHS OF SINE AND COSINE

FUNCTIONS 150

2.6 OTHER TRIGONOMETRIC GRAPHS 162

2.7 INVERSE TRIGONOMETRIC FUNCTIONS 175

2.8 ApPLICATIONS OF TRIGONOMETRY 186

REVIEW EXERCISES 199

CUMULATIVE TEST FOR CHAPTERS 1-2 202

Chapter 3 ANALYTIC TRIGONOMETRY 203

3.1 ApPLICATIONS OF FUNDAMENTAL

IDENTITIES 203

3.2 VERIFYING TRIGONOMETRIC

IDENTITIES 212

3.3 SOLVING TRIGONOMETRIC IDENTITIES 219

3.4 SUM AND DIFFERENCE FORMULAS 231

-3 3.5 MULTIPLE-ANGLE AND PRODUCT-TO-SUM

FORMULAS 239

REVIEW EXERCISES 250

253

CON TEN T 5 XXIII

Chapter 4 ADDITIONAL TOPICS IN TRIGONOMETRY

VECTORS 310

REVIEW EXERCISES 317

Chapter 5 COMPLEX NUMBERS

Imaginary 5.1 COMPLEX NUMBERS 321

axis

5.2 COMPLEX SOLUTION OF EQUATIONS 330

5.3 TRIGONOMETRIC FORM OF A

COMPLEX NUMBER 336

5.4 DEMoIVRE'S THEOREM AND-_-j-t-----+.----+--- Realaxis

NTH ROOTS 344

REVIEW EXERCISES 350

CUMULATIVE TEST FOR CHAPTERS 3-5 351

2 2 2

321

XXIV CON TEN T S

Chapter 6 EXPONENTIAL AND LOGARITH)lIC FrNCTIONS 353

6.1 EXPONENTIAL FUNCTIONS AND THEIR

GRAPHS 353

6.2 LOGARITHMIC FUNCTIONS AND THEIR

GRAPHS 365

6.3 PROPERTIES OF LOGARITHMS 375

6.4 SOLVING EXPONENTIAL AND LOGARITHMIC

EQUATIONS 383

6.5 ApPLICATIONS OF EXPONENTIAL AND

LOGARITHMIC FUNCTIONS 392

6.6 NONLINEAR MODELS 404

REVIEW EXERCISES 413

Chapter 7 SOME TOPICS IN ANALYTIC GEOMETRY

700r---------------~

600

500

2 4 6 8 10 12 J4 Year (1970 - 0)

7.1 LINES 417

7.2 INTRODUCTION TO CON ICS:

PARABOLAS 425

7.3 ELLIPSES 434

7.4 HYPERBOLAS 443

7.5 ROTATION AND SYSTEMS OF

QUADRATIC EQUATIONS 452

7.6 PLANE CURVES AND PARAMETRIC

CURVES 461

7.7 POLAR COORDI NATES 470

7.8 GRAPHS OF POLAR EQUATIONS 478

7.9 POLAR EQUATIONS OF CONICS 487

REVIEW EXERCISES 496

CUMULATIVE TEST FOR CHAPTERS 6-7 498

417

xxv CON TEN T S

APPENDIXES

APPENDIX A: GRAPHING UTILITIES AI

ApPENDIX B: PROGRAMS All

ApPENDIX C: SERIES AND TRIGONOMETRIC FUNCTIONS A39

ApPENDIX D: ADDITIONAL PROBLEM SOLVING WITH TECHNOLOGY A53

ANSWERS

WARM-UPS, ODD-NuMBERED EXERCISES, CUMULATIVE TESTS A71

INDEXES

INDEX OF ApPLICATIONS A139

INDEX A143

CHAPTER 1

1.1 THE REAL NUMBER SYSTEM

1.2 SOLVING EQUATIONS

1.3 THE CARTESIAN PLANE

1.4 GRAPHS AND GRAPHING UTILITIES

1.5 LINES IN THE PLANE AND SLOPE

1.6 FUNCTIONS

1.7 GRAPHS OF FUNCTIONS PREREQUISITES 1.8 SHIFTING, REFLECTING,

AND STRETCHING

GRAPHS FOR 1.9 COMBINATIONS OF

FUNCTIONS

1.10 INVERSE FUNCTIONS TRIGONOMETRY

1.1 THE REAL NUMBER SYSTEM The Real Number System I The Real Number Line I Ordering the Real Numbers I The Absolute Value of a Real Number I The Distance Between Two Real Numbers

The Real Number System

The algebraic techniques that are reviewed in this chapter will be used in the remaining chapters of the text. A clear understanding of the definitions, strategies. and concepts presented here will help you as you study trigonometry. Real numbers are used in everyday life to describe quantities such as age, miles per gallon, container size, population, and so on. To represent real numbers we use symbols such as

9, 5, 0, 43" 0.6666 ... , 28.21, V2, 1T, and

The set of real numbers contains some important subsets with which you should be familiar:

{1,2,3,4, ...} Set of nalllral numbers

{O. I, 2, 3, 4, ...} Set of whole numbers

{... , -3, -2, -1,0,1,2,3, ...} Sel of integers

2 CHAPTER 1 PREREQUISITES FOR TRIGONOMETRY

A real number is rational if it can be written as the ratio p/q of two integers. where q * 0. For instance, the numbers

I I 125 "3 == 0.3333 . . .. 8 0.125. and 1.126126 ...

III

are rational. The decimal representation of a rational number either repeats (as in 3.1454545 ... ) or terminates (as in ~ == 0.5). A real number that cannot be written as the ratio of two integers is called irrational. Irrational numbers have infinite l10nrepeafing decimal representations. For instance, the numbers

V2 = 1.4142135 . .. and 11 = 3.1415926 ... are irrational. (The symbol means "is approximately equal to.")

EXAMPLE 1 Identifying Real Numbers

Consider the following subset of real numbers:

I r:: }{-8,-Vs'I'~'0, 7' V 3, 11,9 . List the numbers in this set that are

A. Natural numbers B. Integers C. Rational numbers D. Irrational numbers

SOLUTION

A. Natural numbers: I, 9

B. Integers: -8, I, 0, 9 2

C. Rational numbers: 8, I, 3 ,0, -:;' 9

D. Irrational numbers: vi 11 " ". " """" .

The Real Number Line

The model used to represent the real number system is called the real number line. It consists of a horizontal line with a point (the origin) labeled 0. Numbers to the right ofare positive, and numbers to the left of are negative, as shown in Figure 1.1. We use the term nonnegative to describe a number that is either positive or zero.

OriginNegative Positive

direction direction

The Real Number Line

FIGURE 1,1

One-to-One Correspondence

0.75

n8 U 1 - I -I 1 -3 -2 o 1 2 3

(a) Every real number corresponds to exactly one point on the real number line.

-2.4 Vi

o o 1 - 1 1 1 - 1-3 -2 -I o 1 2

(b) Every point on the real number line corresponds to exactly one real number.

FIGURE t.2

a 1

-1 o 2

a < b if and only if a lies to the left of b.

FIGURE 1.3

x::52

III ] x 0 2 4

(a)

-2 ::5 x < 3 x[ )

-2 o 2 3

(b)

x> -5 I I .. x

-7 -6 -5 -4 -3

(c)

FIGURE 1.4

SECTION 1.1 THE REAL NUMBER SYSTEM 3

Each point on the real number line corresponds to one and only one real number and each real number corresponds to one and only one point on the real number line. This type of relationship is called a one-to-one correspondence, as shown in Figure 1.2.

The number associated with a point on the real number line is called the coordinate of the point. For example, in Figure I.2(a), - ~ is the coordinate of the leftmost point and 7T is the coordinate of the righ