College Algebra with Trigonometry

896 Pages·086.045 MB·English

by  Raymond A. Barnett, Michael R. Ziegler, Karl E. Byleen, Dave Sobecki

bar19502_fm_i-xxxii.qxd 12/10/09 11:47 PM Page iii
NINTH EDITION
College Algebra
with Trigonometry
Raymond A. Barnett
Merritt College
Michael R. Ziegler
Marquette University
Karl E. Byleen
Marquette University
Dave Sobecki
Miami University Hamilton
bar19502_fm_i-xxxii.qxd 12/10/09 11:47 PM Page iv
COLLEGE ALGEBRA WITH TRIGONOMETRY, NINTH EDITION
Published by McGraw-Hill, a business unit of The McGraw-Hill Companies, Inc., 1221 Avenue of the Americas,
New York, NY 10020. Copyright © 2011 by The McGraw-Hill Companies, Inc. All rights reserved. Previous
editions © 2008, 2001, and 1999. No part of this publication may be reproduced or distributed in any form or
by any means, or stored in a database or retrieval system, without the prior written consent of The McGraw-Hill
Companies, Inc., including, but not limited to, in any network or other electronic storage or transmission, or
broadcast for distance learning.
Someancillaries,includingelectronicandprintcomponents,maynotbeavailabletocustomersoutsidetheUnitedStates.
This book is printed on acid-free paper.
1 2 3 4 5 6 7 8 9 0 DOW/DOW 1 0 9 8 7 6 5 4 3 2 1 0
ISBN 978–0–07–351950–0
MHID 0–07–351950–2
ISBN 978–0–07–729720–6 (Annotated Instructor’s Edition)
MHID 0–07–729720–2
Vice President, Editor-in-Chief: Marty Lange
Vice President, EDP: Kimberly Meriwether David
Editorial Director: Stewart K. Mattson
Sponsoring Editor: John R. Osgood
Director of Development: Kristine Tibbetts
Developmental Editor: Christina A. Lane
Marketing Manager: Kevin M. Ernzen
Lead Project Manager: Sheila M. Frank
Senior Production Supervisor: Kara Kudronowicz
Senior Media Project Manager: Sandra M. Schnee
Designer: Tara McDermott
Cover/Interior Designer: Ellen Pettergell
(USE) Cover Image: © Comstock Images/Getty Images
Senior Photo Research Coordinator: Lori Hancock
Supplement Producer: Mary Jane Lampe
Compositor: Aptara®, Inc.
Typeface: 10/12 Times Roman
Printer: R. R. Donnelley
All credits appearing on page or at the end of the book are considered to be an extension of the copyright page.
Chapter R Opener: © Corbis RF; p. 31: © The McGraw-Hill Companies, Inc./John Thoeming photographer.
Chapter 1 Opener: © Corbis RF; p.56: © Vol. 71/Getty RF; p.92: © Getty RF. Chapter 2 Opener: © Vol.
88/Getty RF; p.142:© Big Stock Photo; p.147:© Corbis RF; p.151:© Vol. 112/Getty RF. Chapter 3 Opener,
p.170:© Getty RF; p.187:© Vol. 88/Getty RF; p.220:© Corbis RF; p.250:© The McGraw-Hill Companies,
Inc./Andrew Resek photographer. Chapter 4 Opener,p.271:© Corbis RF; p.272:© Vol. 4/Getty RF. Chapter
5 Opener: © Getty RF; p.333:© Vol. 68/Getty RF; p.345: © Corbis RF. Chapter 6 Opener: © Corbis RF;
p.395:© Digital Vision Punchstock/RF; p.441:© Corbis RF; p.460(left):© Vol. DV289/Getty RF; p.460(right):
Vol. 44/Getty RF. Chapter 7 Opener:© Big Stock Photos; p.479:© Corbis RF; p.492:© Jacqui Hurst/Corbis.
Chapter 8 Opener:© Digital Vision Punchstock/RF; p.553:© Big Stock Photos. Chapter 9 Opener:© Brand
X/Superstock RF; p.587:© California Institute of Technology. Chapter 10 Opener:© Corbis RF; p.637:Cour-
tesy of Bill Tapenning, USDA; p.641:© Vol. 5/Getty RF; p.658:© Vol. 48/Getty RF; p.662:© Getty RF. Chap-
ter 11 Opener:© Vol. 6/Getty RF; p.733:© ThinkStock/PictureQuest RF; p.745:© Corbis RF.
Library of Congress Cataloging-in-Publication Data
Barnett, Raymond A.
College algebra with trigonometry/Raymond A. Barnett ... [et al.]. — 9th ed.
p. cm. — (Barnett, Ziegler & Byleen’s precalculus series)
Includes index.
ISBN 978-0-07-351950-0 — ISBN 0-07-351950-2 (hard copy : alk. paper) 1. Algebra–Textbooks.
2. Trigonometry–Textbooks. I. Title
QA154.3.B368 2011
512.13–dc22
2009019473
www.mhhe.com
bar19502_fm_i-xxxii.qxd 12/10/09 11:47 PM Page vii
About the Authors
Raymond A. Barnett, a native of and educated in California, received his B.A. in math-
ematical statistics from the University of California at Berkeley and his M.A. in mathe-
matics from the University of Southern California. He has been a member of the Merritt
College Mathematics Department and was chairman of the department for four years. Asso-
ciated with four different publishers, Raymond Barnett has authored or co-authored 18 text-
books in mathematics, most of which are still in use. In addition to international English
editions, a number of the books have been translated into Spanish. Co-authors include
Michael Ziegler, Marquette University; Thomas Kearns, Northern Kentucky University;
Charles Burke, City College of San Francisco; John Fujii, Merritt College; Karl Byleen,
Marquette University; and Dave Sobecki, Miami University Hamilton.
Michael R. Ziegler received his B.S. from Shippensburg State College and his M.S. and
Ph.D. from the University of Delaware. After completing postdoctoral work at the Univer-
sity of Kentucky, he was appointed to the faculty of Marquette University where he held
the rank of Professor in the Department of Mathematics, Statistics, and Computer Science.
Dr. Ziegler published more than a dozen research articles in complex analysis and co-authored
more than a dozen undergraduate mathematics textbooks with Raymond Barnett and Karl
Byleen before passing away unexpectedly in 2008.
Karl E. Byleen received his B.S., M.A., and Ph.D. degrees in mathematics from the Uni-
versity of Nebraska. He is currently an Associate Professor in the Department of Mathe-
matics, Statistics, and Computer Science of Marquette University. He has published a dozen
research articles on the algebraic theory of semigroups and co-authored more than a dozen
undergraduate mathematics textbooks with Raymond Barnett and Michael Ziegler.
Dave Sobecki earned a B.A. in math education from Bowling Green State University, then
went on to earn an M.A. and a Ph.D. in mathematics from Bowling Green. He is an asso-
ciate professor in the Department of Mathematics at Miami University in Hamilton, Ohio.
He has written or co-authored five journal articles, eleven books and five interactive
CD-ROMs. Dave lives in Fairfield, Ohio with his wife (Cat) and dogs (Macleod and Tessa).
His passions include Ohio State football, Cleveland Indians baseball, heavy metal music, travel,
and home improvement projects.
vii
bar19502_fm_i-xxxii.qxd 12/10/09 11:47 PM Page ix
Dedicated to the memory of Michael R. Ziegler,
trusted author, colleague, and friend.
bar19502_fm_i-xxxii.qxd 12/10/09 11:47 PM Page xi
Brief Contents
Preface xiv
Features xvii
Application Index xxviii
R
CHAPTER Basic Algebraic Operations 1
1
CHAPTER Equations and Inequalities 43
2
CHAPTER Graphs 109
3
CHAPTER Functions 161
4
CHAPTER Polynomial and Rational Functions 259
5
CHAPTER Exponential and Logarithmic Functions 327
6
CHAPTER Trigonometric Functions 385
7
CHAPTER Trigonometric Identities and Conditional Equations 461
8
CHAPTER Additional Topics in Trigonometry 509
9
CHAPTER Additional Topics in Analytic Geometry 571
10
CHAPTER Systems of Equations and Matrices 625
11
CHAPTER Sequences, Induction, and Probability 705
Appendix A Cumulative Review Exercises A1
Appendix B Special Topics A17
Appendix C Geometric Formulas A37
Student Answers SA1 Instructor Answers IA1
Subject Index I1
xi
bar19502_fm_i-xxxii.qxd 12/10/09 11:47 PM Page xii
Contents
Preface xiv 4
CHAPTER Polynomial and Rational
Features xvii
Functions 259
Applications Index xxviii
4-1 Polynomial Functions, Division,
and Models 260
4-2 Real Zeros and Polynomial
R
CHAPTER Basic Algebraic Inequalities 278
Operations 1 4-3 Complex Zeros and Rational Zeros of
R-1 Algebra and Real Numbers 2 Polynomials 288
R-2 Exponents and Radicals 11 4-4 Rational Functions and Inequalities 298
R-3 Polynomials: Basic Operations and Factoring 21 4-5 Variation and Modeling 315
R-4 Rational Expressions: Basic Operations 32 Chapter 4 Review 321
Chapter R Review 39 Chapter 4 Review Exercises 323
Chapter R Review Exercises 40 Chapter 4 Group Activity:
Interpolating Polynomials 326
1
CHAPTER Equations and
5
Inequalities 43 CHAPTER Exponential and Logarithmic
1-1 Linear Equations and Applications 44 Functions 327
1-2 Linear Inequalities 56 5-1 Exponential Functions 328
1-3 Absolute Value in Equations and Inequalities 65 5-2 Exponential Models 340
1-4 Complex Numbers 74 5-3 Logarithmic Functions 354
1-5 Quadratic Equations and Applications 84 5-4 Logarithmic Models 365
1-6 Additional Equation-Solving Techniques 97 5-5 Exponential and Logarithmic Equations 372
Chapter 1 Review 104 Chapter 5 Review 379
Chapter 1 Review Exercises 106 Chapter 5 Review Exercises 380
Chapter 1 Group Activity: Solving a Cubic Chapter 5 Group Activity: Comparing
Equation 108 Regression Models 383
2 6
CHAPTER Graphs 109 CHAPTER Trigonometric Functions 385
2-1 Cartesian Coordinate Systems 110 6-1 Angles and Their Measure 386
2-2 Distance in the Plane 122 6-2 Right Triangle Trigonometry 395
2-3 Equation of a Line 132 6-3 Trigonometric Functions: A Unit Circle
2-4 Linear Equations and Models 147 Approach 404
Chapter 2 Review 157 6-4 Properties of Trigonometric Functions 414
Chapter 2 Review Exercises 158 6-5 More General Trigonometric Functions
Chapter 2 Group Activity: Average Speed 160 and Models 428
6-6 Inverse Trigonometric Functions 441
Chapter 6 Review 453
CHAPTER 3 Functions 161 Chapter 6 Review Exercises 456
3-1 Functions 162 Chapter 6 Group Activity: A Predator–Prey
3-2 Graphing Functions 175 Analysis Involving Mountain Lions
3-3 Transformations of Functions 188 and Deer 460
3-4 Quadratic Functions 203
3-5 Operations on Functions; Composition 223
7
3-6 Inverse Functions 235 CHAPTER Trigonometric Identities and
Chapter 3 Review 250 Conditional Equations 461
Chapter 3 Review Exercises 252 7-1 Basic Identities and Their Use 462
Chapter 3 Group Activity: 7-2 Sum, Difference, and Cofunction Identities 471
Mathematical Modeling: Choosing a 7-3 Double-Angle and Half-Angle Identities 480
Cell Phone Plan 257 7-4 Product–Sum and Sum–Product Identities 488
xii
bar19502_fm_i-xxxii.qxd 12/10/09 11:47 PM Page xiii
7-5 Trigonometric Equations 493 10-3 Matrix Operations 659
Chapter 7 Review 504 10-4 Solving Systems of Linear Equations Using Matrix
Chapter 7 Review Exercises 505 Inverse Methods 672
Chapter 7 Group Activity: From M sin Bt ! N 10-5 Determinants and Cramer’s Rule 689
cos Bt to A sin (Bt ! C)—A Harmonic Analysis
Tool 507 Additional Topics Available Online:
(Visit www.mhhe.com/barnett)
10-6 Systems of Nonlinear Equations
8
CHAPTER Additional Topics in 10-7 System of Linear In Equalities in Two Variables
Trigonometry 509 10-8 Linear Programming
8-1 Law of Sines 510 Chapter 10 Review 698
8-2 Law of Cosines 519 Chapter 10 Review Exercises 700
8-3 Vectors in the Plane 527 Chapter 10 Group Activity: Modeling with Systems
8-4 Polar Coordinates and Graphs 540 of Linear Equations 703
8-5 Complex Numbers and De Moivre’s Theorem 553
Chapter 8 Review 563
Chapter 8 Review Exercises 567 11
CHAPTER Sequences, Induction,
Chapter 8 Group Activity: Polar Equations of Conic
and Probability 705
Sections 570
11-1 Sequences and Series 706
11-2 Mathematical Induction 713
CHAPTER 9 Additional Topics in Analytic 11-3 Arithmetic and Geometric Sequences 722
11-4 Multiplication Principle, Permutations,
Geometry 571
9-1 Conic Sections; Parabola 572 and Combinations 733
9-2 Ellipse 581 11-5 Sample Spaces and Probability 745
9-3 Hyperbola 591 11-6 The Binomial Formula 760
9-4 Translation and Rotation of Axes 604 Chapter 11 Review 766
Chapter 9 Review 620 Chapter 11 Review Exercises 768
Chapter 9 Review Exercises 623 Chapter 11 Group Activity: Sequences Specified
Chapter 9 Group Activity: Focal Chords 624 by Recursion Formulas 770
10
CHAPTER Systems of Equations Appendix A Cumulative Review Exercises A1
and Matrices 625 Appendix B Special Topics A17
10-1 Systems of Linear Equations 626 Appendix C Geometric Formulas A37
10-2 Solving Systems of Linear Equations Using Student Answers SA1 Instructor Answers IA1
Gauss-Jordan Elimination 643 Subject Index I1
xiii
bar19502_fm_i-xxxii.qxd 12/10/09 11:47 PM Page xiv
Preface
Enhancing a Tradition of Success
The ninth edition of College Algebra with Trigonometry represents a substantial step for-
ward in student accessibility. Every aspect of the revision of this classic text focuses on
making the text more accessible to students, while retaining the precise presentation of the
mathematics for which the Barnett name is renowned. Extensive work has been done to
enhance the clarity of the exposition, improving to the overall presentation of the content.
This in turn has decreased the length of the text.
Specifically, we concentrated on the areas of writing, exercises, worked examples, design,
and technology. Based on numerous reviews, advice from expert consultants, and direct cor-
respondence with the many users of previous editions, this edition is more relevant and acces-
sible than ever before.
Writing Without sacrificing breadth or depth or coverage, we have rewritten explanations
to make them clearer and more direct. As in previous editions, the text emphasizes compu-
tational skills, essential ideas, and problem solving rather than theory.
Exercises Over twenty percent of the exercises in the ninth edition are new. These exer-
cises encompass both a variety of skill levels as well as increased content coverage, ensur-
ing a gradual increase in difficulty level throughout. In addition, brand new writing exer-
cises have been included at the beginning of each exercise set in order to encourage a more
thorough understanding of key concepts for students.
Examples Color annotations accompany many examples, encouraging the learning process
for students by explaining the solution steps in words. Each example is then followed by a
similar matched problem for the student to solve. Answers to the matched problems are located
at the end of each section for easy reference. This active involvement in learning while reading
helps students develop a more thorough understanding of concepts and processes.
Technology Instructors who use technology to teach college algebra with trigonometry,
whether it be exploring mathematics with a graphing calculator or assigning homework and
quizzes online, will find the ninth edition to be much improved.
Refined “Technology Connections” boxes included at appropriate points in the text illus-
trate how problems previously introduced in an algebraic context may be solved using a graph-
ing calculator. Exercise sets include calculator-based exercises marked with a calculator icon.
Note, however, that the use of graphing technology is completely optional with this text. We
understand that at many colleges a single text must serve the purposes of teachers with widely
divergent views on the proper use of graphing and scientific calculators in college algebra with
trigonometry, and this text remains flexible regarding the degree of calculator integration.
Additionally, McGraw-Hill’s MathZone offers a complete online homework system for
mathematics and statistics. Instructors can assign textbook-specific content as well as cus-
tomize the level of feedback students receive, including the ability to have students show
their work for any given exercise. Assignable content for the ninth edition of College Algebra
with Trigonometry includes an array of videos and other multimedia along with algorithmic
exercises, providing study tools for students with many different learning styles.
A Central Theme
In the Barnett series, the function concept serves as a unifying theme. A brief look at the
table of contents reveals this emphasis. A major objective of this book is the development
of a library of elementary functions, including their important properties and uses.
Employing this library as a basic working tool, students will be able to proceed through
this book with greater confidence and understanding.
xiv
bar19502_fm_i-xxxii.qxd 12/10/09 11:47 PM Page xv
Preface xv
Reflecting trends in the way college algebra with trigonometry is taught, the ninth edi-
tion emphasizes functions modeled in the real world more strongly than previous editions.
In some cases, data are provided and the student is asked to produce an approximate cor-
responding function using regression on a graphing calculator. However, as with previous
editions, the use of a graphing calculator remains completely optional and any such exam-
ples or exercises can be easily omitted without loss of continuity.
Key Features
The revised full-color design gives the book a more contemporary feel and will appeal to
students who are accustomed to high production values in books, magazines, and nonprint
media. The rich color palette, streamlined calculator explorations, and use of color to sig-
nify important steps in problem material work in conjunction to create a more visually
appealing experience for students.
An emphasis on mathematical modeling is evident in section titles such as “Linear
Equations and Models” and “Exponential Models.” These titles reflect a focus on the rela-
tionship between functions and real-world phenomena, especially in examples and exercises.
Modeling problems vary from those where only the function model is given (e.g., when the
model is a physical law such as F " ma), through problems where a table of data and the
function are provided, to cases where the student is asked to approximate a function from
data using the regression function of a calculator or computer.
Matched problems following worked examples encourage students to practice prob-
lem solving immediately after reading through a solution. Answers to the matched problems
are located at the end of each section for easy reference.
Interspersed throughout each section, Explore-Discuss boxes foster conceptual under-
standing by asking students to think about a relationship or process before a result is stated.
Verbalization of mathematical concepts, results, and processes is strongly encouraged in these
explanations and activities. Many Explore-Discuss boxes are appropriate for group work.
Refined Technology Connections boxes employ graphing calculators to show graph-
ical and numerical alternatives to pencil-and-paper symbolic methods for problem solv-
ing—but the algebraic methods are not omitted. Screen shots are from the TI-84 Plus
calculator, but the Technology Connections will interest users of any automated graphing
utility.
Think boxes (color dashed boxes) are used to enclose steps that, with some experi-
ence, many students will be able to perform mentally.
Balanced exercise sets give instructors maximum flexibility in assigning homework. A
wide variety of easy, moderate, and difficult level exercises presented in a range of prob-
lem types help to ensure a gradual increase in difficulty level throughout each exercise set.
The division of exercise sets into A (routine, easy mechanics), B (more difficult mechan-
ics), and C (difficult mechanics and some theory) is explicitly presented only in the Anno-
tated Instructor’s Edition. This is due to our attempt to avoid fueling students’anxiety about
challenging exercises.
This book gives the student substantial experience in modeling and solving applied
problems. Over 500 application exercises help convince even the most skeptical student
that mathematics is relevant to life outside the classroom.
An Applications Index is included following the Guided Tour to help locate particu-
lar applications.
Most exercise sets include calculator-based exercises that are clearly marked with a
calculator icon. These exercises may use real or realistic data, making them computation-
ally heavy, or they may employ the calculator to explore mathematics in a way that would
be impractical with paper and pencil.
As many students will use this book to prepare for a calculus course, examples and
exercises that are especially pertinent to calculus are marked with an icon.
A Group Activity is located at the end of each chapter and involves many of the con-
cepts discussed in that chapter. These activities require students to discuss and write about
mathematical concepts in a complex, real-world context.