Coburn J.W. Algebra and Trigonometry

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John w. Coburn

St. Louis Community College at Florissant Valley

second editionsecond edition

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ALGEBRA AND TRIGONOMETRY, SECOND EDITION

Published by McGraw-Hill, a business unit of The McGraw-Hill Companies, Inc., 1221 Avenue of the Americas, New York, NY 10020.

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.

Some ancillaries, including electronic and print components, may not be available to customers outside the United States.

This book is printed on acid-free paper.

1 2 3 4 5 6 7 8 9 0 DOW/DOW 0 9

ISBN 978–0–07–351952–4

MHID 0–07–351952–9

ISBN 978–0–07–723501–7 (Annotated Instructor’s Edition)

MHID 0–07–723501–0

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Printer: R. R. Donnelley Willard, OH

Chapter R Opener: © Royalty-Free/CORBIS; pg. 12: © Ryan McVay/Getty Images/RF; pg. 19: © Photodisc/Getty Images/RF;

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Library of Congress Cataloging-in-Publication Data

Coburn, John W.

College algebra / John Coburn. — 2nd ed.

p. cm.

Includes index.

ISBN 978–0–07–351952–4 — ISBN 0–07–351952–9 (hard copy : alk. paper)

1. Algebras--Textbooks. 2. Trigonometry—Textbooks. I. Title.

QA154.3.C593 2010

512’.13–dc22 2008050849

www.mhhe.com

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Preface vi

Index of Applications xxxv

CHAPTER R A Review of Basic Concepts and Skills 1

CHAPTER 1 Equations and Inequalities 73

CHAPTER 2 Relations, Functions, and Graphs 151

CHAPTER 3 Polynomial and Rational Functions 293

CHAPTER 4 Exponential and Logarithmic Functions 411

CHAPTER 5 An Introduction to Trigonometric Functions 503

CHAPTER 6 Trigonometric Identities, Inverses, and Equations 615

CHAPTER 7 Applications of Trigonometry 711

CHAPTER 8 Systems of Equations and Inequalities 793

CHAPTER 9 Matrices and Matrix Applications 847

CHAPTER 10 Analytic Geometry and the Conic Sections 919

CHAPTER 11 Additional Topics in Algebra 1017

Appendix I More on Synthetic Division A-1

Appendix II More on Matrices A-3

Appendix III Deriving the Equation of a Conic A-5

Appendix IV Selected Proofs A-7

Appendix V Families of Polar Curves A-13

Student Answer Appendix (SE only) SA-1

Instructor Answer Appendix (AIE only) IA-1

Index I-1

Additional Topics Online

(Visit www.mhhe.com/coburn)

R.7 Geometry Review with Unit Conversions

R.8 Expressions, Tables and Graphing Calculators

5.0 An Introduction to Cycles and Periodic Functions

7.7 Complex Numbers in Exponential Form

7.8 Trigonometry, Complex Numbers and Cubic Equations

11.8 Conditional Probability and Expected Value

11.9 Probability and the Normal Curve with Applications

Brief Contents

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About the Author

Background John Coburn grew up in the Hawaiian Islands, the seventh of sixteen children.

John’s mother and father were both teachers. John’s mother taught English and

his father, as fate would have it, held advanced degrees in physics, chemistry,

and mathematics. Whereas John’s father was well known, well respected, and

a talented mathematician, John had to work very hard to see the connections

so necessary for success in mathematics. In many ways, his writing is born

of this experience.

Education In 1979 John received a bachelor’s degree in education from the University of Hawaii.

After working in the business world for a number of years, John returned to his

first love by accepting a teaching position in high school mathematics and in

1987 was recognized as Teacher of the Year. Soon afterward John decided to

seek a master’s degree, which he received two years later from the University

of Oklahoma.

Teaching Experience John is now a full professor at the Florissant Valley campus of St. Louis Community

College where he has taught mathematics for the last eighteen years. During

his time there he has received numerous nominations as an

outstanding teacher by the local chapter of Phi Theta Kappa,

and was recognized as Post-Secondary Teacher of the Year

in 2004 by Mathematics Educators of Greater St. Louis

(MEGSL). John is a member of the following organizations:

National Council of Teachers of Mathematics (NCTM),

Missouri Council of Teachers of Mathematics (MCTM),

Mathematics Educators of Greater Saint Louis (MEGSL),

American Mathematical Association of two Year Colleges

(AMATYC), Missouri Mathematical Association of two

Year Colleges (MoMATYC), Missouri Community College

Association (MCCA), and Mathematics Association of

America (MAA).

Personal Interests Some of John’s other interests include body surfing, snorkeling, and beach

combing whenever he gets the chance. In addition, John’s loves include his

family, music, athletics, games, and all things beautiful. John hopes that this

love of life comes through in the writing, and serves to make the learning

experience an interesting and engaging one for all students.

(

Dedication

To my wife and best friend Helen, whose love, support,

and willingness to sacrifice never faltered.

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From the Author

This text is the result of a mighty confluence of needs, ideas, desires, and directions. This is

easily understandable, as the intended audience is one of the most diverse in all of education.

Our students come to us with a wide range of backgrounds, varying degrees of preparation,

and interest levels that vary from apathy to excitement. In addition, our classes include those

needing only a general education requirement, as well as our country’s future engineers and

scientists. To say our greatest challenge is meeting the needs of so diverse a population would

be an understatement. In reflecting on this diversity, the image of a coral reef came to mind,

and I was struck by the strength of the analogy. We have a hugely diverse population, with

the reef as a common meeting place, with all the inhabitants depending on the reef for their

purpose, nourishment, and direction.

Writing a text for this course has been one of the most daunting and challenging experiences

in my life. Long before I began, my teaching experience left a nagging sense that most texts on

the market lacked the ability to connect with so diverse an audience. In addition, they appeared

to offer too scant a framework to build concepts, too terse a development to make connections,

and insufficient support in their exercise sets to develop long-term retention or foster a love of

mathematics. In particular, the applications seemed to lack a sense of realism, curious interest,

and/or connections to a student’s everyday experience.

With all of this in mind and a strong desire to write a better text, I set about the task of

creating what I hoped would become a more engaging tool for students, and a more supportive

tool for instructors. Drawing on the diversity of my own educational experience, and an early

exposure to different cultures, views, and perspectives, I believe has contributed to the text’s

unique and engaging style, and I hope in the end, to more and better connections with our diverse

audience. Having feedback from more than 400 people, including manuscript reviewers, focus

group participants, and contributors, was invaluable to helping me hone the connections in the

book. As a collateral outgrowth of this experience, I admit there was also a desire to interest

and engage ourselves, the instructors—to remind us again and again, why we fell in love with

mathematics in the first place. —John Coburn

Coral reefs support an extraordinary biodiversity as they

are home to over 4000 species of tropical or reef fish. In addition, coral reefs are

immensely beneficial to humans; buffering coastal regions from strong waves and

storms, providing millions of people with food and jobs, and prompting advances

in modern medicine.

Similar to a reef, a college algebra and trigonometry course is unique because of

its diverse population of students. Nearly every major is represented in this course,

featuring students with a wide range of backgrounds and skill sets. Just like the

variety of the fish in the sea rely on the coral reefs to survive, the assortment of

students in college algebra and trigonometry rely on succeeding in this course in

order to further pursue their degree, as well as their career goals.

About the Cover

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Making Connections . . .

College algebra and trigonometry tends to be a

challenging course for many students.

They don’t see the

connections that college algebra and trigonometry has to their life or why

it is so critical that they take and pass this course for both technical and

nontechnical careers alike. Others may enter into this course underprepared

or improperly placed and with very little motivation.

Instructors are faced with several challenges as well. They are given the task

of improving pass rates and student retention while energizing a classroom

full of students comprised of nearly every major. Furthermore, it can be

difficult to distinguish between students who are likely to succeed and

students who may struggle until after the first test is given.

The goal of the Coburn series is to provide both students and instructors

with tools to address these challenges, as well as the diversity of the students

taking this course, so that you can experience greater success in college

algebra and trigonometry. For instance, the comprehensive exercise sets have

a range of difficulty that provides very strong support for weaker students,

while advanced students are challenged to reach even further. The rest of

this preface further explains the tools that John Coburn and McGraw-Hill

have developed and how they can be used to connect students to college

algebra and trigonometry and connect instructors to their students.

The Coburn Precalculus Series provides you with strong

tools to achieve better outcomes in your College Algebra

and Trigonometry course as follows:

▶

Better Student Preparedness

▶

Increased Student Engagement

▶

Solid Skill Development

▶

Strong Connections

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vii

▶

Better Student Preparedness

No two students have the same strengths and weaknesses in mathematics. Typically students will

enter any math course with different preparedness levels. For most students who have trouble

retaining or recalling concepts learned in past courses, basic review is simply not enough to sustain

them successfully throughout the course. Moreover, instructors whose main focus is to prepare

students for the next course do not have adequate time in or out of class to individually help each

student with review material.

ALEKS Prep uniquely assesses each student to determine their individual strengths and

weaknesses and informs the student of their capabilities using a personalized pie chart. From there,

students begin learning through ALEKS via a personalized learning path uniquely designed for

each student. ALEKS Prep interacts with students like a private tutor and provides a safe learning

environment to remediate their individual knowledge gaps of the course pre-requisite material

outside of class.

ALEKS Prep is the only learning tool that empowers students by giving them an opportunity to

remediate individual knowledge gaps and improve their chances for success. ALEKS Prep is especially

effective when used in conjunction with ALEKS Placement and ALEKS 3.0 course-based software.

▶

Increased Student Engagement

What makes John Coburn’s applications unique is that he is constantly thinking mathematically.

John’s applications are spawned during a trip to Chicago, a phone call with his brother or sister, or

even while watching the evening news for the latest headlines. John literally takes notes on things

that he sees in everyday life and connects these situations to math. This truly makes for relevant

applications that are born from real-life experiences as opposed to applications that can seem

fictitious or contrived.

▶

Solid Skill Development

The Coburn series intentionally relates the examples to the exercise sets so there is a strong

connection between what students are learning while working through the examples in each section

and the homework exercises that they complete. In turn, students who attempt to work the exercises

first can surely rely on the examples to offer support as needed. Because of how well the examples

and exercises are connected, key concepts are easily understood and students have plenty of help

when using the book outside of class.

There are also an abundance of exercise types to choose from to ensure that homework

challenges a wide variety of skills. Furthermore, John reconnects students to earlier chapter material

with Mid-Chapter Checks; students have praised these exercises for helping them understand what

key concepts require additional practice.

▶

Strong Connections

John Coburn’s experience in the classroom and his strong connections to how students comprehend

the material are evident in his writing style. This is demonstrated by the way he provides a tight weave

from topic to topic and fosters an environment that doesn’t just focus on procedures but illustrates the

big picture, which is something that so often is sacrificed in this course. Moreover, he deploys a clear

and supportive writing style, providing the students with a tool they can depend on when the teacher is

not available, when they miss a day of class, or simply when working on their own.

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viii

Better Student Preparedness . . .

ALEKS is a unique online math tool that uses adaptive questioning and artificial intelligence

to correctly place, prepare, and remediate students . . . all in one product! Institutional case studies have

shown that ALEKS has improved pass rates by over 20% versus traditional online homework and by

over 30% compared to using a text alone.

By offering each student an individualized learning path, ALEKS directs students to work on the math

topics that they are ready to learn. Also, to help students keep pace in their course, instructors can correlate

ALEKS to their textbook or syllabus in seconds.

To learn more about how ALEKS can be used to boost student performance, please visit

www.aleks.com/highered/math or contact your McGraw-Hill representative.

ALAL

EKEK

isis

nini

e online math tool that uses adaptive questioning and artificial intelligence

Experience Student Success!

Easy Graphing Utility!

Students can answer graphing

problems with ease!

ALEKS Pie

Each student is given their own

individualized learning path.

Course Calendar

Instructors can schedule assignments

and reminders for students.

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. . . Through

New ALEKS Instructor Module

Enhanced Functionality and Streamlined Interface Help to Save Instructor Time

The new ALEKS Instructor Module features enhanced functionality and streamlined

interface based on research with ALEKS instructors and homework management instructors. Paired with

powerful assignment driven features, textbook integration, and extensive content  exibility, the new ALEKS

Instructor Module simpli es administrative tasks and makes ALEKS more powerful than ever.

Select topics for each assignment

Gradebook view for all students

radebook view for all students

Gradebook view for an individual student

Learn more about ALEKS by visiting

www.aleks.com/highered/math or

contact your McGraw-Hill representative.

New Gradebook!

Instructors can seamlessly track student

scores on automatically graded assignments.

They can also easily adjust the weighting and

grading scale of each assignment.

earn more about ALEK

by visiting

Automatically Graded Assignments

Instructors can easily assign homework, quizzes,

tests, and assessments to all or select students.

Deadline extensions can also be created for

select students.

Track Student Progress

Through Detailed Reporting

Instructors can track student progress

through automated reports and robust

reporting features.

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