Which number when rounded to the nearest hundredth is 37.62

Trigonometry L I A L | H O R N S B Y | S C H N E I D E R | D A N I E L S

E L E V E N T H E D I T I O N

MyMathLab, Pearson’s online learning management system, creates personalized experiences for students and provides powerful tools for instructors. With a wealth of tested and proven resources, each course can be tailored to fit your specific needs. Talk to your Pearson Representative about ways to integrate MyMathLab into your course for the best results.

Visit www.mymathlab.com and click Get Trained to make sure you're getting the most out of MyMathLab.

Data-Driven Reporting for Instructors • MyMathLab’s comprehensive online

gradebook automatically tracks students’ results to tests, quizzes, homework, and work in the study plan. The Reporting Dashboard, found under More Gradebook Tools, makes it easier than ever to identify topics where students are struggling, or specific students who may need extra help.

Get the most out of MyMathLab®

Learning in Any Environment • Because classroom formats and

student needs continually change and evolve, MyMathLab has built-in flexibility to accommodate various course designs and formats.

• With a new, streamlined, mobile-friendly design, students and instructors can access courses from most mobile devices to work on exercises and review completed assignments.

A00_LHSD7659_11_AIE_FEP2.indd 2 26/11/15 1:57 pm

http://www.mymathlab.com
www.mymathlab.com

Available in MyMathLab® for Your Course

Success in math can make a difference in your life. MyMathLab is a learning experience with resources to help you achieve your potential in this course and beyond. MyMathLab will help you learn the new skills required, and also help you learn the concepts and make connections for future courses and careers.

Achieve Your Potential

Visualization and Conceptual Understanding These MyMathLab resources will help you think visually and connect the concepts.

Updated Videos Updated videos are available to support students outside of the classroom and cover all topics in the text. Quick Review videos cover key definitions and procedures. Example Solution videos offer a detailed solution process for every example in the textbook.

Video Assessment Exercises Assignable in MyMathLab, Example Solution Videos present the detailed solution process for every example in the text. Additional Quick Reviews cover definitions and procedures for each section. Assessment exercises check conceptual understanding of the mathematics.

A00_LHSD7642_11_AIE_FEP3-4.indd 3 04/12/15 9:46 am

http://www.mymathlab.com
Preparedness and Study Skills

For more information on how MyMathLab can help you Achieve Your Potential visit http://www.pearsonhighered.com/achieve-your-potential/

MyMathLab® gives access to many learning resources which refresh knowledge of topics previously learned. Integrated Review MyMathLab Courses, Getting Ready material and Skills for Success Modules are some of the tools available.

Students refresh prerequisite topics through skill review quizzes and personalized home- work integrated in MyMathLab. With Getting Ready content in MyMathLab students get just the help they need to be prepared to learn the new material.

Getting Ready

Skills for Success Modules help foster success in collegiate courses and prepare students for future professions. Topics such as “Time Management,” “Stress Management” and “Financial Literacy” are available within the MyMathLab course.

Skills for Success Modules

A00_LHSD7642_11_AIE_FEP3-4.indd 4 04/12/15 9:46 am

http://www.pearsonhighered.com/achieve-your-potential/
Boston Columbus Indianapolis New York San Francisco Amsterdam Cape Town Dubai London Madrid Milan Munich Paris Montréal Toronto

Delhi Mexico City São Paulo Sydney Hong Kong Seoul Singapore Taipei Tokyo

Margaret L. Lial American River College

John Hornsby University of New Orleans

David I. Schneider University of Maryland

Callie J. Daniels St. Charles Community College

Trigonometry ELEVENTH EDITION

A01_LHSD7642_11_AIE_FM_ppi-xviii.indd 1 01/12/15 9:45 am

To Butch, Peggy, Natalie, and Alexis—and in memory of Mark E.J.H.

To Coach Lonnie Myers—thank you for your leadership on and off the court.

C.J.D.

A01_LHSD7642_11_AIE_FM_ppi-xviii.indd 3 01/12/15 9:45 am

561590_MILL_MICRO_FM_ppi-xxvi.indd 2 24/11/14 5:26 PM

This page intentionally left blank

v

Contents

Preface xi

Resources for Success xvi

1 Trigonometric Functions 1 1.1 Angles 2 Basic Terminology ■ Degree Measure ■ Standard Position ■ Coterminal Angles

1.2 Angle Relationships and Similar Triangles 10 Geometric Properties ■ Triangles

Chapter 1 Quiz (Sections 1.1–1.2) 21

1.3 Trigonometric Functions 22 The Pythagorean Theorem and the Distance Formula ■ Trigonometric Functions ■ Quadrantal Angles

1.4 Using the Definitions of the Trigonometric Functions 30 Reciprocal Identities ■ Signs and Ranges of Function Values ■ Pythagorean Identities ■ Quotient Identities

Test Prep 39 ■ Review Exercises 42 ■ Test 45

2 Acute Angles and Right Triangles 47 2.1 Trigonometric Functions of Acute Angles 48 Right-Triangle-Based Definitions of the Trigonometric Functions ■ Cofunctions ■ How Function Values Change as Angles Change ■ Trigonometric Function Values of Special Angles

2.2 Trigonometric Functions of Non-Acute Angles 56 Reference Angles ■ Special Angles as Reference Angles ■ Determination of Angle Measures with Special Reference Angles

2.3 Approximations of Trigonometric Function Values 64 Calculator Approximations of Trigonometric Function Values ■ Calculator Approximations of Angle Measures ■ An Application

Chapter 2 Quiz ( Sections 2.1–2.3) 71

2.4 Solutions and Applications of Right Triangles 72 Historical Background ■ Significant Digits ■ Solving Triangles ■ Angles of Elevation or Depression

2.5 Further Applications of Right Triangles 82 Historical Background ■ Bearing ■ Further Applications

Test Prep 91 ■ Review Exercises 93 ■ Test 97

A01_LHSD7642_11_AIE_FM_ppi-xviii.indd 5 01/12/15 9:45 am

vi CONTENTS

3 Radian Measure and the Unit Circle 99 3.1 Radian Measure 100 Radian Measure ■ Conversions between Degrees and Radians ■ Trigonometric Function Values of Angles in Radians

3.2 Applications of Radian Measure 106 Arc Length on a Circle ■ Area of a Sector of a Circle

3.3 The Unit Circle and Circular Functions 116 Circular Functions ■ Values of the Circular Functions ■ Determining a Number with a Given Circular Function Value ■ Applications of Circular Functions ■ Function Values as Lengths of Line Segments

Chapter 3 Quiz (Sections 3.1–3.3) 126

3.4 Linear and Angular Speed 127 Linear Speed ■ Angular Speed

Test Prep 133 ■ Review Exercises 135 ■ Test 138

4 Graphs of the Circular Functions 139 4.1 Graphs of the Sine and Cosine Functions 140 Periodic Functions ■ Graph of the Sine Function ■ Graph of the Cosine Function ■ Techniques for Graphing, Amplitude, and Period ■ Connecting Graphs with Equations ■ A Trigonometric Model

4.2 Translations of the Graphs of the Sine and Cosine Functions 153

Horizontal Translations ■ Vertical Translations ■ Combinations of Translations ■ A Trigonometric Model

Chapter 4 Quiz ( Sections 4.1 –4.2) 164

4.3 Graphs of the Tangent and Cotangent Functions 164 Graph of the Tangent Function ■ Graph of the Cotangent Function ■ Techniques for Graphing ■ Connecting Graphs with Equations

4.4 Graphs of the Secant and Cosecant Functions 173 Graph of the Secant Function ■ Graph of the Cosecant Function ■ Techniques for Graphing ■ Connecting Graphs with Equations ■ Addition of Ordinates

Summary Exercises on Graphing Circular Functions 181

4.5 Harmonic Motion 181 Simple Harmonic Motion ■ Damped Oscillatory Motion

Test Prep 187 ■ Review Exercises 189 ■ Test 193

A01_LHSD7642_11_AIE_FM_ppi-xviii.indd 6 01/12/15 9:45 am

viiCONTENTS

5 Trigonometric Identities 195 5.1 Fundamental Identities 196 Fundamental Identities ■ Uses of the Fundamental Identities

5.2 Verifying Trigonometric Identities 202 Strategies ■ Verifying Identities by Working with One Side ■ Verifying Identities by Working with Both Sides

5.3 Sum and Difference Identities for Cosine 211 Difference Identity for Cosine ■ Sum Identity for Cosine ■ Cofunction Identities ■ Applications of the Sum and Difference Identities ■ Verifying an Identity

5.4 Sum and Difference Identities for Sine and Tangent 221 Sum and Difference Identities for Sine ■ Sum and Difference Identities for Tangent ■ Applications of the Sum and Difference Identities ■ Verifying an Identity

Chapter 5 Quiz (Sections 5.1–5.4) 230

5.5 Double-Angle Identities 230 Double-Angle Identities ■ An Application ■ Product-to-Sum and Sum-to-Product Identities

5.6 Half-Angle Identities 238 Half-Angle Identities ■ Applications of the Half-Angle Identities ■ Verifying an Identity

Summary Exercises on Verifying Trigonometric Identities 245

Test Prep 246 ■ Review Exercises 248 ■ Test 250

6 Inverse Circular Functions and Trigonometric Equations 251 6.1 Inverse Circular Functions 252 Inverse Functions ■ Inverse Sine Function ■ Inverse Cosine Function ■ Inverse Tangent Function ■ Other Inverse Circular Functions ■ Inverse Function Values

6.2 Trigonometric Equations I 268 Linear Methods ■ Zero-Factor Property Method ■ Quadratic Methods ■ Trigonometric Identity Substitutions ■ An Application

6.3 Trigonometric Equations II 275 Equations with Half-Angles ■ Equations with Multiple Angles ■ An Application

Chapter 6 Quiz ( Sections 6.1–6.3) 282

A01_LHSD7642_11_AIE_FM_ppi-xviii.indd 7 01/12/15 9:45 am

viii

7 Applications of Trigonometry and Vectors 295 7.1 Oblique Triangles and the Law of Sines 296 Congruency and Oblique Triangles ■ Derivation of the Law of Sines ■ Solutions of SAA and ASA Triangles (Case 1) ■ Area of a Triangle

7.2 The Ambiguous Case of the Law of Sines 306 Description of the Ambiguous Case ■ Solutions of SSA Triangles (Case 2) ■ Analyzing Data for Possible Number of Triangles

7.3 The Law of Cosines 312 Derivation of the Law of Cosines ■ Solutions of SAS and SSS Triangles (Cases 3 and 4) ■ Heron’s Formula for the Area of a Triangle ■ Derivation of Heron’s Formula

Chapter 7 Quiz ( Sections 7.1–7.3) 325

7.4 Geometrically Defined Vectors and Applications 326 Basic Terminology ■ The Equilibrant ■ Incline Applications ■ Navigation Applications

7.5 Algebraically Defined Vectors and the Dot Product 336 Algebraic Interpretation of Vectors ■ Operations with Vectors ■ The Dot Product and the Angle between Vectors

Summary Exercises on Applications of Trigonometry and Vectors 344

Test Prep 346 ■ Review Exercises 349 ■ Test 353

6.4 Equations Involving Inverse Trigonometric Functions 282 Solution for x in Terms of y Using Inverse Functions ■ Solution of Inverse Trigonometric Equations

Test Prep 289 ■ Review Exercises 291 ■ Test 293

CONTENTS

8 Complex Numbers, Polar Equations, and Parametric Equations 355 8.1 Complex Numbers 356 Basic Concepts of Complex Numbers ■ Complex Solutions of Quadratic Equations (Part 1) ■ Operations on Complex Numbers ■ Complex Solutions of Quadratic Equations (Part 2) ■ Powers of i

8.2 Trigonometric (Polar) Form of Complex Numbers 366 The Complex Plane and Vector Representation ■ Trigonometric (Polar) Form ■ Converting between Rectangular and Trigonometric (Polar) Forms ■ An Application of Complex Numbers to Fractals

A01_LHSD7642_11_AIE_FM_ppi-xviii.indd 8 01/12/15 9:45 am

ixCONTENTS

8.3 The Product and Quotient Theorems 372 Products of Complex Numbers in Trigonometric Form ■ Quotients of Complex Numbers in Trigonometric Form

8.4 De Moivre’s Theorem; Powers and Roots of Complex Numbers 378

Powers of Complex Numbers (De Moivre’s Theorem) ■ Roots of Complex Numbers

Chapter 8 Quiz (Sections 8.1–8.4) 385

8.5 Polar Equations and Graphs 385 Polar Coordinate System ■ Graphs of Polar Equations ■ Conversion from Polar to Rectangular Equations ■ Classification of Polar Equations

8.6 Parametric Equations, Graphs, and Applications 398 Basic Concepts ■ Parametric Graphs and Their Rectangular Equivalents ■ The Cycloid ■ Applications of Parametric Equations

Test Prep 406 ■ Review Exercises 409 ■ Test 412

Appendices 413 Appendix A Equations and Inequalities 413

Basic Terminology of Equations ■ Linear Equations ■ Quadratic Equations ■ Inequalities ■ Linear Inequalities and Interval Notation ■ Three-Part Inequalities

Appendix B Graphs of Equations 422 The Rectangular Coordinate System ■ Equations in Two Variables ■ Circles

Appendix C Functions 428 Relations and Functions ■ Domain and Range ■ Determining Whether Relations Are Functions ■ Function Notation ■ Increasing, Decreasing, and Constant Functions

Appendix D Graphing Techniques 438 Stretching and Shrinking ■ Reflecting ■ Symmetry ■ Translations

Answers to Selected Exercises A-1 Photo Credits C-1 Index I-1

A01_LHSD7642_11_AIE_FM_ppi-xviii.indd 9 01/12/15 9:45 am

561590_MILL_MICRO_FM_ppi-xxvi.indd 2 24/11/14 5:26 PM

This page intentionally left blank

xi

Preface

WELCOME TO THE 11TH EDITION In the eleventh edition of Trigonometry, we continue our ongoing commitment to providing the best possible text to help instructors teach and students succeed. In this edition, we have remained true to the pedagogical style of the past while staying focused on the needs of today’s students. Support for all classroom types (traditional, hybrid, and online) may be found in this classic text and its supplements backed by the power of Pearson’s MyMathLab.

In this edition, we have drawn upon the extensive teaching experience of the Lial team, with special consideration given to reviewer suggestions. General updates include enhanced readability with improved layout of examples, better use of color in displays, and language written with students in mind. All calculator screenshots have been updated and now provide color displays to enhance students’ conceptual understanding. Each homework section now begins with a group of Concept Preview exercises, assignable in MyMathLab, which may be used to ensure students’ understanding of vocabulary and basic concepts prior to beginning the regular homework exercises.

Further enhancements include numerous current data examples and exercises that have been updated to reflect current information. Additional real-life exercises have been included to pique student interest; answers to writing exercises have been provided; better consistency has been achieved between the directions that introduce examples and those that introduce the corresponding exercises; and better guidance for rounding of answers has been provided in the exercise sets.

The Lial team believes this to be our best Trigonometry edition yet, and we sin- cerely hope that you enjoy using it as much as we have enjoyed writing it. Additional textbooks in this series are as follows:

College Algebra, Twelfth Edition College Algebra & Trigonometry, Sixth Edition Precalculus, Sixth Edition

HIGHLIGHTS OF NEW CONTENT ■ Discussion of the Pythagorean theorem and the distance formula has been

moved from an appendix to Chapter 1.

■ In Chapter 2, the two sections devoted to applications of right triangles now begin with short historical vignettes, to provide motivation and illustrate how trigonometry developed as a tool for astronomers.

■ The example solutions of applications of angular speed in Chapter 3 have been rewritten to illustrate the use of unit fractions.

■ In Chapter 4, we have included new applications of periodic functions. They involve modeling monthly temperatures of regions in the southern hemisphere and fractional part of the moon illuminated for each day of a particular month. The example of addition of ordinates in Section 4.4 has been rewritten, and a new example of analysis of damped oscillatory motion has been included in Section 4.5.

■ Chapter 5 now presents a derivation of the product-to-sum identity for the product sin A cos B.

■ In Chapter 6, we include several new screens of periodic function graphs that differ in appearance from typical ones. They pertain to the music phenomena of pressure of a plucked spring, beats, and upper harmonics.

A01_LHSD7642_11_AIE_FM_ppi-xviii.indd 11 04/12/15 9:48 am

xii PREFACE

■ The two sections in Chapter 7 on vectors have been reorganized but still cover the same material as in the previous edition. Section 7.4 now introduces geometrically defined vectors and applications, and Section 7.5 follows with algebraically defined vectors and the dot product.

■ In Chapter 8, the examples in Section 8.1 have been reordered for a better flow with respect to solving quadratic equations with complex solutions.

■ For visual learners, numbered Figure and Example references within the text are set using the same typeface as the figure number itself and bold print for the example. This makes it easier for the students to identify and connect them. We also have increased our use of a “drop down” style, when appropri- ate, to distinguish between simplifying expressions and solving equations, and we have added many more explanatory side comments. Guided Visual- izations, with accompanying exercises and explorations, are now available and assignable in MyMathLab.

■ Trigonometry is widely recognized for the quality of its exercises. In the eleventh edition, nearly 500 are new or modified, and many present updated real-life data. Furthermore, the MyMathLab course has expanded coverage of all exercise types appearing in the exercise sets, as well as the mid-chapter Quizzes and Summary Exercises.

FEATURES OF THIS TEXT SUPPORT FOR LEARNING CONCEPTS We provide a variety of features to support students’ learning of the essential topics of trigonometry. Explanations that are written in understandable terms, figures and graphs that illustrate examples and concepts, graphing technology that supports and enhances algebraic manipulations, and real-life applications that enrich the topics with meaning all provide opportunities for students to deepen their understanding of mathematics. These features help students make mathematical connections and expand their own knowledge base.

■ Examples Numbered examples that illustrate the techniques for working exercises are found in every section. We use traditional explanations, side comments, and pointers to describe the steps taken—and to warn students about common pitfalls. Some examples provide additional graphing calcula- tor solutions, although these can be omitted if desired.

■ Now Try Exercises Following each numbered example, the student is directed to try a corresponding odd-numbered exercise (or exercises). This feature allows for quick feedback to determine whether the student has understood the principles illustrated in the example.

■ Real-Life Applications We have included hundreds of real-life applica- tions, many with data updated from the previous edition. They come from fields such as sports, biology, astronomy, geology, music, and environmental studies.

■ Function Boxes Special function boxes offer a comprehensive, visual introduction to each type of trigonometric function and also serve as an excellent resource for reference and review. Each function box includes a table of values, traditional and calculator-generated graphs, the domain, the range, and other special information about the function. These boxes are assignable in MyMathLab.

■ Figures and Photos Today’s students are more visually oriented than ever before, and we have updated the figures and photos in this edition to

A01_LHSD7642_11_AIE_FM_ppi-xviii.indd 12 03/12/15 12:50 pm

promote visual appeal. Guided Visualizations with accompanying exercises and explorations are now available and assignable in MyMathLab.

■ Use of Graphing Technology We have integrated the use of graphing calculators where appropriate, although this technology is completely op- tional and can be omitted without loss of continuity. We continue to stress that graphing calculators support understanding but that students must first master the underlying mathematical concepts. Exercises that require the use of a graphing calculator are marked with the icon .

■ Cautions and Notes Text that is marked CAUTION warns students of common errors, and NOTE comments point out explanations that should receive particular attention.

■ Looking Ahead to Calculus These margin notes offer glimpses of how the topics currently being studied are used in calculus.

SUPPORT FOR PRACTICING CONCEPTS This text offers a wide variety of exercises to help students master trigonometry. The extensive exercise sets provide ample opportunity for practice, and the exercise problems generally increase in difficulty so that students at every level of under- standing are challenged. The variety of exercise types promotes understanding of the concepts and reduces the need for rote memorization.

■ NEW Concept Preview Each exercise set now begins with a group of CONCEPT PREVIEW exercises designed to promote understanding of vo- cabulary and basic concepts of each section. These new exercises are assign- able in MyMathLab and will provide support especially for hybrid, online, and flipped courses.

■ Exercise Sets In addition to traditional drill exercises, this text includes writing exercises, optional graphing calculator problems , and multiple- choice, matching, true/false, and completion exercises. Concept Check exer- cises focus on conceptual thinking. Connecting Graphs with Equations exercises challenge students to write equations that correspond to given graphs.

■ Relating Concepts Exercises Appearing at the end of selected exer- cise sets, these groups of exercises are designed so that students who work them in numerical order will follow a line of reasoning that leads to an un- derstanding of how various topics and concepts are related. All answers to these exercises appear in the student answer section, and these exercises are assignable in MyMathLab.

■ Complete Solutions to Selected Exercises Complete solutions to all exercises marked are available in the eText. These are often exercises that extend the skills and concepts presented in the numbered examples.

SUPPORT FOR REVIEW AND TEST PREP Ample opportunities for review are found within the chapters and at the ends of chapters. Quizzes that are interspersed within chapters provide a quick assessment of students’ understanding of the material presented up to that point in the chapter. Chapter “Test Preps” provide comprehensive study aids to help students prepare for tests.

■ Quizzes Students can periodically check their progress with in-chapter quizzes that appear in all chapters. All answers, with corresponding section references, appear in the student answer section. These quizzes are assign- able in MyMathLab.

xiiiPREFACE

A01_LHSD7642_11_AIE_FM_ppi-xviii.indd 13 01/12/15 9:45 am

xiv

■ Summary Exercises These sets of in-chapter exercises give students the all-important opportunity to work mixed review exercises, requiring them to synthesize concepts and select appropriate solution methods.

■ End-of-Chapter Test Prep Following the final numbered section in each chapter, the Test Prep provides a list of Key Terms, a list of New Symbols (if applicable), and a two-column Quick Review that includes a section-by-section summary of concepts and examples. This feature con- cludes with a comprehensive set of Review Exercises and a Chapter Test. The Test Prep, Review Exercises, and Chapter Test are assignable in MyMathLab. Additional Cumulative Review homework assignments are available in MyMathLab, following every chapter.

PREFACE

A01_LHSD7642_11_AIE_FM_ppi-xviii.indd 14 01/12/15 9:45 am

MyMathLab® Get the most out of

MyMathLab is the world’s leading online resource for teaching and learning mathemat-

ics. MyMathLab helps students and instructors improve results, and it provides engag-

ing experiences and personalized learning for each student so learning can happen in

any environment. Plus, it offers flexible and time-saving course management features

to allow instructors to easily manage their classes while remaining in complete control,

regardless of course format.

Personalized Support for Students

student performance to provide personalized recommendations on what to work

through concepts they need to master.

tailored to each student’s specific needs and focused on the topics they have not yet mastered.

products delivers consistent, measurable gains in student learning outcomes, retention,

www.mymathlab.com

A01_LHSD7642_11_AIE_FM_ppi-xviii.indd 15 01/12/15 9:45 am

http://www.mymathlab.com
www.mymathlab.com

MyMathLab® Online Course for Trigonometry by Lial, Hornsby, Schneider, and Daniels

to give students the practice they need to develop a conceptual understanding of

classroom formats (traditional, hybrid, and online).

Concept Preview Exercises

Exercise sets now begin with a group of Concept Preview Exer-

cises, assignable in MyMathLab and also available in Learning

ensure that students understand the related vocabulary and basic

concepts before beginning the regu- lar homework problems.

Learning Catalytics is a “bring your own device” system of prebuilt

student engagement and facilitate assessment.

MyNotes and MyClassroomExamples MyNotes provide a note-taking structure for students to use while they read the text or watch the MyMathLab videos. MyClassroom Examples offer structure for notes taken during lecture and are for use with the Classroom Examples found in

Both sets of notes are available in MyMathLab and can be customized by the instructor.

A01_LHSD7642_11_AIE_FM_ppi-xviii.indd 16 01/12/15 9:45 am

http://www.mymathlab.com
Resources for Success Student Supplements Student’s Solutions Manual By Beverly Fusfield

Provides detailed solutions to all odd-numbered text exercises

ISBN: 0-13-431021-7 & 978-0-13-431021-3

Video Lectures with Optional Captioning

Feature Quick Reviews and Example Solutions: Quick Reviews cover key definitions and procedures from each section. Example Solutions walk students through the detailed solution process for every example in the textbook.

Ideal for distance learning or supplemental instruction at home or on campus

Include optional text captioning Available in MyMathLab®

MyNotes Available in MyMathLab and offer structure for

students as they watch videos or read the text Include textbook examples along with ample space

for students to write solutions and notes Include key concepts along with prompts for

students to read, write, and reflect on what they have just learned

Customizable so that instructors can add their own examples or remove examples that are not covered in their courses

MyClassroomExamples Available in MyMathLab and offer structure for

classroom lecture Include Classroom Examples along with ample space

for students to write solutions and notes Include key concepts along with fill in the blank

opportunities to keep students engaged Customizable so that instructors can add their

own examples or remove Classroom Examples that are not covered in their courses

Instructor Supplements Annotated Instructor’s Edition

Provides answers in the margins to almost all text exercises, as well as helpful Teaching Tips and Classroom Examples

Includes sample homework assignments indicated by exercise numbers underlined in blue within each end-of-section exercise set

Sample homework exercises assignable in MyMathLab

ISBN: 0-13-421764-0 & 978-0-13-421764-2

Online Instructor’s Solutions Manual By Beverly Fusfield

Provides complete solutions to all text exercises Available in MyMathLab or downloadable from

Pearson Education’s online catalog

Online Instructor’s Testing Manual By David Atwood

Includes diagnostic pretests, chapter tests, final exams, and additional test items, grouped by section, with answers provided

Available in MyMathLab or downloadable from Pearson Education’s online catalog

TestGen® Enables instructors to build, edit, print, and administer

tests Features a computerized bank of questions developed

to cover all text objectives Available in MyMathLab or downloadable from

Pearson Education’s online catalog

Online PowerPoint Presentation and Classroom Example PowerPoints

Written and designed specifically for this text Include figures and examples from the text Provide Classroom Example PowerPoints that include

full worked-out solutions to all Classroom Examples Available in MyMathLab or downloadable from

Pearson Education’s online catalog

www.mymathlab.com

A01_LHSD7642_11_AIE_FM_ppi-xviii.indd 17 03/12/15 12:50 pm

http://www.mymathlab.com
xviii ACKNOWLEDGMENTS

ACKNOWLEDGMENTS We wish to thank the following individuals who provided valuable input into this edition of the text.

John E. Daniels – Central Michigan University Mary Hill – College of DuPage Rene Lumampao – Austin Community College Randy Nichols – Delta College Patty Schovanec – Texas Tech University Deanna M. Welsch – Illinois Central College

Our sincere thanks to those individuals at Pearson Education who have sup- ported us throughout this revision: Anne Kelly, Christine O’Brien, Joe Vetere, and Danielle Simbajon. Terry McGinnis continues to provide behind-the-scenes guid- ance for both content and production. We have come to rely on her expertise during all phases of the revision process. Marilyn Dwyer of Cenveo® Publishing Services, with the assistance of Carol Merrigan, provided excellent production work. Special thanks go out to Paul Lorczak and Perian Herring for their excellent accuracy- checking. We thank Lucie Haskins, who provided an accurate index, and Jack Hornsby, who provided assistance in creating calculator screens, researching data updates, and proofreading.

As an author team, we are committed to providing the best possible college algebra course to help instructors teach and students succeed. As we continue to work toward this goal, we welcome any comments or suggestions you might send, via e-mail, to math@pearson.com.

Margaret L. Lial John Hornsby David I. Schneider Callie J. Daniels

A01_LHSD7642_11_AIE_FM_ppi-xviii.indd 18 01/12/15 9:45 am

mailto:math@pearson.com
1

A sequence of similar triangles, a topic covered in this introductory chapter, can be used to approximate the spiral of the chambered nautilus.

Angles

Angle Relationships and Similar Triangles

Chapter 1 Quiz

Trigonometric Functions

Using the Definitions of the Trigonometric Functions

1.1

1.2

1.3

1.4

Trigonometric Functions1

M01_LHSD7642_11_AIE_C01_pp001-046.indd 1 27/10/15 6:27 pm

2 CHAPTER 1 Trigonometric Functions

Degree Measure The most common unit for measuring angles is the degree. Degree measure was developed by the Babylonians 4000 yr ago. To use degree measure, we assign 360 degrees to a complete rotation of a ray.* In Figure 4, notice that the terminal side of the angle corresponds to its initial side when it makes a complete rotation.

One degree, written 1°, represents 1

360 of a complete rotation.

Therefore, 90° represents 90360 = 1 4 of a complete rotation, and 180° represents

180 360 =

1 2 of a complete rotation.

An angle measuring between 0° and 90° is an acute angle. An angle mea- suring exactly 90° is a right angle. The symbol m is often used at the vertex of a right angle to denote the 90° measure. An angle measuring more than 90° but less than 180° is an obtuse angle, and an angle of exactly 180° is a straight angle.

1.1 Angles

Basic Terminology Two distinct points A and B determine a line called line AB. The portion of the line between A and B, including points A and B them- selves, is line segment AB, or simply segment AB. The portion of line AB that starts at A and continues through B, and on past B, is the ray AB. Point A is the endpoint of the ray. See Figure 1.

In trigonometry, an angle consists of two rays in a plane with a common endpoint, or two line segments with a common endpoint. These two rays (or segments) are the sides of the angle, and the common endpoint is the vertex of the angle. Associated with an angle is its measure, generated by a rotation about the vertex. See Figure 2. This measure is determined by rotating a ray starting at one side of the angle, the initial side, to the position of the other side, the terminal side. A counterclockwise rotation generates a positive measure, and a clockwise rotation generates a negative measure. The rotation can consist of more than one complete revolution.

Figure 3 shows two angles, one positive and one negative.

■ Basic Terminology ■ Degree Measure ■ Standard Position ■ Coterminal Angles

A B Line AB

Segment AB

A B

Ray AB

A B

Figure 1

Terminal side

Vertex A

Angle A

Initial side

Figure 2 Positive angle Negative angle

B

C A

Figure 3

An angle can be named by using the name of its vertex. For example, the angle on the right in Figure 3 can be named angle C. Alternatively, an angle can be named using three letters, with the vertex letter in the middle. Thus, the angle on the right also could be named angle ACB or angle BCA.

*The Babylonians were the first to subdivide the circumference of a circle into 360 parts. There are various theories about why the number 360 was chosen. One is that it is approximately the number of days in a year, and it has many divisors, which makes it convenient to work with in computations.

A complete rotation of a ray gives an angle whose measure

is 360°. of a complete

rotation gives an angle whose measure is 1°.

360 1

Figure 4

M01_LHSD7642_11_AIE_C01_pp001-046.indd 2 27/10/15 6:27 pm

3 1.1 Angles

In Figure 5, we use the Greek letter U (theta)* to name each angle. The table in the margin lists the upper- and lowercase Greek letters, which are often used in trigonometry.

* In addition to u (theta), other Greek letters such as a (alpha) and b (beta) are used to name angles.

Acute angle 0° < u < 90°

u

Right angle u = 90°

u

Obtuse angle

90° < u < 180°

u

Straight angle u = 180°

u

Figure 5

If the sum of the measures of two positive angles is 90°, the angles are comple- mentary and the angles are complements of each other. Two positive angles with measures whose sum is 180° are supplementary, and the angles are supplements.

The Greek Letters

Α a alpha Β b beta Γ g gamma ∆ d delta Ε e epsilon Ζ z zeta Η h eta ϴ u theta Ι i iota Κ k kappa Λ l lambda Μ m mu Ν n nu Ξ j xi Ο o omicron Π p pi Ρ r rho Σ s sigma Τ t tau Υ y upsilon Φ f phi Χ x chi Ψ c psi Ω v omega

EXAMPLE 1 Finding the Complement and the Supplement of an Angle

Find the measure of (a) the complement and (b) the supplement of an angle measuring 40°.

SOLUTION

(a) To find the measure of its complement, subtract the measure of the angle from 90°.

90° - 40° = 50° Complement of 40° (b) To find the measure of its supplement, subtract the measure of the angle

from 180°. 180° - 40° = 140° Supplement of 40°

■✔ Now Try Exercise 11.

EXAMPLE 2 Finding Measures of Complementary and Supplementary Angles

Find the measure of each marked angle in Figure 6.

SOLUTION

(a) Because the two angles in Figure 6(a) form a right angle, they are comple- mentary angles.

6x + 3x = 90 Complementary angles sum to 90°. 9x = 90 Combine like terms.

x = 10 Divide by 9.Don’t stop here.

Be sure to determine the measure of each angle by substituting 10 for x in 6x and 3x. The two angles have measures of 61102 = 60° and 31102 = 30°.

(b) The angles in Figure 6(b) are supplementary, so their sum must be 180°.

4x + 6x = 180 Supplementary angles sum to 180°. 10x = 180 Combine like terms.

x = 18 Divide by 10. The angle measures are 4x = 41182 = 72° and 6x = 61182 = 108°.

■✔ Now Try Exercises 23 and 25.

(6x)° (3x)°

(a)

(4x)° (6x)°

(b)

Figure 6

M01_LHSD7642_11_AIE_C01_pp001-046.indd 3 27/10/15 6:27 pm

4 CHAPTER 1 Trigonometric Functions

The measure of angle A in Figure 7 is 35°. This measure is often expressed by saying that m 1angle A 2 is 35°, where m1angle A2 is read “the measure of angle A.” The symbolism m1angle A2 = 35° is abbreviated as A = 35°.

Traditionally, portions of a degree have been measured with minutes and seconds. One minute, written 1′, is 160 of a degree.

1′ = 1 60 ° or 60′ = 1°

One second, 1″, is 160 of a minute.

1″ = 1 60 ′ =

1 3600

° or 60″ = 1′ and 3600″ = 1°

The measure 12° 42′ 38″ represents 12 degrees, 42 minutes, 38 seconds.

A = 35° x

y

0

Figure 7

EXAMPLE 4 Converting between Angle Measures

(a) Convert 74° 08′ 14″ to decimal degrees to the nearest thousandth.

(b) Convert 34.817° to degrees, minutes, and seconds to the nearest second.

SOLUTION

(a) 74° 08′ 14″

= 74° + 8 60 ° + 14

3600 ° 08′ # 1°60′ = 860° and 14″ # 1°3600″ = 143600°

≈ 74° + 0.1333° + 0.0039° Divide to express the fractions as decimals. ≈ 74.137° Add and round to the nearest thousandth.

An alternative way to measure angles involves decimal degrees. For example,

12.4238° represents 12 4238

10,000 ° .

EXAMPLE 3 Calculating with Degrees, Minutes, and Seconds

Perform each calculation.

(a) 51° 29′ + 32° 46′ (b) 90° - 73° 12′

SOLUTION

(a) 51° 29′ + 32° 46′ 83° 75′

Add degrees and minutes separately.

The sum 83° 75′ can be rewritten as follows.

83° 75′

= 83° + 1° 15′ 75′ = 60′ + 15′ = 1° 15′ = 84° 15′ Add.

(b) 90° 89° 60′ Write 90° as 89° 60′. - 73° 12′ can be written - 73° 12′ 16° 48′

■✔ Now Try Exercises 41 and 45.

M01_LHSD7642_11_AIE_C01_pp001-046.indd 4 27/10/15 6:27 pm

5 1.1 Angles

(b) 34.817°

= 34° + 0.817° Write as a sum. = 34° + 0.817160′2 0.817° # 60′1° = 0.817160′2 = 34° + 49.02′ Multiply. = 34° + 49′ + 0.02′ Write 49.02′ as a sum. = 34° + 49′ + 0.02160″2 0.02′ # 60″1′ = 0.02160″2 = 34° + 49′ + 1.2″ Multiply. ≈ 34° 49′ 01″ Approximate to the nearest second.

■✔ Now Try Exercises 61 and 71.

This screen shows how the TI-84 Plus performs the conversions in Example 4. The ▶DMS option is found in the ANGLE Menu.

Standard Position An angle is in standard position if its vertex is at the origin and its initial side lies on the positive x-axis. The angles in Figures 8(a) and 8(b) are in standard position. An angle in standard position is said to lie in the quadrant in which its terminal side lies. An acute angle is in quadrant I (Figure 8(a)) and an obtuse angle is in quadrant II (Figure 8(b)). Figure 8(c) shows ranges of angle measures for each quadrant when 0° 6 u 6 360°.

0 x

y

Terminal side

Vertex Initial side

Q I

50°

x

y

0

Q II

160°

0° 360°180°

90°

270°

Q II 90° < u < 180°

Q I 0° < u < 90°

Q III 180° < u < 270°

Q IV 270° < u < 360°

(a) (b) (c)

Figure 8

Angles in standard position

Coterminal Angles A complete rotation of a ray results in an angle meas- uring 360°. By continuing the rotation, angles of measure larger than 360° can be produced. The angles in Figure 9 with measures 60° and 420° have the same initial side and the same terminal side, but different amounts of rotation. Such angles are coterminal angles. Their measures differ by a multiple of 360°. As shown in Figure 10, angles with measures 110° and 830° are coterminal.

0 x

y

60° 420°

Coterminal angles

Figure 9

0 x

y

110° 830°

Coterminal angles

Figure 10

Quadrantal Angles

Angles in standard position whose terminal sides lie on the x-axis or y-axis, such as angles with measures 90°, 180°, 270°, and so on, are quadrantal angles.

M01_LHSD7642_11_AIE_C01_pp001-046.indd 5 27/10/15 6:27 pm

6 CHAPTER 1 Trigonometric Functions

EXAMPLE 5 Finding Measures of Coterminal Angles

Find the angle of least positive measure that is coterminal with each angle.

(a) 908° (b) -75° (c) -800°

SOLUTION

(a) Subtract 360° as many times as needed to obtain an angle with measure greater than 0° but less than 360°.

908° - 2 # 360° = 188° Multiply 2 # 360°. Then subtract. An angle of 188° is coterminal with an angle of 908°. See Figure 11.

0 x

y

188° 908°

Figure 11

0

285° –75°

x

y

Figure 12

(b) Add 360° to the given negative angle measure to obtain the angle of least positive measure. See Figure 12.

-75° + 360° = 285°

(c) The least integer multiple of 360° greater than 800° is

3 # 360° = 1080°. Add 1080° to -800° to obtain

-800° + 1080° = 280°. ■✔ Now Try Exercises 81, 91, and 95.