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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

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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

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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.

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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

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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

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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

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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

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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

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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.

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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

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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

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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

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MyMathLab® Get the most out of

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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-

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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

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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