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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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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

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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

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

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

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Personalized Support for Students

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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,

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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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concepts before beginning the regu- lar homework problems.

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

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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

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

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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

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

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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

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

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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

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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

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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

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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

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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

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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.

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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.

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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.

Sometimes it is necessary to find an expression that will generate all angles coterminal with a given angle. For example, we can obtain any angle coterminal with 60° by adding an integer multiple of 360° to 60°. Let n represent any inte- ger. Then the following expression represents all such coterminal angles.

60° + n # 360° Angles coterminal with 60° The table below shows a few possibilities.

Examples of Angles Coterminal with 60°

Value of n Angle Coterminal with 60° 2 60° + 2 # 360° = 780° 1 60° + 1 # 360° = 420° 0 60° + 0 # 360° = 60° (the angle itself) -1 60° + 1-12 # 360° = -300° -2 60° + 1-22 # 360° = -660°

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7 1.1 Angles

This table shows some examples of coterminal quadrantal angles.

Examples of Coterminal Quadrantal Angles

Quadrantal Angle U Coterminal with U

0° {360°, {720° 90° -630°, -270°, 450° 180° -180°, 540°, 900° 270° -450°, -90°, 630°

EXAMPLE 6 Analyzing Revolutions of a Disk Drive

A constant angular velocity disk drive spins a disk at a constant speed. Suppose a disk makes 480 revolutions per min. Through how many degrees will a point on the edge of the disk move in 2 sec?

SOLUTION The disk revolves 480 times in 1 min, or 480 60 times = 8 times per sec

(because 60 sec = 1 min). In 2 sec, the disk will revolve 2 # 8 = 16 times. Each revolution is 360°, so in 2 sec a point on the edge of the disk will revolve

16 # 360° = 5760°. A unit analysis expression can also be used.

480 rev 1 min

* 1 min 60 sec

* 360° 1 rev

* 2 sec = 5760° Divide out common units. ■✔ Now Try Exercise 123.

CONCEPT PREVIEW Fill in the blank(s) to correctly complete each sentence.

1. One degree, written 1°, represents of a complete rotation.

2. If the measure of an angle is x°, its complement can be expressed as - x°.

3. If the measure of an angle is x°, its supplement can be expressed as - x°.

4. The measure of an angle that is its own complement is .

5. The measure of an angle that is its own supplement is .

6. One minute, written 1′, is of a degree.

7. One second, written 1″, is of a minute.

8. 12° 30′ written in decimal degrees is .

9. 55.25° written in degrees and minutes is .

10. If n represents any integer, then an expression representing all angles coterminal with 45° is 45° + .

1.1 Exercises

Find the measure of (a) the complement and (b) the supplement of an angle with the given measure. See Examples 1 and 3.

11. 30° 12. 60° 13. 45° 14. 90°

15. 54° 16. 10° 17. 1° 18. 89°

19. 14° 20′ 20. 39° 50′ 21. 20° 10′ 30″ 22. 50° 40′ 50″

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8 CHAPTER 1 Trigonometric Functions

Find the measure of each marked angle. See Example 2.

23.

(7x)° (11x)°

24. (20x + 10)° (3x + 9)°

25.

(2x)°

(4x)°

26.

(5x + 5)°

(3x + 5)°

27. (–4x)°

(–14x)°

28.

(9x)°

(9x)°

29. supplementary angles with measures 10x + 7 and 7x + 3 degrees

30. supplementary angles with measures 6x - 4 and 8x - 12 degrees

31. complementary angles with measures 9x + 6 and 3x degrees

32. complementary angles with measures 3x - 5 and 6x - 40 degrees

Find the measure of the smaller angle formed by the hands of a clock at the following times.

33. 34.

35. 3:15 36. 9:45 37. 8:20 38. 6:10

Perform each calculation. See Example 3.

39. 62° 18′ + 21° 41′ 40. 75° 15′ + 83° 32′ 41. 97° 42′ + 81° 37′

42. 110° 25′ + 32° 55′ 43. 47° 29′ - 71° 18′ 44. 47° 23′ - 73° 48′

45. 90° - 51° 28′ 46. 90° - 17° 13′ 47. 180° - 119° 26′

48. 180° - 124° 51′ 49. 90° - 72° 58′ 11″ 50. 90° - 36° 18′ 47″

51. 26° 20′ + 18° 17′ - 14° 10′ 52. 55° 30′ + 12° 44′ - 8° 15′

Convert each angle measure to decimal degrees. If applicable, round to the nearest thou- sandth of a degree. See Example 4(a).

53. 35° 30′ 54. 82° 30′ 55. 112° 15′ 56. 133° 45′

57. -60° 12′ 58. -70° 48′ 59. 20° 54′ 36″ 60. 38° 42′ 18″

61. 91° 35′ 54″ 62. 34° 51′ 35″ 63. 274° 18′ 59″ 64. 165° 51′ 09″

Convert each angle measure to degrees, minutes, and seconds. If applicable, round to the nearest second. See Example 4(b).

65. 39.25° 66. 46.75° 67. 126.76° 68. 174.255°

69. -18.515° 70. -25.485° 71. 31.4296° 72. 59.0854°

73. 89.9004° 74. 102.3771° 75. 178.5994° 76. 122.6853°

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9 1.1 Angles

Find the angle of least positive measure (not equal to the given measure) that is coterminal with each angle. See Example 5.

77. 32° 78. 86° 79. 26° 30′ 80. 58° 40′

81. -40° 82. -98° 83. -125° 30′ 84. -203° 20′

85. 361° 86. 541° 87. -361° 88. -541°

89. 539° 90. 699° 91. 850° 92. 1000°

93. 5280° 94. 8440° 95. -5280° 96. -8440°

Give two positive and two negative angles that are coterminal with the given quadrantal angle.

97. 90° 98. 180° 99. 0° 100. 270°

Write an expression that generates all angles coterminal with each angle. Let n represent any integer.

101. 30° 102. 45° 103. 135° 104. 225°

105. -90° 106. -180° 107. 0° 108. 360°

109. Why do the answers to Exercises 107 and 108 give the same set of angles?

110. Concept Check Which two of the following are not coterminal with r°?

A. 360° + r° B. r° - 360° C. 360° - r° D. r° + 180°

Concept Check Sketch each angle in standard position. Draw an arrow representing the correct amount of rotation. Find the measure of two other angles, one positive and one negative, that are coterminal with the given angle. Give the quadrant of each angle, if applicable.

111. 75° 112. 89° 113. 174° 114. 234°

115. 300° 116. 512° 117. -61° 118. -159°

119. 90° 120. 180° 121. -90° 122. -180°

Solve each problem. See Example 6.

123. Revolutions of a Turntable A turntable in a shop makes 45 revolutions per min. How many revolutions does it make per second?

124. Revolutions of a Windmill A windmill makes 90 revolutions per min. How many revolutions does it make per second?

125. Rotating Tire A tire is rotating 600 times per min. Through how many degrees does a

point on the edge of the tire move in 12 sec?

126. Rotating Airplane Propeller An airplane propeller rotates 1000 times per min. Find the number of degrees that a point on the edge of the propeller will rotate in 2 sec.

127. Rotating Pulley A pulley rotates through 75° in 1 min. How many rotations does the pulley make in 1 hr?

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10 CHAPTER 1 Trigonometric Functions

128. Surveying One student in a surveying class measures an angle as 74.25°, while another student measures the same angle as 74° 20′. Find the difference between these measurements, both to the nearest minute and to the nearest hundredth of a degree.

129. Viewing Field of a Telescope As a consequence of Earth’s rotation, celestial objects such as the moon and the stars appear to move across the sky, rising in the east and setting in the west. As a result, if a telescope on Earth remains stationary while viewing a celestial object, the object will slowly move outside the viewing field of the telescope. For this reason, a motor is often attached to telescopes so that the telescope rotates at the same rate as Earth. Determine how long it should take the motor to turn the telescope through an angle of 1 min in a direction per- pendicular to Earth’s axis.

130. Angle Measure of a Star on the American Flag Determine the measure of the angle in each point of the five-pointed star appearing on the American flag. (Hint: Inscribe the star in a circle, and use the following theorem from geometry: An angle whose vertex lies on the circumference of a circle is equal to half the central angle that cuts off the same arc. See the figure.)

74.25°

u

2u

u

1.2 Angle Relationships and Similar Triangles

Geometric Properties In Figure 13, we extended the sides of angle NMP to form another angle, RMQ. The pair of angles NMP and RMQ are vertical angles. Another pair of vertical angles, NMQ and PMR, are also formed. Vertical angles have the following important property.

■ Geometric Properties ■ Triangles

Q R

N P

M

Vertical angles

Figure 13

Vertical Angles

Vertical angles have equal measures.

Parallel lines are lines that lie in the same plane and do not intersect. Figure 14 shows parallel lines m and n. When a line q intersects two parallel lines, q is called a transversal. In Figure 14, the transversal intersecting the parallel lines forms eight angles, indicated by numbers.

m

q

n

1 2 3 4

5 6 7 8

Transversal

Parallel lines

Figure 14

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111.2 Angle Relationships and Similar Triangles

We learn in geometry that the degree measures of angles 1 through 8 in Figure 14 possess some special properties. The following table gives the names of these angles and rules about their measures.

Angle Pairs of Parallel Lines Intersected by a Transversal

Name Sketch Rule

Alternate interior angles

45

(also 3 and 6)

q

m

n

Angle measures are equal.

Alternate exterior angles 1

8 (also 2 and 7)

q

m

n

Angle measures are equal.

Interior angles on the same side of a transversal

46

(also 3 and 5)

q

m

n

Angle measures add to 180°.

Corresponding angles

6

2

(also 1 and 5, 3 and 7, 4 and 8)

q

m

n

Angle measures are equal.

m

n

(3x + 2)°

(5x – 40)°

1 2 3

4

Figure 15

EXAMPLE 1 Finding Angle Measures

Find the measures of angles 1, 2, 3, and 4 in Figure 15, given that lines m and n are parallel.

SOLUTION Angles 1 and 4 are alternate exterior angles, so they are equal.

3x + 2 = 5x - 40 Alternate exterior angles have equal measures. 42 = 2x Subtract 3x and add 40. 21 = x Divide by 2.

Angle 1 has measure

3x + 2 = 3 # 21 + 2 Substitute 21 for x. = 65°. Multiply, and then add.

Angle 4 has measure

5x - 40 = 5 # 21 - 40 Substitute 21 for x. = 65°. Multiply, and then

subtract.

Angle 2 is the supplement of a 65° angle, so it has measure

180° - 65° = 115°.

Angle 3 is a vertical angle to angle 1, so its measure is also 65°. (There are other ways to determine these measures.)

■✔ Now Try Exercises 11 and 19.

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12 CHAPTER 1 Trigonometric Functions

Triangles An important property of triangles, first proved by Greek geom- eters, deals with the sum of the measures of the angles of any triangle.

Angle Sum of a Triangle

The sum of the measures of the angles of any triangle is 180°.

A rather convincing argument for the truth of this statement uses any size trian- gle cut from a piece of paper. Tear each corner from the triangle, as suggested in Figure 16(a). We should be able to rearrange the pieces so that the three angles form a straight angle, which has measure 180°, as shown in Figure 16(b).

1

2

3

1 3

2

(a)

(b)

Figure 16

EXAMPLE 2 Applying the Angle Sum of a Triangle Property

The measures of two of the angles of a triangle are 48° and 61°. See Figure 17. Find the measure of the third angle, x.

SOLUTION 48° + 61° + x = 180° The sum of the angles is 180°. 109° + x = 180° Add. x = 71° Subtract 109°.

The third angle of the triangle measures 71°. ■✔ Now Try Exercises 13 and 23.

48° 61°

x

Figure 17

Types of Triangles

All acute One right angle One obtuse angle

Angles

Acute triangle Right triangle Obtuse triangle

All sides equal Two sides equal No sides equal

Sides

Equilateral triangle Isosceles triangle Scalene triangle

Similar triangles are triangles of exactly the same shape but not neces- sarily the same size. Figure 18 on the next page shows three pairs of similar triangles. The two triangles in Figure 18(c) have not only the same shape but also the same size. Triangles that are both the same size and the same shape are congruent triangles. If two triangles are congruent, then it is possible to pick one of them up and place it on top of the other so that they coincide.

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131.2 Angle Relationships and Similar Triangles

(a) (b) (c)

Figure 18

Conditions for Similar Triangles

Triangle ABC is similar to triangle DEF if the following conditions hold.

1. Corresponding angles have the same measure.

2. Corresponding sides are proportional. (That is, the ratios of their corre- sponding sides are equal.)

The triangular supports for a child’s swing set are congruent (and thus simi- lar) triangles, machine-produced with exactly the same dimensions each time. These supports are just one example of similar triangles. The supports of a long bridge, all the same shape but increasing in size toward the center of the bridge, are examples of similar (but not congruent) figures. See the photo.

Consider the correspondence between triangles ABC and DEF in Figure 19.

Angle A corresponds to angle D.

Angle B corresponds to angle E.

Angle C corresponds to angle F.

Side AB corresponds to side DE.

Side BC corresponds to side EF.

Side AC corresponds to side DF.

The small arcs found at the angles in Figure 19 denote the corresponding angles in the triangles.

CA

B

E

D F

Figure 19

EXAMPLE 3 Finding Angle Measures in Similar Triangles

In Figure 20, triangles ABC and NMP are similar. Find all unknown angle measures.

SOLUTION First, we find the measure of angle M using the angle sum prop- erty of a triangle.

104° + 45° + M = 180° The sum of the angles is 180°. 149° + M = 180° Add.

M = 31° Subtract 149°.

The triangles are similar, so corresponding angles have the same measure. Because C corresponds to P and P measures 104°, angle C also measures 104°. Angles B and M correspond, so B measures 31°.

■✔ Now Try Exercise 49.

Figure 20

A

45°

C B

M P

N 45°

104°

If two triangles are congruent, then they must be similar. However, two similar triangles need not be congruent.

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14 CHAPTER 1 Trigonometric Functions

EXAMPLE 4 Finding Side Lengths in Similar Triangles

Given that triangle ABC and triangle DFE in Figure 21 are similar, find the lengths of the unknown sides of triangle DFE.

SOLUTION Similar triangles have corre- sponding sides in proportion. Use this fact to find the unknown side lengths in triangle DFE.

Side DF of triangle DFE corresponds to side AB of triangle ABC, and sides DE and AC correspond. This leads to the following proportion.

8 16

= DF 24

Recall this property of proportions from algebra.

If a b =

c d

, then ad = bc.

We use this property to solve the equation for DF.

8 16

= DF 24

Corresponding sides are proportional.

8 # 24 = 16 # DF If ab = cd , then ad = bc. 192 = 16 # DF Multiply. 12 = DF Divide by 16.

Side DF has length 12. Side EF corresponds to CB. This leads to another proportion.

8 16

= EF 32

Corresponding sides are proportional.

8 # 32 = 16 # EF If ab = cd , then ad = bc. 16 = EF Solve for EF.

Side EF has length 16. ■✔ Now Try Exercise 55.

➡➡

➡➡

16

BA

C

32

24 FD

E

8

Figure 21

EXAMPLE 5 Finding the Height of a Flagpole

Workers must measure the height of a building flagpole. They find that at the instant when the shadow of the building is 18 m long, the shadow of the flagpole is 27 m long. The building is 10 m high. Find the height of the flagpole.

SOLUTION Figure 22 shows the information given in the problem. The two triangles are similar, so corresponding sides are in proportion.

MN 10

= 27 18

Corresponding sides are proportional.

MN 10

= 3 2

Write 2718 in lowest terms.

MN # 2 = 10 # 3 Property of proportions MN = 15 Solve for MN.

The flagpole is 15 m high. ■✔ Now Try Exercise 59.

10

18

27 M

N

Figure 22

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151.2 Angle Relationships and Similar Triangles

CONCEPT PREVIEW Fill in the blank(s) to correctly complete each sentence.

1. The sum of the measures of the angles of any triangle is .

2. An isosceles right triangle has one angle and equal sides.

3. An equilateral triangle has equal sides.

4. If two triangles are similar, then their corresponding are proportional and their corresponding have equal measure.

1.2 Exercises

CONCEPT PREVIEW In each figure, find the measures of the numbered angles, given that lines m and n are parallel.

5.

131° 1 2 3

4 5 6 7

m

n

6.

m

n

120° 10

3 1 2

9

4 55°5

8 67

CONCEPT PREVIEW Name the corresponding angles and the corresponding sides of each pair of similar triangles.

7.

AC

B

P R

Q 8.

A C

B Q

RP

9. (EA is parallel to CD.) 10. (HK is parallel to EF.)

E

B

D A

C

G

H K

FE

Find the measure of each marked angle. In Exercises 19–22, m and n are parallel. See Examples 1 and 2.

11.

(5x – 129)° (2x – 21)°

12.

(11x – 37)° (7x + 27)°

13.

(210 – 3x)°

(x + 20)°

x°

14.

(10x – 20)°

(x + 15)°

(x + 5)°

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16 CHAPTER 1 Trigonometric Functions

15. (2x – 120)°

(x – 30)°

x + 15 °1_2( )

16. (2x + 16)°

(5x – 50)°

(3x – 6)°

17. (6x + 3)°

(9x + 12)° (4x – 3)°

18. (–5x)°

(7 – 12x)° (–8x + 3)°

19.

(2x – 5)° (x + 22)°

m

n

20.

(2x + 61)°

(6x – 51)°

m

n

21.

(4x – 56)°

nm

(x + 1)°

22.

(10x + 11)°

nm

(15x – 54)°

The measures of two angles of a triangle are given. Find the measure of the third angle. See Example 2.

23. 37°, 52° 24. 29°, 104° 25. 147° 12′, 30° 19′

26. 136° 50′, 41° 38′ 27. 74.2°, 80.4° 28. 29.6°, 49.7°

29. 51° 20′ 14″, 106° 10′ 12″ 30. 17° 41′ 13″, 96° 12′ 10″

31. Concept Check Can a triangle have angles of measures 85° and 100°?

32. Concept Check Can a triangle have two obtuse angles?

Concept Check Classify each triangle as acute, right, or obtuse. Also classify each as equilateral, isosceles, or scalene. See the discussion following Example 2.

33. 34.

120°

35.

60° 60°

60°

88

8

36.

9

6

9

37. 90°

3

4 5

38.

130°

8

8

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171.2 Angle Relationships and Similar Triangles

39.

4

4

40.

60°

41.

96°

14

10

42.

rr

r 60° 60°

60° 43.

50° 50°

44.

45. Angle Sum of a Triangle Use this figure to dis- cuss why the measures of the angles of a triangle must add up to the same sum as the measure of a straight angle.

46. Carpentry Technique The following tech- nique is used by carpenters to draw a 60° angle with a straightedge and a compass. Why does this technique work? (Source: Hamilton, J. E. and M. S. Hamilton, Math to Build On, Con- struction Trades Press.)

“Draw a straight line segment, and mark a point near the midpoint. Now place the tip on the marked point, and draw a semicircle. Without changing the setting of the compass, place the tip at the right intersection of the line and the semicircle, and then mark a small arc across the semicircle. Finally, draw a line segment from the marked point on the original segment to the point where the arc crosses the semicircle. This will form a 60° angle with the original segment.”

90°

13

12

5

m

n

1 2

3

2 514

T

P

QR m and n are parallel.

60°

Point marked on line

Tip placed here

Find all unknown angle measures in each pair of similar triangles. See Example 3.

47.

C B

A

42°

PR

Q 48.

49.

C 30°

B

A

N

K

M 30°

106°

50.

C

78°

46°

B

A

M

N

P

Z

X

74°

Y V

T

28° W

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18 CHAPTER 1 Trigonometric Functions

51. X

Z

Y

38°

M

N

P 38°

52.

64°

20°

Q R

P

V U

T

Find the unknown side lengths in each pair of similar triangles. See Example 4.

53.

a

b

25

8

6

10

54. a

75

b

25

2010

55.

15

12 12 a

b

6

56.

9

6 3

a

57.

x

6

9

4

58. 21

14

12m

Solve each problem. See Example 5.

59. Height of a Tree A tree casts a shadow 45 m long. At the same time, the shadow cast by a vertical 2-m stick is 3 m long. Find the height of the tree.

60. Height of a Lookout Tower A forest fire lookout tower casts a shadow 180 ft long at the same time that the shadow of a 9-ft truck is 15 ft long. Find the height of the tower.

61. Lengths of Sides of a Triangle On a photograph of a triangular piece of land, the lengths of the three sides are 4 cm, 5 cm, and 7 cm, respectively. The shortest side of the actual piece of land is 400 m long. Find the lengths of the other two sides.

62. Height of a Lighthouse The Biloxi lighthouse in the figure casts a shadow 28 m long at 7 a.m. At the same time, the shadow of the lighthouse keeper, who is 1.75 m tall, is 3.5 m long. How tall is the lighthouse?

28 m

3.5 m

NOT TO SCALE

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191.2 Angle Relationships and Similar Triangles

63. Height of a Building A house is 15 ft tall. Its shadow is 40 ft long at the same time that the shadow of a nearby building is 300 ft long. Find the height of the building.

64. Height of a Carving of Lincoln Assume that Lincoln was 6 13 ft tall and his head 3 4 ft long. Knowing that the carved head of Lincoln at Mt. Rushmore is 60 ft tall, find how tall his entire body would be if it were carved into the mountain.

60 ft

In each figure, there are two similar triangles. Find the unknown measurement. Give approximations to the nearest tenth.

65.

x 50

120100

66.

y

40160

60

67. c

10 90

100

68. m

5 75

80

Solve each problem.

69. Solar Eclipse on Earth The sun has a diameter of about 865,000 mi with a maxi- mum distance from Earth’s surface of about 94,500,000 mi. The moon has a smaller diam- eter of 2159 mi. For a total solar eclipse to occur, the moon must pass between Earth and the sun. The moon must also be close enough to Earth for the moon’s umbra (shadow) to reach the surface of Earth. (Source: Karttunen, H., P. Kröger, H. Oja, M. Putannen, and K. Donners, Editors, Funda- mental Astronomy, Fourth Edition, Springer-Verlag.)

(a) Calculate the maximum distance, to the nearest thousand miles, that the moon can be from Earth and still have a total solar eclipse occur. (Hint: Use similar triangles.)

(b) The closest approach of the moon to Earth’s surface was 225,745 mi and the far- thest was 251,978 mi. (Source: World Almanac and Book of Facts.) Can a total solar eclipse occur every time the moon is between Earth and the sun?

70. Solar Eclipse on Neptune (Refer to Exercise 69.) The sun’s distance from Neptune is approximately 2,800,000,000 mi (2.8 billion mi). The largest moon of Neptune is Triton, with a diameter of approximately 1680 mi. (Source: World Almanac and Book of Facts.)

(a) Calculate the maximum distance, to the nearest thousand miles, that Triton can be from Neptune for a total eclipse of the sun to occur on Neptune. (Hint: Use similar triangles.)

(b) Triton is approximately 220,000 mi from Neptune. Is it possible for Triton to cause a total eclipse on Neptune?

EarthMoon

Umbra

Sun

NOT TO SCALE

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20 CHAPTER 1 Trigonometric Functions

2°

10° 20°

71. Solar Eclipse on Mars (Refer to Exercise 69.) The sun’s distance from the surface of Mars is approximately 142,000,000 mi. One of Mars’ two moons, Phobos, has a maximum diameter of 17.4 mi. (Source: World Almanac and Book of Facts.)

(a) Calculate the maximum distance, to the nearest hundred miles, that the moon Phobos can be from Mars for a total eclipse of the sun to occur on Mars.

(b) Phobos is approximately 5800 mi from Mars. Is it possible for Phobos to cause a total eclipse on Mars?

72. Solar Eclipse on Jupiter (Refer to Exercise 69.) The sun’s distance from the sur- face of Jupiter is approximately 484,000,000 mi. One of Jupiter’s moons, Gany- mede, has a diameter of 3270 mi. (Source: World Almanac and Book of Facts.)

(a) Calculate the maximum distance, to the nearest thousand miles, that the moon Ganymede can be from Jupiter for a total eclipse of the sun to occur on Jupiter.

(b) Ganymede is approximately 665,000 mi from Jupiter. Is it possible for Ganymede to cause a total eclipse on Jupiter?

73. Sizes and Distances in the Sky Astronomers use degrees, minutes, and seconds to measure sizes and distances in the sky along an arc from the horizon to the zenith point directly overhead. An adult observer on Earth can judge distances in the sky using his or her hand at arm’s length. An outstretched hand will be about 20 arc degrees wide from the tip of the thumb to the tip of the little finger. A clenched fist at arm’s length measures about 10 arc degrees, and a thumb corresponds to about 2 arc degrees. (Source: Levy, D. H., Skywatching, The Nature Company.)

(a) The apparent size of the moon is about 31 arc minutes. Approximately what part of your thumb would cover the moon?

(b) If an outstretched hand plus a fist cover the distance between two bright stars, about how far apart in arc degrees are the stars?

74. Estimates of Heights There is a relatively simple way to make a reasonable esti- mate of a vertical height.

Step 1 Hold a 1-ft ruler vertically at arm’s length and approach the object to be measured.

Step 2 Stop when one end of the ruler lines up with the top of the object and the other end with its base.

Step 3 Now pace off the distance to the object, taking normal strides. The number of paces will be the approximate height of the object in feet.

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