Math Trigonometry …… due tomorrow…….
This assignment is due tomorrow…… no late work….. due 6pm california time…
Use the attachment which is my Trigonometry book to do the following:
Sec 4.1 & 4.2 Graphs of sine and cosine Functions (Homework 4.1 # 27,31,33,35) (Homework 4.2 #29-35 odds, 57, 58)
Sec 4.3 Graphs of Tangent and Cotangent Functions ( Homework 4.3 # 15, 19, 21, 25, 35)
Sec 4.3 Graphs of Tangent and Cotangent Functions ( Homework 4.3 # 15, 19, 21, 25, 35)
Sec 4.4 Graphs of the Secant and Cosecant Functions ( Homework 4.4 # 7,8,11)
Sec 4.4 Graphs of the Secant and Cosecant Functions ( Homework 4.4 # 7,8,11)
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
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Boston Columbus Indianapolis New York San Francisco
Amsterdam Cape Town Dubai London Madrid Milan Munich Paris Montral Toronto
Delhi Mexico City So 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 Alexisand in memory of Mark
E.J.H.
To Coach Lonnie Myersthank 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 1.11.2) 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 2.12.3) 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 3.13.3) 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 5.15.4) 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 6.16.3) 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) Herons Formula for the Area of a Triangle Derivation
of Herons Formula
Chapter 7 Quiz ( Sections 7.17.3) 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 Moivres Theorem; Powers and Roots of Complex
Numbers 378
Powers of Complex Numbers (De Moivres Theorem) Roots of Complex
Numbers
Chapter 8 Quiz (Sections 8.18.4) 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 todays 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 Pearsons 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 takenand 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 Todays 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 worlds 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
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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-
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
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structure for students to use while
they read the text or watch the
MyMathLab videos. MyClassroom
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the instructor.
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Resources for Success
Student Supplements
Students 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 Instructors 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 Instructors Solutions Manual
By Beverly Fusfield
Provides complete solutions to all text exercises
Available in MyMathLab or downloadable from
Pearson Educations online catalog
Online Instructors 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 Educations 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 Educations 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 Educations online catalog
www.mymathlab.com
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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 OBrien, 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 [emailprotected]
Margaret L. Lial
John Hornsby
David I. Schneider
Callie J. Daniels
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mailto:[emailprotected]
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 segm
