Algebra-and-Trigonometry-Solution-by-Sheldon-Axler

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1

Algebra &
Trigonometry
with

Student Solutions Manual

Sheldon Axler
San Francisco State University

JOHN WILEY & SONS, INC.

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ISBN-13
ISBN-13
ISBN-13

978-0470-58579-5 (hardcover)
978-0470-47081-7 (softcover)
978-0470-47082-4 (binder ready)

Printed in the United States of America
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About the Author

Sheldon Axler is Dean of
the College of Science & Engineering at San Francisco
State University, where he
joined the faculty as Chair of the Mathematics Department in 1997.
Axler was valedictorian of his high school in Miami, Florida. He received
his AB from Princeton University with highest honors, followed by a PhD in
Mathematics from the University of California at Berkeley.
As a postdoctoral Moore Instructor at MIT, Axler received a university-wide

teaching award. Axler was then an assistant professor, associate professor,
and professor at Michigan State University, where he received the first J.
Sutherland Frame Teaching Award and the Distinguished Faculty Award.
Axler received the Lester R. Ford Award for expository writing from the
Mathematical Association of America in 1996. In addition to publishing
numerous research papers, Axler is the author of Linear Algebra Done Right
(which has been adopted as a textbook at over 240 universities and colleges),
College Algebra, and Precalculus: A Prelude to Calculus; he is also co-author
of Harmonic Function Theory (a graduate/research-level book).
Axler has served as Editor-in-Chief of the Mathematical Intelligencer and as
Associate Editor of the American Mathematical Monthly. He has been a member of the Council of the American Mathematical Society and a member of
the Board of Trustees of the Mathematical Sciences Research Institute. Axler
currently serves on the editorial board of Springer’s series Undergraduate
Texts in Mathematics, Graduate Texts in Mathematics, and Universitext.

v

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Contents

About the Author

v

Preface to the Instructor
WileyPLUS

xvi

xxii

Acknowledgments

xxiii

Preface to the Student
1 The Real Numbers

1

1.1 The Real Line

2

xxvi

Construction of the Real Line

2

Is Every Real Number Rational?
Problems

3

6

1.2 Algebra of the Real Numbers

7

Commutativity and Associativity

7

The Order of Algebraic Operations
The Distributive Property

8

10

Additive Inverses and Subtraction

11

Multiplicative Inverses and the Algebra of Fractions
Symbolic Calculators

16

Exercises, Problems, and Worked-out Solutions

1.3 Inequalities, Intervals, and Absolute Value
Positive and Negative Numbers
Lesser and Greater
Intervals

13

19

24

24

25

27

Absolute Value

30

Exercises, Problems, and Worked-out Solutions

33

Chapter Summary and Chapter Review Questions
vi

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40


Contents vii

2 Combining Algebra and Geometry
2.1 The Coordinate Plane
Coordinates

41

42

42

Graphs of Equations

44

Distance Between Two Points

46

Length, Perimeter, and Circumference

48

Exercises, Problems, and Worked-out Solutions

2.2 Lines

57

Slope

57

The Equation of a Line
Parallel Lines
Midpoints

58

61

Perpendicular Lines

62

64

Exercises, Problems, and Worked-out Solutions

2.3 Quadratic Expressions and Conic Sections
Completing the Square

75

The Quadratic Formula

77

Circles

66

75

79

Ellipses

81

Parabolas
Hyperbolas

83
85

Exercises, Problems, and Worked-out Solutions

2.4 Area

50

88

98

Squares, Rectangles, and Parallelograms
Triangles and Trapezoids
Stretching

98

99

101

Circles and Ellipses

102

Exercises, Problems, and Worked-out Solutions

105

Chapter Summary and Chapter Review Questions
3 Functions and Their Graphs
3.1 Functions

117

118

Definition and Examples

118

The Graph of a Function

121

The Domain of a Function

The Range of a Function

124
126

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115


viii Contents

Functions via Tables

128

Exercises, Problems, and Worked-out Solutions

3.2 Function Transformations and Graphs

129

142

Vertical Transformations: Shifting, Stretching, and Flipping
Horizontal Transformations: Shifting, Stretching, Flipping
Combinations of Vertical Function Transformations
Even Functions

152

Odd Functions

153

Exercises, Problems, and Worked-out Solutions

3.3 Composition of Functions

165

Definition of Composition

166

Order Matters in Composition

149

154

169

170

Composing More than Two Functions

171

Function Transformations as Compositions

172

Exercises, Problems, and Worked-out Solutions

3.4 Inverse Functions

174

180

The Inverse Problem

180

One-to-one Functions

181

The Definition of an Inverse Function

182

The Domain and Range of an Inverse Function

184

The Composition of a Function and Its Inverse

185

Comments About Notation

187

Exercises, Problems, and Worked-out Solutions

3.5 A Graphical Approach to Inverse Functions
The Graph of an Inverse Function

Increasing and Decreasing Functions
Inverse Functions via Tables

189

197

197

Graphical Interpretation of One-to-One

199

200

203

Exercises, Problems, and Worked-out Solutions

204

Chapter Summary and Chapter Review Questions
4 Polynomial and Rational Functions
4.1 Integer Exponents

145

165

Combining Two Functions

Decomposing Functions

142

214

Positive Integer Exponents

214

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213

209


Contents ix

Properties of Exponents
Defining x

0

215

217

Negative Integer Exponents

218

Manipulations with Exponents

219

Exercises, Problems, and Worked-out Solutions

4.2 Polynomials

221

227

The Degree of a Polynomial

227

The Algebra of Polynomials

228

Zeros and Factorization of Polynomials

230

The Behavior of a Polynomial Near ±∞

234

Graphs of Polynomials

237

Exercises, Problems, and Worked-out Solutions

4.3 Rational Functions

239

245

Ratios of Polynomials

245

The Algebra of Rational Functions
Division of Polynomials

246

247

The Behavior of a Rational Function Near ±∞
Graphs of Rational Functions

250

253

Exercises, Problems, and Worked-out Solutions

4.4 Complex Numbers

255

262

The Complex Number System

262

Arithmetic with Complex Numbers

263

Complex Conjugates and Division of Complex Numbers
Zeros and Factorization of Polynomials, Revisited
Exercises, Problems, and Worked-out Solutions

271

Chapter Summary and Chapter Review Questions
5 Exponents and Logarithms

276

279

5.1 Exponents and Exponential Functions
Roots

264

268

280

280

Rational Exponents
Real Exponents

284

285

Exponential Functions

286

Exercises, Problems, and Worked-out Solutions

287

5.2 Logarithms as Inverses of Exponential Functions
Logarithms Base 2

293

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293


x Contents

Logarithms with Any Base

295

Common Logarithms and the Number of Digits
Logarithm of a Power

297

Radioactive Decay and Half-Life

299

Exercises, Problems, and Worked-out Solutions

5.3 Applications of Logarithms
Logarithm of a Product

301

310

310

Logarithm of a Quotient

311

Earthquakes and the Richter Scale
Sound Intensity and Decibels

312

313

Star Brightness and Apparent Magnitude
Change of Base

297

315

316

Exercises, Problems, and Worked-out Solutions

5.4 Exponential Growth

328

Functions with Exponential Growth
Population Growth

319

329

333

Compound Interest

335

Exercises, Problems, and Worked-out Solutions

340

Chapter Summary and Chapter Review Questions
6 e and the Natural Logarithm
6.1 Defining e and ln

349

350

Estimating Area Using Rectangles
Defining e

350

352

Defining the Natural Logarithm

355

Properties of the Exponential Function and ln
Exercises, Problems, and Worked-out Solutions

356
358

6.2 Approximations and area with e and ln 366
Approximation of the Natural Logarithm

366

Approximations with the Exponential Function
An Area Formula

368

369

Exercises, Problems, and Worked-out Solutions

6.3 Exponential Growth Revisited

376

Continuously Compounded Interest
Continuous Growth Rates
Doubling Your Money

377

378

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376

372

347


Contents xi

Exercises, Problems, and Worked-out Solutions

380

Chapter Summary and Chapter Review Questions
7 Systems of Equations

385

387

7.1 Equations and Systems of Equations
Solving an Equation

388

388

Solving a System of Equations Graphically

391

Solving a System of Equations by Substitution
Exercises, Problems, and Worked-out Solutions

7.2 Solving Systems of Linear Equations
Linear Equations: How Many Solutions?
Systems of Linear Equations
Gaussian Elimination

392
393

399
399

402

404

Exercises, Problems, and Worked-out Solutions

406

7.3 Solving Systems of Linear Equations Using Matrices
Representing Systems of Linear Equations by Matrices
Gaussian Elimination with Matrices

411

411

413

Systems of Linear Equations with No Solutions

415

Systems of Linear Equations with Infinitely Many Solutions
How Many Solutions, Revisited

Exercises, Problems, and Worked-out Solutions

7.4 Matrix Algebra
Matrix Size

419

424

424

Adding and Subtracting Matrices

426

Multiplying a Matrix by a Number
Multiplying Matrices

427

428

The Inverse of a Matrix

433

Exercises, Problems, and Worked-out Solutions

440

Chapter Summary and Chapter Review Questions
8 Sequences, Series, and Limits

8.1 Sequences

416

418

447

448

Introduction to Sequences

448

Arithmetic Sequences

450

Geometric Sequences

451

Recursively Defined Sequences

454

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445

xii Contents

Exercises, Problems, and Worked-out Solutions

8.2 Series

456

463

Sums of Sequences

463

Arithmetic Series

463

Geometric Series

466

Summation Notation

468

The Binomial Theorem

470

Exercises, Problems, and Worked-out Solutions

8.3 Limits

476

483

Introduction to Limits
Infinite Series

483

487

Decimals as Infinite Series
Special Infinite Series

489

491

Exercises, Problems, and Worked-out Solutions

493

Chapter Summary and Chapter Review Questions
9 Trigonometric Functions
9.1 The Unit Circle

497

498

The Equation of the Unit Circle
Angles in the Unit Circle
Negative Angles

498

499

501

Angles Greater Than 360◦
Length of a Circular Arc

502
503

Special Points on the Unit Circle

504

Exercises, Problems, and Worked-out Solutions

9.2 Radians

506

514

A Natural Unit of Measurement for Angles
The Radius Corresponding to an Angle
Length of a Circular Arc
Area of a Slice

514

517

520

521

Special Points on the Unit Circle

522

Exercises, Problems, and Worked-out Solutions

9.3 Cosine and Sine

523

529

Definition of Cosine and Sine

529

The Signs of Cosine and Sine

532

The Key Equation Connecting Cosine and Sine

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534

496


Contents xiii

The Graphs of Cosine and Sine

535

Exercises, Problems, and Worked-out Solutions

9.4 More Trigonometric Functions
Definition of Tangent
The Sign of Tangent

537

542

542
544

Connections Among Cosine, Sine, and Tangent
The Graph of Tangent

545

545

Three More Trigonometric Functions

547

Exercises, Problems, and Worked-out Solutions

9.5 Trigonometry in Right Triangles

555

Trigonometric Functions via Right Triangles
Two Sides of a Right Triangle

555

557

One Side and One Angle of a Right Triangle

558

Exercises, Problems, and Worked-out Solutions

9.6 Trigonometric Identities

549

559

566

The Relationship Among Cosine, Sine, and Tangent

566

Trigonometric Identities for the Negative of an Angle
Trigonometric Identities with

π
2

568

570

Trigonometric Identities Involving a Multiple of π
Exercises, Problems, and Worked-out Solutions

572

575

Chapter Summary and Chapter Review Questions
10 Trigonometric Algebra and Geometry

583

10.1 Inverse Trigonometric Functions

584

The Arccosine Function
The Arcsine Function

580

584

587

The Arctangent Function

590

Exercises, Problems, and Worked-out Solutions

10.2 Inverse Trigonometric Identities

593

599

Composition of Trigonometric Functions and Their Inverses

599

The Arccosine, Arcsine, and Arctangent of −t: Graphical
Approach 600
The Arccosine, Arcsine, and Arctangent of −t: Algebraic
Approach 602
Arccosine Plus Arcsine
The Arctangent of

1
t

603

604

More Compositions with Inverse Trigonometric Functions

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605


xiv Contents

Exercises, Problems, and Worked-out Solutions

10.3 Using Trigonometry to Compute Area
The Area of a Triangle via Trigonometry
Ambiguous Angles

613
613

614

The Area of a Parallelogram via Trigonometry
The Area of a Polygon

616

617

Trigonometric Approximations

619

Exercises, Problems, and Worked-out Solutions

10.4 The Law of Sines and the Law of Cosines
The Law of Sines

The Law of Cosines

628

629

631

Using the Law of Cosines
When to Use Which Law

632
634

Exercises, Problems, and Worked-out Solutions

10.5 Double-Angle and Half-Angle Formulas
The Sine of 2θ

636

644

644

645

The Tangent of 2θ

646

The Cosine and Sine of
The Tangent of

622

628

Using the Law of Sines

The Cosine of 2θ

608

θ
2

θ
2

647

649

Exercises, Problems, and Worked-out Solutions

10.6 Addition and Subtraction Formulas
The Cosine of a Sum and Difference
The Sine of a Sum and Difference

650

658

658

660

The Tangent of a Sum and Difference

661

Exercises, Problems, and Worked-out Solutions

662

Chapter Summary and Chapter Review Questions
11 Applications of Trigonometry
11.1 Parametric Curves

671

672

Curves in the Coordinate Plane

672

Graphing Inverse Functions as Parametric Curves

677

Shifting, Stretching, or Flipping a Parametric Curve

Exercises, Problems, and Worked-out Solutions

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681

678

668


Contents xv

11.2 Transformations of Trigonometric Functions
Amplitude
Period

687

687

689

Phase Shift

692

Fitting Transformations of Trigonometric Functions to Data
Exercises, Problems, and Worked-out Solutions

11.3 Polar Coordinates

696

705

Defining Polar Coordinates

705

Converting from Polar to Rectangular Coordinates

706

Converting from Rectangular to Polar Coordinates

707

Graphs of Polar Equations

711

Exercises, Problems, and Worked-out Solutions

11.4 Vectors

694

715

718

An Algebraic and Geometric Introduction to Vectors
Vector Addition

720

Vector Subtraction

723

Scalar Multiplication
The Dot Product

718

725

726

Exercises, Problems, and Worked-out Solutions

11.5 The Complex Plane

728

732

Complex Numbers as Points in the Plane

732

Geometric Interpretation of Complex Multiplication and Division
De Moivre’s Theorem

737

Finding Complex Roots

738

Exercises, Problems, and Worked-out Solutions

739

Chapter Summary and Chapter Review Questions
Photo Credits
Index

743

745

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741

734

Preface to the Instructor
Goals
This book aims to provide students with the algebraic and trigonometric
skills and understanding needed for other coursework and for participating
as an educated citizen in a complex society.
Mathematics faculty frequently complain that many students do not read
the textbook. When doing homework, a typical student may look only at
the relevant section of the textbook or the student solutions manual for
an example similar to the homework problem at hand. The student reads
enough of that example to imitate the procedure and then does the homework
problem. Little understanding may take place.
In contrast, this book is designed to be read by students. The writing
style and layout are meant to induce students to read and understand the
material. Explanations are more plentiful than typically found in algebra
and trigonometry books. Examples of the concepts make the ideas concrete
whenever possible.

Exercises and Problems
Each exercise has a
unique correct answer, usually a number or a function;
most problems have
multiple correct answers, usually explanations or examples.

This book contains
what is usually
a separate book
called the student
solutions manual.

Students learn mathematics by actively working on a wide range of exercises

and problems. Ideally, a student who reads and understands the material in
a section of this book should be able to do the exercises and problems in
that section without further help. However, some of the exercises require
application of the ideas in a context that students may not have seen before;
many students will need help with these exercises. This help is available
from the complete worked-out solutions to all the odd-numbered exercises
that appear at the end of each section.
Because the worked-out solutions were written solely by the author of
the textbook, students can expect a consistent approach to the material.
Furthermore, students will save money by not having to purchase a separate
student solutions manual.
The exercises (but not the problems) occur in pairs, so that an oddnumbered exercise is followed by an even-numbered exercise whose solution
uses the same ideas and techniques. A student stumped by an even-numbered
exercise should be able to tackle it after reading the worked-out solution to
the corresponding odd-numbered exercise. This arrangement allows the text
to focus more centrally on explanations of the material and examples of the
concepts.
xvi

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Preface to the Instructor xvii

Most students will read the student solutions manual when they are
assigned homework, even though they are reluctant to read the main text.
The integration of the student solutions manual within this book should
encourage students to drift over and also read the main text. To reinforce
this tendency, the worked-out solutions to the odd-numbered exercises at
the end of each section are intentionally typeset with a slightly less appealing

style (smaller type, two-column format, and not right justified) than the main
text. The reader-friendly appearance of the main text might nudge students
to spend some time there.
Exercises and problems in this book vary greatly in difficulty and purpose.
Some exercises and problems are designed to hone algebraic manipulation
skills; other exercises and problems are designed to push students to genuine
understanding beyond rote algorithmic calculation.
Some exercises and problems intentionally reinforce material from earlier
in the book and require multiple steps. For example, Exercise 30 in Section 5.3
asks students to find all numbers x such that
log5 (x + 4) + log5 (x + 2) = 2.
To solve this exercise, students will need to use the formula for a sum of
logarithms as well as the quadratic formula; they will also need to eliminate
one of the potential solutions produced by the quadratic formula because it
would lead to the evaluation of the logarithm of a negative number. Although
such multi-step exercises require more thought than most exercises in the
book, they allow students to see crucial concepts more than once, sometimes
in unexpected contexts.

The Calculator Issue
The issue of whether and how calculators should be used by students has
generated immense controversy.
Some sections of this book have many exercises and problems designed
for calculators (for example Section 5.4 on exponential growth), but some
sections deal with material not as amenable to calculator use. The text seeks
to provide students with both understanding and skills. Thus the book
does not aim for an artificially predetermined percentage of exercises and
problems in each section requiring calculator use.
Some exercises and problems that require a calculator are intentionally
designed to make students realize that by understanding the material, they

can overcome the limitations of calculators. As one example among many,
Exercise 83 in Section 5.3 asks students to find the number of digits in the
decimal expansion of 74000 . Brute force with a calculator will not work with
this problem because the number involved has too many digits. However, a
few moments’ thought should show students that they can solve this problem
by using logarithms (and their calculators!).

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To aid instructors in
presenting the kind of
course they want, the
symbol
appears
with exercises and
problems that require
students to use a calculator.


xviii Preface to the Instructor
Regardless of what
level of calculator use
an instructor expects,
students should not
turn to a calculator
to compute something like log 1, because then log has
become just a button on the calculator.

The calculator icon
can be interpreted for some exercises, depending

on the instructor’s preference, to mean that the solution should be a decimal
approximation rather than the exact answer. For example, Exercise 3 in
Section 6.3 asks how much would need to be deposited in a bank account
paying 4% interest compounded continuously so that at the end of 10 years
the account would contain $10,000. The exact answer to this exercise is
10000/e0.4 dollars, but it may be more satisfying to the student (after obtaining the exact answer) to use a calculator to see that approximately $6,703
needs to be deposited.
For exercises such as the one described in the paragraph above, instructors
can decide whether to ask for exact answers or decimal approximations or
both (the worked-out solutions for the odd-numbered exercises will usually
contain both). If an instructor asks for only an exact answer, then a calculator
may not be needed despite the presence of the calculator icon.
Symbolic processing programs such as Mathematica and Maple offer appealing alternatives to hand-held calculators because of their ability to solve
equations and deal with symbols as well as numbers. Furthermore, the larger
size, better resolution, and color on a computer screen make graphs produced by such software more informative than graphs on a typical hand-held
graphing calculator.
Your students may not use a symbolic processing program because of the
complexity or expense of such software. However, easy-to-use free web-based
symbolic programs are becoming available. Occasionally this book shows
how students can use Wolfram|Alpha, which has almost no learning curve, to
go beyond what can be done easily by hand.
Even if you do not tell your students about such free tools, knowledge
about such web-based homework aids is likely to spread rapidly among
students.

Distinctive Approaches
Half-life and Exponential Growth
Almost all algebra and trigonometry books present radioactive decay as an
example of exponential decay. Amazingly, the typical algebra and trigonometry textbook states that if a radioactive isotope has half-life h, then the
amount left at time t will equal e−(t ln 2)/h times the amount present at time 0.

A much clearer formulation would state, as this textbook does, that the
amount left at time t will equal 2−t/h times the amount present at time 0. The
unnecessary use of e and ln 2 in this context may suggest to students that e
and natural logarithms have only contrived and artificial uses, which is not
the message a textbook should send. Using 2−t/h helps students understand
the concept of half-life, with a formula connected to the meaning of the
concept.
Similarly, many algebra and trigonometry textbooks consider, for example,
a colony of bacteria doubling in size every 3 hours, with the textbook then

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Preface to the Instructor xix

producing the formula e(t ln 2)/3 for the growth factor after t hours. The
simpler and more natural formula 2t/3 seems not to be mentioned in such
books. This book presents the more natural approach to such issues of
exponential growth and decay.
Algebraic Properties of Logarithms
The base for logarithms in Chapter 5 is arbitrary. Most of the examples and
motivation use logarithms base 2 or logarithms base 10. Students will see
how the algebraic properties of logarithms follow easily from the properties
of exponents.
The crucial concepts of e and natural logarithms are saved for Chapter 6.
Thus students can concentrate in Chapter 5 on understanding logarithms
(arbitrary base) and their properties without at the same time worrying about
grasping concepts related to e. Similarly, when natural logarithms arise
naturally in Chapter 6, students should be able to concentrate on issues
surrounding e without at the same time learning properties of logarithms.

The initial separation
of logarithms and e
should help students
master both concepts.

Area
Section 2.4 in this book builds the intuitive notion of area starting with
squares, and then quickly derives formulas for the area of rectangles, triangles, parallelograms, and trapezoids. A discussion of the effects of stretching
either horizontally or vertically easily leads to the familiar formula for the
area enclosed by a circle. Similar ideas are then used to find the formula for
the area inside an ellipse (without calculus!).
Section 6.1 deals with the question of estimating the area under parts of
the curve y = x1 by using rectangles. This easy nontechnical introduction,
with its emphasis on ideas without the clutter of the notation of Riemann
sums, gives students a taste of an important idea from calculus.
e, The Exponential Function, and the Natural Logarithm
Most algebra and trigonometry textbooks either present no motivation for e
or motivate e via continuously compounding interest or through the limit of
an indeterminate expression of the form 1∞ ; these concepts are difficult for
students at this level to understand.
Chapter 6 presents a clean and well-motivated approach to e and the
natural logarithm. We do this by looking at the area (intuitively defined)
1
under the curve y = x , above the x-axis, and between the lines x = 1 and
x = c.
A similar approach to e and the natural logarithm is common in calculus courses. However, this approach is not usually adopted in algebra and
trigonometry textbooks. Using basic properties of area, the simple presentation given here shows how these ideas can come through clearly without the
technicalities of calculus or Riemann sums.

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The approach taken
here to the exponential function and the
natural logarithm
shows that a good
understanding of
these subjects need
not wait until a calculus course.


xx Preface to the Instructor

The approach taken here also has the advantage that it easily leads, as
we will see in Chapter 6, to the approximation ln(1 + h) ≈ h for |h| small.
Furthermore, the same methods show that if r is any number, then
1+

r x
x

≈ er

for large values of x. A final bonus of this approach is that the connection
between continuously compounding interest and e becomes a nice corollary
of natural considerations concerning area.
Inverse Functions

With the approach
taken in this book, the

statement “logb y = x
means bx = y” is
consistent with the
notation used for
inverse functions.

The unifying concept of inverse functions is introduced in Section 3.4. This
crucial idea has its first major use in this book in the definition of y 1/m as
the number x such that x m = y (in other words, the function y → y 1/m is
the inverse of the function x → x m ; see Section 5.1). The second major use
of inverse functions occurs in the definition of logb y as the number x such
that bx = y (in other words, the function y → logb y is the inverse of the
function x → bx ; see Section 5.2).
Thus students should be comfortable with using inverse functions by
the time they reach the inverse trigonometric functions (arccosine, arcsine,
and arctangent) in Section 10.1. For students who go on to calculus, this
familiarity with inverse functions should help when dealing with inverse
operations such as anti-differentiation.
This book emphasizes that f −1 (y) = x means f (x) = y. Thus this book
states that to find f −1 (y), solve the equation f (x) = y for x.
In contrast, many books at this level unfortunately instruct the reader
wanting to find f −1 to start with the equation y = f (x), then “interchange
the variables x and y to obtain x = f (y)”, then solve for y in terms of
x. This “interchange” method ends up with notation expressing f −1 as a
function of x.
However, the “interchange” method makes no sense when trying to find
the value of an inverse function at a specific number instead of at a variable
name. Consider, for example, the problem of finding f −1 (11) if f is the
function defined by f (x) = 2x + 3. The student mechanically following
the “interchange” method as it is stated in many books would start with the

equation 11 = 2x + 3 and then interchange x and 11, getting the equation
x = 2 · 11 + 3. This is, of course, completely wrong.
In contrast, this book does this problem by solving the equation 11 = 2x+3
for x, getting x = 4 and concluding that f −1 (11) = 4.
The “interchange” method will also be confusing to students when the
variables names have meaning. For example, in an applied problem the
variables might be t (for time) and d (for distance) rather than x and y, and
we might have a function that gives distance in terms of time: d = f (t).
The inverse function should then give time in terms of distance: t = f −1 (d).
Interchanging the variable names here would be quite confusing.

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Preface to the Instructor xxi

Trigonometry
This book defines cos θ and sin θ as the first and second coordinates of the
radius of the unit circle corresponding to θ (see Section 9.3). In contrast
to this definition using only one symbol, many books at this level require
students to juggle at least four symbols—θ (or t), x, y, and P —to parse the
definitions of the trigonometric functions. These books define cos θ = x, and
students become accustomed to thinking of cos θ as the x-coordinate. When
students encounter cos x, as often happens within a dozen pages of the
initial definition, they think that cos x is the x-coordinate of . . . oops, that is
a different use of x. No wonder so many students struggle with trigonometric
functions.
This book defines sine and cosine in one section, then defines the tangent
function (and the other three trigonometric functions that have less importance) in another section. This gentle approach contrasts with most books
that define all six trigonometric functions on the same page. Students have

difficulty assimilating so many definitions simultaneously.

What to Cover
Different instructors will want to cover different sections of this book. Many
instructors will want to cover Chapter 1 (The Real Numbers), even though it
should be review, because it deals with familiar topics in a deeper fashion
than students may have previously seen.
Some instructors will cover Section 4.3 (Rational Functions) only lightly
because graphing rational functions, and in particular finding local minima
and maxima, is better done with calculus. Many instructors will prefer to skip
Chapter 8 (Sequences, Series, and Limits), leaving that material to a calculus
course.
The inverse trigonometric identities (Section 10.2) are given more space
in this book than in most books at this level. This material is included not
so much for its intrinsic importance but as a way for students to obtain a
deeper understanding of the trigonometric functions. Instructors can skip
this material or cover it lightly.

Comments Welcome
I seek your help in making this a better book. Please send me your comments
and your suggestions for improvements. Thanks!
Sheldon Axler
San Francisco State University
e-mail:
web site: algebraTrig.axler.net
Twitter: @AxlerAlgebra

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This book emphasizes cos, sin, tan and

places little emphasis
on sec, csc, cot.


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xxii

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Acknowledgments
As usual in a textbook, as opposed to a research article, little attempt has
been made to provide proper credit to the original creators of the ideas
presented in this book. Where possible, I have tried to improve on standard
approaches to this material. However, the absence of a reference does not
imply originality on my part. I thank the many mathematicians who have
created and refined our beautiful subject.
I chose Wiley as the publisher of this book because of the company’s
commitment to excellence. The people at Wiley have made outstanding
contributions to this project, providing astute editorial advice, superb design
expertise, high-level production skill, and insightful marketing savvy. I
am truly grateful to the following Wiley folks, all of whom helped make
this a better and more successful book than it would have been otherwise:
Jonathan Cottrell, Joanna Dingle, Melissa Edwards, Jessica Jacobs, Ellen
Keohane, Madelyn Lesure, Beth Pearson, Mary Ann Price, Laurie Rosatone,
Lisa Sabatini, Ken Santor, Anne Scanlan-Rohrer, Jennifer Wreyford.
Celeste Hernandez, the accuracy checker, and Katrina Avery, the copy
editor, excelled at catching mathematical and linguistic errors.
The instructors and students who used the earlier versions of this book
provided wonderfully useful feedback. Numerous reviewers gave me terrific
suggestions as the book progressed through various stages of development.

I am grateful to all the class testers and reviewers whose names are listed on
the following page, with special thanks to Michael Price.
Like most mathematicians, I owe thanks to Donald Knuth, who invented
TEX, and to Leslie Lamport, who invented LATEX, which I used to typeset this
book. I am grateful to the authors of the many open-source LATEX packages I
used to improve the appearance of the book, especially to Hàn Th´
ê Thành
for pdfLATEX, Robert Schlicht for microtype, and Frank Mittelbach for multicol.
Thanks also to Wolfram Research for producing Mathematica, which is the
software I used to create the graphics in this book.
My awesome partner Carrie Heeter deserves considerable credit for her
wise advice and continual encouragement throughout the long book-writing
process.
Many thanks to all of you!

xxiii

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Most of the results in
this book belong to
the common heritage
of mathematics, created over thousands
of years by clever and
curious people.