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Algebra and Trigonometry

March 2011, ©2012
Algebra and Trigonometry (EHEP001844) cover image
Axler Algebra & Trigonometry is written for the two semester course. The text provides students with the skill and understanding needed for their coursework and for participating as an educated citizen in a complex society.

Axler Algebra & Trigonometry focuses on depth, not breadth of topics by exploring necessary topics in greater detail. Readers will benefit from the straightforward definitions and plentiful examples of complex concepts.

The Student Solutions Manual is integrated at the end of every section. The proximity of the solutions encourages students to go back and read the main text as they are working through the problems and exercises. The inclusion of the manual also saves students money.

Axler Algebra & Trigonometry is available with WileyPLUS; an innovative, research-based, online environment for effective teaching and learning.

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About the Author

Preface to the Instructor

Acknowledgments

Preface to the Student

1 The Real Numbers

1.1 The Real Line

Construction of the Real Line

Is Every Real Number Rational?

Problems

1.2 Algebra of the Real Numbers

Commutativity and Associativity

The Order of Algebraic Operations

The Distributive Property

Additive Inverses and Subtraction

Multiplicative Inverses and the Algebra of Fractions

Symbolic Calculators

Exercises, Problems, and Worked-out Solutions

1.3 Inequalities

Positive and Negative Numbers

Lesser and Greater

Intervals

Absolute Value

Exercises, Problems, and Worked-out Solutions

Chapter Summary and Chapter Review Questions

2 Combining Algebra and Geometry

2.1 The Coordinate Plane

Coordinates

Graphs of Equations

Distance Between Two Points

Length, Perimeter, and Circumference

Exercises, Problems, and Worked-out Solutions

2.2 Lines

Slope

The Equation of a Line

Parallel Lines

Perpendicular Lines

Midpoints

Exercises, Problems, and Worked-out Solutions

2.3 Quadratic Expressions and Conic Sections

Completing the Square

The Quadratic Formula

Circles

Ellipses

Parabolas

Hyperbolas

Exercises, Problems, and Worked-out Solutions

2.4 Area

Squares, Rectangles, and Parallelograms

Triangles and Trapezoids

Stretching

Circles and Ellipses

Exercises, Problems, and Worked-out Solutions

Chapter Summary and Chapter Review Questions

3 Functions and Their Graphs

3.1 Functions

Definition and Examples

The Graph of a Function

The Domain of a Function

The Range of a Function

Functions via Tables

Exercises, Problems, and Worked-out Solutions

3.2 Function Transformations and Graphs

Vertical Transformations: Shifting, Stretching, and Flipping

Horizontal Transformations: Shifting, Stretching, Flipping

Combinations of Vertical Function Transformations

Even Functions

Odd Functions

Exercises, Problems, and Worked-out Solutions

3.3 Composition of Functions

Combining Two Functions

Definition of Composition

Order Matters in Composition

Decomposing Functions

Composing More than Two Functions

Function Transformations as Compositions

Exercises, Problems, and Worked-out Solutions

3.4 Inverse Functions

The Inverse Problem

One-to-one Functions

The Definition of an Inverse Function

The Domain and Range of an Inverse Function

The Composition of a Function and Its Inverse

Comments about Notation

Exercises, Problems, and Worked-out Solutions

3.5 A Graphical Approach to Inverse Functions

The Graph of an Inverse Function

Graphical Interpretation of One-to-One

Increasing and Decreasing Functions

Inverse Functions via Tables

Exercises, Problems, and Worked-out Solutions

Chapter Summary and Chapter Review Questions

4 Polynomial and Rational Functions

4.1 Integer Exponents

Positive Integer Exponents

Properties of Exponents

Defining x0

Negative Integer Exponents

Manipulations with Exponents

Exercises, Problems, and Worked-out Solutions

4.2 Polynomials

The Degree of a Polynomial

The Algebra of Polynomials

Zeros and Factorization of Polynomials

The Behavior of a Polynomial Near _1

Graphs of Polynomials

Exercises, Problems, and Worked-out Solutions

4.3 Rational Functions

Ratios of Polynomials

The Algebra of Rational Functions

Division of Polynomials

The Behavior of a Rational Function Near _1

Graphs of Rational Functions

Exercises, Problems, and Worked-out Solutions

4.4 Complex Numbers

The Complex Number System

Arithmetic with Complex Numbers

Complex Conjugates and Division of Complex Numbers

Zeros and Factorization of Polynomials, Revisited

Exercises, Problems, and Worked-out Solutions

Chapter Summary and Chapter Review Questions

5 Exponents and Logarithms

5.1 Exponents and Exponential Functions

Roots

Rational Exponents

Real Exponents

Exponential Functions

Exercises, Problems, and Worked-out Solutions

5.2 Logarithms as Inverses of Exponential Functions

Logarithms Base 2

Logarithms with Any Base

Common Logarithms and the Number of Digits

Logarithm of a Power

Radioactive Decay and Half-Life

Exercises, Problems, and Worked-out Solutions

5.3 Applications of Logarithms

Logarithm of a Product

Logarithm of a Quotient

Earthquakes and the Richter Scale

Sound Intensity and Decibels

Star Brightness and Apparent Magnitude

Change of Base

Exercises, Problems, and Worked-out Solutions

5.4 Exponential Growth

Functions with Exponential Growth

Population Growth

Compound Interest

Exercises, Problems, and Worked-out Solutions

Chapter Summary and Chapter Review Questions

6 e and the Natural Logarithm

6.1 Defining e and ln

Estimating Area Using Rectangles

Defining e

Defining the Natural Logarithm

Properties of the Exponential Function and ln

Exercises, Problems, and Worked-out Solutions

6.2 Approximations with e and ln

Approximation of the Natural Logarithm

Inequalities with the Natural Logarithm

Approximations with the Exponential Function

An Area Formula

Exercises, Problems, and Worked-out Solutions

6.3 Exponential Growth Revisited

Continuously Compounded Interest

Continuous Growth Rates

Doubling Your Money

Exercises, Problems, and Worked-out Solutions

Chapter Summary and Chapter Review Questions

7 Trigonometric Functions

7.1 The Unit Circle

The Equation of the Unit Circle

Angles in the Unit Circle

Negative Angles

Angles Greater Than 360_

Length of a Circular Arc

Special Points on the Unit Circle

Exercises, Problems, and Worked-out Solutions

7.2 Radians

A Natural Unit of Measurement for Angles

Negative Angles

Angles Greater Than 2_

Length of a Circular Arc

Area of a Slice

Special Points on the Unit Circle

Exercises, Problems, and Worked-out Solutions

7.3 Cosine and Sine

Definition of Cosine and Sine

Cosine and Sine of Special Angles

The Signs of Cosine and Sine

The Key Equation Connecting Cosine and Sine

The Graphs of Cosine and Sine

Exercises, Problems, and Worked-out Solutions

7.4 More Trigonometric Functions

Definition of Tangent

Tangent of Special Angles

The Sign of Tangent

Connections between Cosine, Sine, and Tangent

The Graph of Tangent

Three More Trigonometric Functions

Exercises, Problems, and Worked-out Solutions

7.5 Trigonometry in Right Triangles

Trigonometric Functions via Right Triangles

Two Sides of a Right Triangle

One Side and One Angle of a Right Triangle

Exercises, Problems, and Worked-out Solutions

7.6 Trigonometric Identities

The Relationship Between Cosine and Sine

Trigonometric Identities for the Negative of an Angle

Trigonometric Identities with

Trigonometric Identities Involving a Multiple of

Exercises, Problems, and Worked-out Solutions

Chapter Summary and Chapter Review Questions

8 Trigonometric Algebra and Geometry

8.1 Inverse Trigonometric Functions

The Arccosine Function

The Arcsine Function

The Arctangent Function

Exercises, Problems, and Worked-out Solutions

8.2 Inverse Trigonometric Identities

The Arccosine, Arcsine, and Arctangent of

t: Graphical

Approach

The Arccosine, Arcsine, and Arctangent of

t: Algebraic

Approach

Arccosine Plus Arcsine

The Arctangent of 1t

Composition of Trigonometric Functions and Their Inverses

More Compositions with Inverse Trigonometric Functions

Exercises, Problems, and Worked-out Solutions

8.3 Using Trigonometry to Compute Area

The Area of a Triangle via Trigonometry

Ambiguous Angles

The Area of a Parallelogram via Trigonometry

The Area of a Polygon

Exercises, Problems, and Worked-out Solutions

8.4 The Law of Sines and the Law of Cosines

The Law of Sines

Using the Law of Sines

The Law of Cosines

Using the Law of Cosines

When to Use Which Law

Exercises, Problems, and Worked-out Solutions

8.5 Double-Angle and Half-Angle Formulas

The Cosine of 2_

The Sine of 2_

The Tangent of 2_

The Cosine and Sine of _2

The Tangent of _2

Exercises, Problems, and Worked-out Solutions

8.6 Addition and Subtraction Formulas

The Cosine of a Sum and Difference

The Sine of a Sum and Difference

The Tangent of a Sum and Difference

Exercises, Problems, and Worked-out Solutions

Chapter Summary and Chapter Review Questions

9 Applications of Trigonometry

9.1 Parametric Curves

Curves in the Coordinate Plane

Graphing Inverse Functions as Parametric Curves

Shifting, Stretching, or Flipping a Parametric Curve

Exercises, Problems, and Worked-out Solutions

9.2 Transformations of Trigonometric Functions

Amplitude

Period

Phase Shift

Exercises, Problems, and Worked-out Solutions

9.3 Polar Coordinates

Defining Polar Coordinates

Converting from Polar to Rectangular Coordinates

Converting from Rectangular to Polar Coordinates

Graphs of Polar Equations

Exercises, Problems, and Worked-out Solutions

9.4 Vectors

An Algebraic and Geometric Introduction to Vectors

Vector Addition

Vector Subtraction

The Dot Product

Exercises, Problems, and Worked-out Solutions

9.5 The Complex Plane

Complex Numbers as Points in the Plane

Geometric Interpretation of Complex Multiplication and Division

De Moivre’s Theorem

Finding Complex Roots

Exercises, Problems, and Worked-out Solutions

Chapter Summary and Chapter Review Questions

10 Systems of Equations and Inequalities

10.1 Equations and Systems of Equations

Solving an Equation

Solving a System of Equations

Systems of Linear Equations

Matrices

Exercises, Problems, and Worked-out Solutions

10.2 Solving Systems of Linear Equations

Gaussian Elimination

Gaussian Elimination with Matrices

Special Cases—No Solutions

Special Cases—Infinitely Many Solutions

Exercises, Problems, and Worked-out Solutions

10.3 Matrix Algebra

Adding and Subtracting Matrices

Multiplying Matrices

The Inverse of a Matrix

Exercises, Problems, and Worked-out Solutions

Chapter Summary and Chapter Review Questions

11 Sequences, Series, and Limits

11.1 Sequences

Introduction to Sequences

Arithmetic Sequences

Geometric Sequences

Recursively-Defined Sequences

Exercises, Problems, and Worked-out Solutions

11.2 Series

Sums of Sequences

Arithmetic Series

Geometric Series

Summation Notation

The Binomial Theorem

Exercises, Problems, and Worked-out Solutions

11.3 Limits

Introduction to Limits

Infinite Series

Decimals as Infinite Series

Special Infinite Series

Exercises, Problems, and Worked-out Solutions

Chapter Summary and Chapter Review Questions

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  • Depth, Not Breadth: Topics have been carefully selected to get at the heart of algebraic weakness by narrowing down to key sets of skills which are regularly revisited from varied perspectives.
  • Unit-circle Approach: This approach is used because calculus requires the unit-circle approach and it allows for a well-motivated introduction to radian measure. Once the approach has been introduced, applications to right triangles are given.
  • Exercises and Problems: The difference between an exercise and a problem is that each exercise has a unique correct answer that is a mathematical object such as a number or a function, while the solutions to problems consist of explanations or examples. The solutions to the odd-numbered exercises appear directly behind the relevant section.
  • Variety: 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. Applications are written to reflect real scenarios, not artificial examples.
  • Integrated Student's Solutions Manual: The solutions manual encourages students to read the main text and students will save money by not having to purchase a separate solutions manual.
  • Designed to be Read: The writing style and layout are meant to induce students to read and understand the material. Explanations are more plentiful than typically found in College Algebra books, with examples of concepts making the ideas concrete whenever possible.
  • Calculator Problems: A symbol appears next to problems that require a calculator, and some exercises and problems are designed to make students realize that by understanding the material, they can overcome the limitations of calculators.
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Instructors Resources
Wiley Instructor Companion Site
Instructor's Solutions Manual
The Instructor's Solutions Manual contains the solutions to all the problems in College Algebra and Trigonometry.
Lecture Slides
Computerized Test Bank
This Computerized Test Bank includes a collection of open response, multiple-choice, true/false, and free-response questions.
Printed Test Bank
This Printed Test Bank includes a collection of open response, multiple-choice, true/false, and free-response questions.
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Algebra and Trigonometry
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