Cliff Notes - QuickReview Trigonometry

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Trigonometry
Trigonometry
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CliffsQuickReview™
Trigonometry
By David A. Kay, MS
Best-Selling Books • Digital Downloads • e-Books • Answer Networks • e-Newsletters • Branded Web Sites • e-Learning
New York, NY • Cleveland, OH • Indianapolis, IN
6389-0 FM.F 7/27/01 8:40 AM Page i
CliffsQuickReview™Trigonometry
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About the Author
David A. Kay has been teaching Mathematics
and Computer Information Systems at the college
level for over 33 years. In addition, he has been
working as a test prep instructor for the last 24
years.
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6389-0 FM.F 7/27/01 8:40 AM Page ii
Table of Contents
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1
Why You Need This Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
How to Use This Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Visit Our Web Site . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
Chapter 1: Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3
Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
Functions of Acute Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
Functions of General Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
Tables of Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
Chapter 2: Trigonometry of Triangles . . . . . . . . . . . . . . . . . . . . . . . . . . .22
Solving Right Triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
Law of Cosines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
Law of Sines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
Solving General Triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
Areas of Triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
Chapter 3: Graphs of Trigonometric Functions . . . . . . . . . . . . . . . . . . .51
Radians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
Circular Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
Periodic and Symmetric Trigonometric Functions . . . . . . . . . . . . . . . 60
Graphs of the Sine and Cosine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
Graphs of Other Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . 70
Graphs of Special Trigonometric Functions . . . . . . . . . . . . . . . . . . . . 76
Chapter 4: Trigonometric Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . .79
Fundamental Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
Addition Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
Double-Angle and Half-Angle Identities . . . . . . . . . . . . . . . . . . . . . . 87
Tangent Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
Product-Sum and Sum-Product Identities . . . . . . . . . . . . . . . . . . . . . 95
Chapter 5: Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .99
Vector Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
Vectors in the Rectangular Coordinate System . . . . . . . . . . . . . . . . . 105
Chapter 6: Polar Coordinates and Complex Numbers . . . . . . . . . . . .113
Polar Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
Geometry of Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
De Moivre’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
Chapter 7: Inverse Functions and Equations . . . . . . . . . . . . . . . . . . . .127
Inverse Cosine and Inverse Sine . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
Other Inverse Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . 132
Trigonometric Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
6389-0 FM.F 7/27/01 8:40 AM Page iii
iv CliffsQuickReview Trigonometry
Chapter 8: Additional Topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .140
The Expression M sin Bt + N cos Bt . . . . . . . . . . . . . . . . . . . . . . . . 140
Uniform Circular Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
Simple Harmonic Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
CQR Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .148
CQR Resource Center . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .154
Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .156
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .161
6389-0 FM.F 7/27/01 8:40 AM Page iv
INTRODUCTION
The word trigonometry comes from Greek words meaning measurement
of triangles. Solving triangles is one of many aspects of trigonometry
that you study today. To develop methods to solve triangles, trigonomet-
ric functions are constructed. The study of the properties of these func-
tions and related applications form the subject matter of trigonometry.
Trigonometry has applications in navigation, surveying, construction, and
many other branches of science, including mathematics and physics.
Why You Need This Book
Can you answer yes to any of these questions?
■Do you need to review the fundamentals of trigonometry fast?
■Do you need a course supplement to trigonometry?
■Do you need a concise, comprehensive reference for trigonometry?
If so, then CliffsQuickReview Trigonometry is for you!
How to Use This Book
You’re in charge here. You get to decide how to use this book. You can
either read the book from cover to cover or just look for the information
you need right now. However, here are a few recommended ways to search
for topics:
■Flip through the book looking for your topics in the running heads.
■Look in the Glossary for all the important terms and definitions.
■Look for your topic in the Table of Contents in the front of the book.
■Look at the Chapter Check-In list at the beginning of each chapter.
■Look at the Chapter Check-Out questions at the end of each
chapter.
■Test your knowledge with the CQR Review at the end of the book.
6389-0 Intro.F 7/27/01 8:40 AM Page 1
2CliffsQuickReview Trigonometry
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6389-0 Intro.F 7/27/01 8:40 AM Page 2
Chapter 1
TRIGONOMETRIC FUNCTIONS
Chapter Checkin
❑Understanding angles and angle measurements
❑Finding out about trigonometric functions of acute angles
❑Defining trigonometric functions of general angles
❑Using inverse notation and linear interpolation
Historically, trigonometry was developed to help find the measurements
in triangles as an aid in navigation and surveying. Recently, trigonom-
etry is used in numerous sciences to help explain natural phenomena. In
this chapter, I define angle measure and basic trigonometric relationships
and introduce the use of inverse trigonometric functions.
6389-0 Ch01.F 7/27/01 8:40 AM Page 3
4CliffsQuickReview Trigonometry
Angles
An angle is a measure of rotation. Angles are measured in degrees. One
complete rotation is measured as 360°. Angle measure can be positive or
negative, depending on the direction of rotation. The angle measure is the
amount of rotation between the two rays forming the angle. Rotation is
measured from the initial side to the terminal side of the angle. Positive
angles (Figure 1-1a) result from counterclockwise rotation, and negative
angles (Figure 1-1b) result from clockwise rotation. An angle with its ini-
tial side on the x-axis is said to be in standard position.
Figure 1-1 (a) A positive angle and (b) a negative angle.
Angles that are in standard position are said to be quadrantal if their ter-
minal side coincides with a coordinate axis. Angles in standard position
that are not quadrantal fall in one of the four quadrants, as shown in
Figure 1-2.
(a) (b)
150°-245°
6389-0 Ch01.F 7/27/01 8:40 AM Page 4
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