Trigonometry For Dummies

387 Pages·2014·7.2 MB·English

by  Mary Jane Sterling

Mary Jane Sterling is the author of Algebra I For Dummies and many
other For Dummies titles. She has been teaching mathematics at
Bradley University in Peoria, Illinois, for more than 30 years and has
loved working with future business executives, physical therapists,
teachers, and many others.

Cover Image: ©Peter Hermes Furian/Alamy

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Open the book and find:

• Definitions of trigonometry
functions

• How to get down with
Pythagoras and his theorem

• Ways to use Cartesian
coordinates

• The lowdown on the sine,
cosine, and tangent functions

• How to go 'round and 'round
with circular functions

• The scoop on inverse
trigonometry

• Details on graphing trig
functions

• Ten basic identities and ten
not-so-basic identities

$19.99 USA / $23.99 CAN / £15.99 UK

9 781118 827413

51999

ISBN:978-1-118-82741-3

Mathematics/Trigonometry

Your guide to getting
an angle on sines, cosines,
and tangents

Confused by cosines? Perplexed by polynomials?
Don’t worry! This friendly guide takes the torture out
of trigonometry by explaining everything in plain English,
offering lots of easy-to-grasp examples, and adding
a dash of humor and fun. You’ll see the ways trig applies
to everyday life, and how it’s an important tool for a
variety of careers.

• Start with basics — get up and running with an overview of trig

functions, radians, and more

• Learn tricky trig functions — take things to the next level with

SOH-CAH-TOA and circular functions

• Identify identities — dig into reciprocal, ratio, opposite-angle,

and Pythagorean identities

• Make trig work — get practical with trig, find out how to use your

calculator for complex solutions, and solve trig equations

• Graph functions — figure out the basics of graphing sine, cosine,

tangent, cotangent, and more

Trigonometry

Sterling

2nd Edition

Mary Jane Sterling
Author of Algebra I For Dummies

Learn to:
• Solve tricky trig equations

• Graph functions and figure
out formulas

• Use trigonometry to solve
practical problems

Trigonometry

2nd Edition

Making Everything Easier!™

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by Mary Jane Sterling

Trigonometry

2nd Edition

Trigonometry For Dummies®, 2nd Edition

Published by: John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030-5774, www.wiley.com

Copyright © 2014 by John Wiley & Sons, Inc., Hoboken, New Jersey

Published simultaneously in Canada

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Library of Congress Control Number: 2013954199

ISBN 978-1-118-82741-3 (pbk); ISBN 978-1-118-82746-8 (ebk); ISBN 978-1-118-82757-4 (ebk)

Manufactured in the United States of America

10 9 8 7 6 5 4 3 2 1

Contents at a Glance

Introduction ................................................................ 1

Part I: Getting Started with Trigonometry...................... 5

Chapter 1: Trouncing Trig Technicalities.......................................................................7
Chapter 2: Coordinating Your Efforts with Cartesian Coordinates..........................29
Chapter 3: Functioning Well ...........................................................................................47
Chapter 4: Getting Your Degree.....................................................................................57
Chapter 5: Dishing Out the Pi: Radians.........................................................................67
Chapter 6: Getting It Right with Triangles ....................................................................81
Part II: Trigonometric Functions.................................. 91

Chapter 7: Doing Right by Trig Functions ....................................................................93
Chapter 8: Trading Triangles for Circles: Circular Functions ..................................109
Chapter 9: Defining Trig Functions Globally ..............................................................121
Chapter 10: Applying Yourself to Trig Functions ......................................................135
Part III: Identities ................................................... 155

Chapter 11: Identifying Basic Identities ......................................................................157
Chapter 12: Operating on Identities ............................................................................171
Chapter 13: Proving Identities: The Basics.................................................................189
Chapter 14: Sleuthing Out Identity Solutions.............................................................207
Part IV: Equations and Applications.......................... 223

Chapter 15: Investigating Inverse Trig Functions......................................................225
Chapter 16: Making Inverse Trig Work for You..........................................................233
Chapter 17: Solving Trig Equations .............................................................................243
Chapter 18: Obeying the Laws .....................................................................................265
Part V: The Graphs of Trig Functions ......................... 289

Chapter 19: Graphing Sine and Cosine........................................................................291
Chapter 20: Graphing Tangent and Cotangent...........................................................307
Chapter 21: Graphing Other Trig Functions...............................................................317
Chapter 22: Topping Off Trig Graphs..........................................................................329

Part VI: The Part of Tens .......................................... 343

Chapter 23: Ten Basic Identities . . . Plus Some Bonuses .........................................345
Chapter 24: Ten Not-So-Basic Identities......................................................................349

Appendix: Trig Functions Table ................................. 353

Index ...................................................................... 357

Table of Contents

Introduction................................................................. 1

About This Book..............................................................................................1
Foolish Assumptions.......................................................................................2
Icons Used in This Book .................................................................................3
Beyond the Book .............................................................................................3
Where to Go from Here...................................................................................4

Part I: Getting Started with Trigonometry....................... 5

Chapter 1: Trouncing Trig Technicalities . . . . . . . . . . . . . . . . . . . . . . . . . 7

Taking Trig for a Ride: What Trig Is ..............................................................7
Sizing up the basic figures....................................................................8
Angling for position ...............................................................................9
Triangulating your position................................................................11
Circling the wagons .............................................................................12
Understanding Trig Speak............................................................................18
Defining trig functions.........................................................................18
Taming the radicals.............................................................................21
Equating and Identifying...............................................................................23
Graphing for Gold..........................................................................................25
Describing graphing scales.................................................................25
Recognizing basic graphs ...................................................................25

Chapter 2: Coordinating Your Efforts with
Cartesian Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

Starting Out Simple: Plotting Points............................................................29
Axes, axes, we all fall down ................................................................29
Determining the origin of it all ...........................................................30
Plotting x versus y ...............................................................................30
Cutting the graph into four parts.......................................................31
From Here to There: Calculating Distances ...............................................32
Counting on vertical and horizontal distances................................32
Another slant: Diagonal distances.....................................................34
Using exact values or estimating distances .....................................35
Getting to the Center of It All .......................................................................36
Finding the midpoint of a line segment.............................................36
Locating the center of a circle............................................................37
Partitioning line segments further.....................................................38
Pinpointing the center of a triangle...................................................41

Trigonometry For Dummies, 2nd Edition
vi

Racing Down the Slope .................................................................................43
Slaloming slope formula......................................................................43
Recognizing parallel and perpendicular lines..................................45
Defining Circles with Numbers ....................................................................45
Centering circles at the origin............................................................46
Wandering centers...............................................................................46

Chapter 3: Functioning Well . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

Relations versus Functions ..........................................................................47
Function junction, what’s your function?.........................................48
Using function notation.......................................................................48
Determining domain and range..........................................................49
In-Verse Functions: Rhyme or Reason? ......................................................50
Which functions have inverses? ........................................................50
Finding an inverse function ................................................................52
Transforming Functions ...............................................................................54
Translating a function .........................................................................54
Reflecting like a mirror........................................................................56

Chapter 4: Getting Your Degree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

Angles, Angles Everywhere: Measuring in Degrees ..................................57
Slicing a coordinate plane...................................................................57
Looking elsewhere for degree measures ..........................................59
Graphing Angles in Standard Position........................................................62
Positioning initial and terminal sides................................................62
Measuring by quadrants.....................................................................62
What’s Your Angle? Labeling in Various Ways..........................................64
Using negative angle measures..........................................................64
Comingling with coterminal angles ...................................................64
Renaming angles: So many aliases.....................................................65

Chapter 5: Dishing Out the Pi: Radians . . . . . . . . . . . . . . . . . . . . . . . . . . 67

What’s in a Radian? .......................................................................................67
Relating to a circle...............................................................................68
Converting degrees and radians........................................................68
Highlighting favorites..........................................................................72
Making a Clone of Arc ...................................................................................73
Taking chunks out of circles ..............................................................73
Sweeping hands ...................................................................................76
Going out and about............................................................................78

Chapter 6: Getting It Right with Triangles . . . . . . . . . . . . . . . . . . . . . . . 81

Sizing Up Right Triangles..............................................................................81
What’s so right about them? ..............................................................81
The anatomy of a right triangle..........................................................82

vii
Table of Contents

Pythagoras Schmythagoras: Demystifying the Pythagorean Theorem..84
Hitting a Pythagorean triple ...............................................................85
Solving for a missing length................................................................86
In a League of Their Own: Special Right Triangles....................................88
30-60-90 right triangles........................................................................89
Isosceles right triangles ......................................................................90

Part II: Trigonometric Functions .................................. 91

Chapter 7: Doing Right by Trig Functions . . . . . . . . . . . . . . . . . . . . . . . . 93

SohCahToa to the Rescue: How Trig Functions Work..............................93
The name game: A right triangle’s three sides.................................94
The six ratios: Relating the three sides.............................................94
The sine function: Opposite over hypotenuse.................................95
The cosine function: Adjacent over hypotenuse.............................96
The tangent function: Opposite over adjacent ................................97
All together, now: Using one function to solve for another ...........99
Similar right triangles within a right triangle ...................................99
Socking the rules away: The legend of SohCahToa.......................101
Taking It a Step Further: Reciprocal Functions .......................................101
The cosecant function: Sine flipped upside down.........................102
The secant function: Cosine on its head.........................................103
The cotangent function: Tangent, tails side up .............................103
Angling In on Your Favorites......................................................................104
Identifying the most popular angles................................................104
Determining the exact values of functions.....................................105

Chapter 8: Trading Triangles for Circles: Circular Functions . . . . . . 109

Getting Acquainted with the Unit Circle...................................................109
Placing points on the unit circle......................................................110
Finding a missing coordinate ...........................................................112
Sticking to rational coordinates.......................................................114
Going Full Circle with the Angles...............................................................116
Staying positive..................................................................................116
Being negative or multiplying your angles .....................................117
Locating and computing reference angles......................................118

Chapter 9: Defining Trig Functions Globally . . . . . . . . . . . . . . . . . . . . 121

Defining Trig Functions for All Angles ......................................................121
Putting reference angles to use........................................................122
Labeling the optimists and pessimists............................................122
Combining all the rules .....................................................................123
Using Coordinates of Circles to Solve for Trig Functions ......................125
Calculating with coordinates on the unit circle.............................127
Calculating with coordinates on any circle at the origin..............128

Trigonometry For Dummies, 2nd Edition
viii

Defining Domains and Ranges of Trig Functions.....................................130
Friendly functions: Sine and cosine.................................................132
Close cousins of their reciprocals: Cosecant and secant.............132
Brothers out on their own: Tangent and cotangent......................133

Chapter 10: Applying Yourself to Trig Functions . . . . . . . . . . . . . . . . . 135

First Things First: Elevating and Depressing ...........................................135
Measuring Tall Buildings with a Single Bound.........................................137
Rescuing a damsel from a tower......................................................137
Determining the height of a tree......................................................139
Measuring the distance between buildings....................................140
Measuring Slope ..........................................................................................142
The Sky’s (Not) the Limit............................................................................143
Spotting a balloon..............................................................................144
Tracking a rocket...............................................................................146
Measuring the view of satellite cameras.........................................148
Calculating Odd Shapes and Maneuvering Corners................................149
Finding the area of a triangular piece of land ................................149
Using Heron’s Formula......................................................................151
Moving an object around a corner ..................................................152

Part III: Identities .................................................... 155

Chapter 11: Identifying Basic Identities . . . . . . . . . . . . . . . . . . . . . . . . 157

Flipping Functions on Their Backs: Reciprocal Identities......................157
Function to Function: Ratio Identities ......................................................159
Opposites Attract: Opposite-Angle Identities..........................................159
Revisiting the Classic Theorem: Pythagorean Identities........................162
The mother of all Pythagorean identities.......................................163
Extending to tangent and secant .....................................................164
Finishing up with cotangent and cosecant.....................................165
Rearranging the Pythagorean identities .........................................166
Combining the Identities ............................................................................168
The many faces of sine......................................................................168
Working out the versions..................................................................168

Chapter 12: Operating on Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

Summing It Up..............................................................................................171
Overcoming the Differences.......................................................................176
Doubling Your Money .................................................................................179
One plus one equals two sines.........................................................180
Three’s a crowd .................................................................................182
Halving Fun Yet? ..........................................................................................184
Explaining the ± .................................................................................... 185
Half a tangent is double the fun .......................................................185
Using half-angle identities.................................................................186

ix
Table of Contents

Chapter 13: Proving Identities: The Basics . . . . . . . . . . . . . . . . . . . . . . 189

Lining Up the Players..................................................................................189
Picking Sides ................................................................................................191
Working on Both Sides................................................................................195
Going Back to Square One..........................................................................198
Changing to sines and cosines.........................................................198
Factoring .............................................................................................201
Using a little bit of both.....................................................................203

Chapter 14: Sleuthing Out Identity Solutions . . . . . . . . . . . . . . . . . . . . 207

Fracturing Fractions....................................................................................207
Breaking up is hard to do .................................................................207
Finding a common denominator......................................................210
Using Tricks of the Trig Trade...................................................................212
Multiplying by a conjugate ...............................................................212
Squaring both sides...........................................................................214
Identifying With the Operations ................................................................216
Adding it up ........................................................................................216
What difference does it make?.........................................................217
Multiplying your fun..........................................................................219
Halving fun, wish you were here......................................................221

Part IV: Equations and Applications........................... 223

Chapter 15: Investigating Inverse Trig Functions . . . . . . . . . . . . . . . . 225

Writing It Right.............................................................................................225
Using the notation .............................................................................226
Distinguishing between the few and the many ..............................227
Determining Domain and Range of Inverse Trig Functions....................229
Inverse sine function .........................................................................229
Inverse cosine function.....................................................................230
Inverse tangent function...................................................................230
Inverse cotangent function...............................................................230
Inverse secant function.....................................................................230
Inverse cosecant function.................................................................231
Summarizing domain and range ......................................................231

Chapter 16: Making Inverse Trig Work for You . . . . . . . . . . . . . . . . . . 233

Working with Inverses ................................................................................233
Getting Friendly with Your Calculator......................................................235
Changing the mode............................................................................235
Interpreting notation on the calculator ..........................................236
Multiplying the Input...................................................................................238
Solving Some Mixed Problems...................................................................240

Trigonometry For Dummies, 2nd Edition
x

Chapter 17: Solving Trig Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243

Generating Simple Solutions ......................................................................244
Factoring In the Solutions...........................................................................245
Finding a greatest common factor...................................................246
Factoring quadratics .........................................................................247
Increasing the degrees in factoring .................................................249
Factoring by grouping.......................................................................252
Using the Quadratic Formula.....................................................................253
Incorporating Identities..............................................................................254
Finding Multiple-Angle Solutions...............................................................258
Squaring Both Sides ....................................................................................259
Multiplying Through ...................................................................................262
Solving with a Graphing Calculator...........................................................262

Chapter 18: Obeying the Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265

Describing the Parts of Triangles..............................................................265
Standardizing the parts.....................................................................266
Determining a triangle.......................................................................266
Following the Law of Sines .........................................................................268
Continuing with the Law of Cosines..........................................................272
Defining the law of cosines...............................................................272
Law of cosines for SAS ......................................................................273
Law of cosines for SSS.......................................................................276
Being ambiguous................................................................................277
Finding the Areas of Triangles...................................................................281
Finding area with base and height...................................................282
Finding area with three sides...........................................................284
Finding area with SAS........................................................................285
Finding area with ASA .......................................................................287

Part V: The Graphs of Trig Functions.......................... 289

Chapter 19: Graphing Sine and Cosine . . . . . . . . . . . . . . . . . . . . . . . . . 291

The ABCs of Graphing.................................................................................291
Waving at the Sine.......................................................................................292
Describing amplitude and period ....................................................292
Formalizing the sine equation..........................................................295
Translating the sine...........................................................................296
Graphing Cosine ..........................................................................................298
Comparing cosine to sine .................................................................298
Using properties to graph cosine ....................................................299

xi
Table of Contents

Applying the Sines of the Times ................................................................300
Sunning yourself ................................................................................300
Averaging temperature .....................................................................301
Taking your temperature..................................................................303
Making a goal......................................................................................304
Theorizing with biorhythms.............................................................305

Chapter 20: Graphing Tangent and Cotangent . . . . . . . . . . . . . . . . . . . 307

Checking Out Tangent.................................................................................307
Determining the period.....................................................................308
Assigning the asymptotes.................................................................308
Fiddling with the tangent..................................................................310
Confronting the Cotangent.........................................................................314

Chapter 21: Graphing Other Trig Functions . . . . . . . . . . . . . . . . . . . . . 317

Seeing the Cosecant for What It Is.............................................................317
Identifying the asymptotes...............................................................317
Using the sine graph..........................................................................318
Varying the cosecant.........................................................................320
Unveiling the Secant....................................................................................321
Determining the asymptotes............................................................321
Sketching the graph of secant..........................................................322
Fooling around with secant..............................................................323
Laying Out the Inverse Functions..............................................................324
Graphing inverse sine and cosine....................................................325
Taking on inverse tangent and cotangent ......................................326
Crafting inverse secant and cosecant .............................................327

Chapter 22: Topping Off Trig Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . 329

The Basics of Trig Equations .....................................................................329
Flipping over a horizontal line .........................................................330
Interpreting the equation..................................................................331
Graphing with the General Form ...............................................................332
Adding and Subtracting Functions............................................................337
Applying Yourself to the Task ...................................................................339
Measuring the tide.............................................................................339
Tracking the deer population...........................................................340
Measuring the movement of an object on a spring.......................341

Trigonometry For Dummies, 2nd Edition
xii

Part VI: The Part of Tens........................................... 343

Chapter 23: Ten Basic Identities . . . Plus Some Bonuses . . . . . . . . . 345

Reciprocal Identities ...................................................................................345
Ratio Identities.............................................................................................346
Pythagorean Identities................................................................................347
Opposite-Angle Identities ...........................................................................347
Multiple-Angle Identities.............................................................................348

Chapter 24: Ten Not-So-Basic Identities . . . . . . . . . . . . . . . . . . . . . . . 349

Product-to-Sum Identities...........................................................................349
Sum-to-Product Identities...........................................................................350
Reduction Formula......................................................................................351
Mollweide’s Equations................................................................................352

Appendix: Trig Functions Table.................................. 353

Index....................................................................... 357

Introduction

M

any of the more practical and exciting accomplishments of early man
were performed using trigonometry. Even before trigonometry was
formalized into a particular topic to study or used to solve problems, trigo-
nometry helped people to sail across large bodies of water, build gigantic
structures, plot out land, and measure heights and distances — even to the
stars.

We still use trigonometry for all these reasons and more. If you’re going to get
your pilot’s license, you’ll need trigonometry. Trigonometry is also the basis
for many courses in mathematics — starting in grade school with geometric
shapes and map reading and moving on through calculus. Trig is all over the
place.

You can get as deeply into this topic or as little into it as you want, and you’ll
still come out of it thinking, “Gee, I didn’t realize that trigonometry was used
to do this! Wasn’t that just loads of fun!” Well, maybe I’m pushing it a bit —
loads may be a slight exaggeration.

Whether you’re pursuing trigonometry so that you can go on in calculus or
prepare for architecture or drafting or do that piloting thing, or even if you’re
just curious, you’ll find what you need here. You can get as technical as you
want. You can skip through the stuff you don’t need. Just know that you’ll be
on the same adventure as that early man — you’ll just have the advantage of
a few more tools.

About This Book

So, what’s in it for you? What’s in a book on trigonometry that’ll ring your bell
or strike your fancy or just make you pretty happy? Where do I begin?

You can start anywhere in the book, jump around, and just go any direction
you please. If you’re really excited about triangles and how they can be used
to your advantage, check them out. Everything you need to read about them
is here.

What if you’ve got another angle? Or, maybe you didn’t have one to begin
with but wish you did. If you’re looking for angles, you’ve come to the right