Master Math Trigonometry

369 Pages·2016·19.77 MB·English

by  Ross Debbra Anne.

Math:
Master
lrigonometry
Including Everything From
Trigonometric Functions, Equations,
Triangle, and Graphs
to Identities, Coordinate Systems,
and Complex Numbers
BY
Debm Anne Ross
THEC AREEPRR ESSIN C.
Franklin Lakes, NJ
Copyright 0 2002 by Debra Anne Ross
All rights reserved under the Pan-American and International
Copyright Conventions. This book may not be reproduced, in
whole or in part, in any form or by any means electronic or
mechanical, including photocopying, recording, or by any
information storage and retrieval system now known or hereafter
invented, without written permission from the publisher, The
Career Press.
MASTEMR ATHT RIGONOMETRY
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dI ll
C A R E E R
R E s s
The Career Press, Inc., 3 Tice Road, PO Box 687
Franklin Lakes, NJ 07417
www.careerpress.com
Library of Congress Cataloging-in-PublicationD ata
ROSS,D ebra, 1958-
Master math : trigonometry including everything from
trigonometric functions, equations, triangles, and graphs to
identities, coordinate systems, and complex numbers / by Debra
Ross
p. cm.
Includes index.
ISBN 1-56414-527-1 (paper)
1. Trigonometry. I. Title.
QA531 .R84 2002
5 16.24-dc21 2001052970
Acknowledpments
I am indebted to Cyndy Lakowske for her careful reading of
this book and for all of her helpful comments and suggestions. I am also
indebted to Professor Melanie McNeil for her carefid reading of th~s
book and for all of her helpful comments. I am eternally indebted to
both Dr. Melanie McNeil and Cyndy Lakowske for all the time they
have generously spent on reviewing the books in the Master Math
series. I am particularly grateful to Professor Channing Robertson for
reviewing this book and the previous four books in the series and for all
hs suggestions and, in general, for all his advice.
I would sincerely like to thank my agent, Dr. Sidney Kramer,
and the staff of Mews Books. Without Dr. Kramer’s interest in the
Master Math series, it would not have been published. I would like to
thank the entire staff of Career Press, especially Stacey Farkas and J d
Brandon, who worked closely with me. I would also like to thank my
mother for her editorial assistance with ths book and the previous
books in the series.
In addition, I would like to thank both of my parents for their
encouragement and my aunt and uncle for their assistance during my
education.
...
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Table of Contents
Introduction 1
Chapter 1. Review of Numbers and Coordinate Systems 3
1.1. Review of numbers, including natural, whole, integers, zero,
rational, irrational, real, complex, and imaginary numbers 3
1.2. Absolute value 7
1.3. Significant digits and rounding numbers and decimals 7
1.4. Review of coordinate systems, including twe and three-
dimensional rectangular coordinates, polar coordinates,
cylindrical coordinates, and spherical coordinates 9
1.5. Chapter 1 summary and highlights 14
Chapter 2. Review of Geometry 15
2.1. Introduction 15
2.2. Lines and angles 17
2.3. Triangles 20
2.4. Polygons and quadrilaterals 25
2.5. Conic sections, including circles, arcs and angles, ellipses,
parabolas, and hyperbolas 28
2.6. Three-dimensional objects, including cubes, rectangular
solids, cylinders, spheres, cones, and pyramids 40
2.7. Chapter 2 summary and highlights 42
Chapter 3. Triangles and Trigonometric Functions 44
3.1. Right triangles and the trigonometric hnctions 44
3.2. Solving right triangles 48
3.3. Examples and applications of right triangles 49
3.4. Oblique triangles and the Law of Sines and Law of Cosines 55
3.5. Solving oblique triangles 59
3.6. Examples and applications of oblique triangles 63
3.7. Finding the area of a triangle 67
3.8. Chapter 3 summary and highlights 75
V
Chapter 4. Trigonometric Functions in a Coordinate System
and Circular Functions 78
4.1. Review of functions and their properties 78
4.2. Types of hnctions, including composite, inverse, linear,
nonlinear, even, odd, exponential, logarithmic, identity,
absolute value, squaring, cubing, square root, cube root,
reciprocal, and fbnctions with more than one variable 81
4.3. Coordinate systems, radians, degrees, and arc length 90
4.4. Angles in standard position and coterminal angles 92
4.5. The trigonometric hnctions defined in a coordinate system
in standard position, quadrant signs, and quadrantal angles 94
4.6. Reference angles and reference triangles 98
4.7. Negative angles 103
4.8. Reciprocal functions and cofunction relationships 104
4.9. Circular fhctions and the unit circle 105
4.10. Linear and angular velocity 109
4.11. Chapter 4 summary and highlights 112
Chapter 5. Graphs of Trigonometric and Circular Functions
and their Periodic Nature 114
5.1. Circular motion 114
5.2. Graphs of sine and cosine 119
5.3. Transforming graphs of sine and cosine through changes
in amplitude, period, and vertical and horizontal shifting 125
5.4. Applications of sinusoids 139
5.5. Graphs of secant and cosecant 146
5.6. Graphs of tangent and cotangent 150
5.7. Chapter 5 summary and highlights 154
Chapter 6. Inverse Trigonometric Functions 156
6. I. Review of general inverse functions 156
6.2. Inverse trigonometric fbnctions 161
6.3. Inverse sine and inverse cosine 166
6.4. Inverse tangent 174
6.5. Inverse cotangent, inverse secant, and inverse cosecant 180
6.6. Chapter 6 summary and highlights 189
Chapter 7. Trigonometric Identities 191
7.1. Surnmary of identities 191
7.2. Quotient identities and reciprocal identities 193
7.3. Pythagorean identities 194
7.4. Negative numbedangle identities 195
vi
7.5. VeriQing trigonometric identities 197
7.6. Sum and difference of angleshumbers identities, also
called addition and subtraction identities 200
7.7. Cofbnction identities 205
7.8. Supplementary angle relations 208
7.9. Dou ble-anglelnumber identities 209
7.10. Half-angle identities 213
7.11. Product-to-sum identities 216
7.12. Sumldifference-to-producti dentities 218
7.13. Squared formulas 22 1
7.14. Chapter 7 summary and highlights 222
Chapter 8. Trigonometric Functions in Equations and
Inequalities 224
8.1. Review of solving algebraic equations 224
8.2. Review of solving algebraic quadratic equations 228
8.3. Review of solving algebraic inequalities 234
8.4. Solving algebraic equations and inequalities using graphing 235
8.5. Introduction to solving trigonometric equations and
inequalities 237
8.6. Solving simple trigonometric equations using standard
position angles, reference triangles, and identities 238
8.7. Solving trigonometric equations involving powers using
hctoring, a unit circle, and identities 240
8.8. Solving trigonometric equations and inequalities using the
quadratic formula, identities, unit circles, factoring, and
gaph ing 244
8.9. Estimating solutions to trigonometric equations and
inequalities using graphing 249
8.10. Chapter 8 summary and highlights 252
Chapter 9. Trigonometric Functions and Vectors 254
9.1. Definitions of vectors 254
9.2. Representing vectors in terms of their components in a
coordinate system 256
9.3. Representing vectors in terms of their components in a
coordinate system using the unit vectors i, j, and k 259
9.4. Addition and subtraction of vectors 261
9.5. Simple vector problems 264
9.6. Multiplying a vector with a scalar 269
9.7. Dot or scalar products 269
9.8. Vector or cross product 273
vii
9.9. Chapter 9 summary and highlights 276
Chapter 10. Trigonometric Functions In Polar Coordinates
and Equations, and Parametric Equations 278
10.1. Polar coordinates defined 278
10.2. Converting between rectangular and polar coordinate
systems and equations 282
10.3. Graphing polar equations 288
10.4. Parametric equations 298
10.5. Chapter 10 summary and highlights 308
Chapter 11. Complex Numbers and The Complex Plane 318
1 1.1. Complex numbers defined 310
1 1.2. The complex plane in rectangular form 313
1 1.3. Addition and subtraction of complex numbers in
rectangular form 313
1 1.4. Complex numbers in polar form and the complex plane 3 14
11 .5. Converting between rectangular and polar form 317
11 .6. Multiplication and division of complex numbers in
rectangular and polar forms 319
1 1.7. Powers and roots of complex numbers 325
11.8. Chapter 11 summary and highlights 33 1
Chapter 12. Relationships Between Trigonometric Functions,
Exponential Functions, Hyperbolic Functions
and Series Expansions 333
I 2.1. Relationships between trigonometric and exponential
fbnctions 333
12.2. Background: summary of sequences, progressions and
series, and expanding a function into a series 335
12.3. Hyperbolic hnctions 340
12.4. Chapter 12 summary and highlights 343
Chapter 13. Spherical Trigonometry 344
1 3.1. Definitions and properties 344
13.2. Measuring spherical triangles 347
13.3. The Law of Sines and the Law of Cosines for spherical
triangles for calculating sides and angles 349
13.4. Celestial sphere 353
13.5. Chapter 13 summary and highlights 354
Index 356
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Introduction
Master Math: Trigonometry is part of the Master Math series,
which includes Master Math: Basic Math and Pre-Algebra, Master
Math: Algebra, Master Math: Pre-Calculus and Geometry, and Master
Math: Calculus. Master Math: Trigonomev and the Master Math series
as a whole are clear, concise, and comprehensive reference sources pro-
viding easy-to-find, easy-to-understand explanations of concepts and
principles, definitions, examples, and applications. Master Math: Trigo-
nometry is written for students, tutors, parents, and teachers, as well as
for scientists and engineers who need to look up principles, definitions,
explanations of concepts, and examples pertaining to the field of
trigonometry.
Trigonometry is a visual and application-oriented field of
mathematics that was developed by early astronomers and scientists to
understand, model, measure, and navigate the physical world around
them. Today, trigonometry has applications in numerous fields, includ-
ing mathematics, astronomy, engineering, physics, chemistry, geo-
graphy, navigation, surveying, architecture, and the study of electricity,
light, sound, and phenomena with periodic and wave properties. Trig-
onometry is one of the more interesting and usefbl areas of mathematics
for the non-mathematician. This book provides detailed, comprehensive
explanations of the findamentals of trigonometry and also provides
applications and examples, which will hopefblly provide motivation for
students to learn and become familiar with this truly interesting field of
mathematics.
Trigonometry involves measurements of angles, distances,
triangles, arc lengths, circles, planes, spheres, and phenomena that
exhibit a periodic nature. The six trigonometric fbnctions, sine, cosine,
tangent, cotangent, secant, and cosecant, can be defined using three
different approaches: as ratios of the sides of a right triangle (Chapter
3), in a coordinate system using angles in standard position (Chapter 4),
and as arc lengths on a unit circle, called circular finctions (Chapter 4).
Trigonometric fbnctions are found, described, and illustrated in numer-
I