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Master Math Trigonometry PDF

369 Pages·2016·19.77 MB·English
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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 Cover design by The Visual Group Printed in the U.S.A. by Book-mart Press To order this title, please call toll-free 1-800-CAREER-1 (NJ and Canada: 201-848-0310) to order using VISA or Mastercard, or for further information on books from Career Press. 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. ... 111 This page intentionally left blank 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 ... Vlll ~~ ~~ 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

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