algebra and trigonometry 3rd edition pdf

www.FreeEngineeringbooksPdf.com ffirs.qxd 11/14/12 10:09 AM Page iv www.FreeEngineeringbooksPdf.com ffirs.qxd 11/14/12 10:09 AM Page i WileyPLUS is a research-based online environment for effective teaching and learning WileyPLUS builds students’ confidence because it takes the guesswork out of studying by providing students with a clear roadmap: • • • what to how to it if they did it right It offers interactive resources along with a complete digital textbook that help students learn more With WileyPLUS, students take more initiative so you’ll have greater impact on their achievement in the classroom and beyond Now available for For more information, visit www.wileyplus.com www.FreeEngineeringbooksPdf.com ffirs.qxd 11/14/12 10:09 AM Page ii ALL THE HELP, RESOURCES, AND PERSONAL SUPPORT YOU AND YOUR STUDENTS NEED! www.wileyplus.com/resources Student Partner Program 2-Minute Tutorials and all of the resources you and your students need to get started Student support from an experienced student user Collaborate with your colleagues, find a mentor, attend virtual and live events, and view resources www.WhereFacultyConnect.com Quick Start © Courtney Keating/ iStockphoto Pre-loaded, ready-to-use assignments and presentations created by subject matter experts Technical Support 24/7 FAQs, online chat, and phone support www.wileyplus.com/support www.FreeEngineeringbooksPdf.com Your WileyPLUS Account Manager, providing personal training and support ffirs.qxd 11/14/12 10:09 AM Page iii Algebra and Trigonometry Third Edition www.FreeEngineeringbooksPdf.com ffirs.qxd 11/14/12 10:09 AM Page iv www.FreeEngineeringbooksPdf.com ffirs.qxd 11/14/12 10:09 AM Page v Algebra and Trigonometry Third Edition C YNTH IA Y YOU NG | Professor of Mathematics U N IVE RSITY OF C E NTRAL F LOR IDA www.FreeEngineeringbooksPdf.com ffirs.qxd 11/15/12 8:35 AM Page vi PUBLISHER ACQUISITIONS EDITOR PROJECT EDITOR ASSOCIATE CONTENT EDITOR EDITORIAL ASSISTANT SENIOR PRODUCTION EDITOR DESIGNER OPERATIONS MANAGER ILLUSTRATION EDITOR SENIOR PHOTO EDITOR COVER DESIGN COVER PHOTO Laurie Rosatone Joanna Dingle Jennifer Brady Beth Pearson Elizabeth Baird Kerry Weinstein/Ken Santor Madelyn Lesure Melissa Edwards Sandra Rigby; Electronic illustrations provided by Techsetters, Inc Jennifer MacMillan Madelyn Lesure Front Cover: Combined image: Ty Milford/Masterfile and Leslie Banks/iStockphoto Back Cover: Rustem Gurler/iStockphoto “Modeling Our World” image: © David Woodfall/Getty Images This book was set in 10/12 Times by MPS Limited, and printed and bound by Quad/Graphics, Inc The cover was printed by Quad/Graphics, Inc This book is printed on acid free paper ϱ Founded in 1807, John Wiley & Sons, Inc has been a valued source of knowledge and understanding for more than 200 years, helping people around the world meet their needs and fulfill their aspirations Our company is built on a foundation of principles that include responsibility to the communities we serve and where we live and work In 2008, we launched a Corporate Citizenship Initiative, a global effort to address the environmental, social, economic, and ethical challenges we face in our business Among the issues we are addressing are carbon impact, paper specifications and procurement, ethical conduct within our business and among our vendors, and community and charitable support For more information, please visit our website: www.wiley.com/go/citizenship Copyright © 2013, 2009, 2006 John Wiley & Sons, Inc All rights reserved No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning or otherwise, except as permitted under Sections 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, website www.copyright.com Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030-5774, (201) 748-6011, fax (201) 748-6008, website www.wiley.com/go/permissions Evaluation copies are provided to qualified academics and professionals for review purposes only, for use in their courses during the next academic year These copies are licensed and may not be sold or transferred to a third party Upon completion of the review period, please return the evaluation copy to Wiley Return instructions and a free of charge return mailing label are available at www.wiley.com/go/returnlabel If you have chosen to adopt this textbook for use in your course, please accept this book as your complimentary desk copy Outside of the United States, please contact your local sales representative ISBN: 978-0-470-64803-2 BRV ISBN: 978-1-118-12930-2 Printed in the United States of America 10 www.FreeEngineeringbooksPdf.com ffirs.qxd 11/14/12 10:09 AM Page vii The Wiley Faculty Network The Place Where Faculty Connect The Wiley Faculty Network (WFN) is a global community of faculty connected by a passion for teaching and a drive to learn, share, and collaborate.Whether you’re seeking guidance, training, and resources or simply looking to re-energize your course, you’ll find what you need with the WFN The WFN also partners with institutions to provide customized professional development opportunities Connect with the Wiley Faculty Network to collaborate with your colleagues, find a Mentor, attend virtual and live events, and view a wealth of resources all designed to help you grow as an educator Attend Discover innovative ideas and gain knowledge you can use Learn from instructors around the world, as well as recognized leaders across disciplines Join thousands of faculty just like you who participate in virtual and live events each semester You’ll connect with fresh ideas, best practices, and practical tools for a wide range of timely topics View Explore your resources and development opportunities See all that is available to you when you connect with the Wiley Faculty Network From Learning Modules and archived Guest Lectures to faculty-development and peer-reviewed resources, there is a wealth of materials at your fingertips Collaborate Connect with colleagues—your greatest resource Tap into your greatest resource—your peers Exchange ideas and teaching tools, while broadening your perspective Whether you choose to blog, join interest groups, or connect with a Mentor—you’ve come to the right place! 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Web: www.WhereFacultyConnect.com EMAIL: FacultyNetwork@wiley.com PHONE: 1-866-4FACULTY (1-866-432-2858) www.FreeEngineeringbooksPdf.com ffirs.qxd 11/14/12 10:09 AM Page viii For Christopher and Caroline www.FreeEngineeringbooksPdf.com bindsub.qxd 1398 11/28/12 11:42 AM Page 1398 SUBJECT INDEX Exponential growth, models, 487–491, 532–533 Exponents, 18–25 See also Logarithms and complex numbers, 77–78, 918–919 defined, 18 evaluating expressions with, 18–21 properties of, 20–21, 81–82, 512 rational, 69–70 scientific notation and, 22–24 Expressions algebraic, exponential, 18–20 rational, 48–59 Extraneous solutions defined, 96 determining, 170–171 to logarithmic equations, 525–528 to radical equations, 96–97, 130–133 F Factorial notation, 1220–1221, 1256 Factoring defined, 38 polynomials See Polynomial factoring quadratic equations, 116–118 solving equations with, 136 Factors See also Partial-fraction decomposition common, 38–39, 54–58 linear, 424 Factor theorem defined, 420 and polynomials, 419–422 Falling lines, characteristics of, 207–208 Feasible solutions defined, 1010 in linear programming, 1010–1015 Fermat’s last theorem, 1250 Fibonacci sequence, properties of, 1221–1222 Finite sequences, defined, 1218 Finite series arithmetic, 1232–1234 defined, 1222 evaluating, 1223, 1241–1244 geometric, 1241–1242 First-degree equations See also Linear equations defined, 93 Focus/foci, defined, 1113, 1127, 1140 FOIL method, for multiplying binomials, 31–34 Four-leaved rose, graphing, 934 Fractions partial-fraction decomposition, 986–995 properties and operations, 13–14 Functions applications involving, 279–280 average rate of change, 293–296 circular, 654–658 common, 287–291 See also specific function composition of, 266, 325–329 defined, 269 domains of, 268–272, 277–280 evaluating, 273–277 even/odd, 290–291 even/odd functions, 665, 667 exponential See Exponential functions expression of (Rule of 4), 272 graphing, 287–299, 308–317 increasing/decreasing, 292–293 inverse, 337–344, 793 logarithmic See Logarithmic functions modeling using variations, 350–355 notation, 273–277, 280 one-to-one, 334–337, 793 operations on, 323–329 piecewise-defined, 296–299 power, 396–397 quadratic See Quadratic functions rational See Rational functions and relations, 268–270, 280 summary table, 300 vertical line test for, 271–272 Fundamental counting principle defined, 1263 using, 1263–1265 Fundamental theorem of algebra, defined, 436 G Gaussian (normal) distribution model, 532, 535–536 defined, 535 Gaussian elimination, solving systems of linear equations with, 1035–1037 Gauss-Jordan elimination, solving systems of linear equations with, 1038–1041, 1071 www.FreeEngineeringbooksPdf.com General form of equation of circle, 224 of quadratic functions, 380 of straight line, 204 General second degree equation, 1111–1112 General term of arithmetic sequence, 1231 defined, 1218 Geometric sequences applications involving, 1245–1246 defined, 1238 properties of, 1238–1240 and series, 1241–1244 Geometric series, applications involving, 1245–1246 Geometry problems See also Applications Index modeling and formulas, 104–105, 125 Graphs/graphing absolute value equations, 163–165 absolute value inequalities, 165–166 asymptotes See Asymptotes and best fit lines, 240–247 circles, 223 conics from polar equations, 1187–1190 continuous and noncontinuous, 396 cosecant function, 697–698, 702–704 cosine function, 666–667 cotangent function, 695–696, 699–702 curves, 1195–1196 ellipses, 1129–1130, 1132–1133 of equations, 190–192 exponential functions, 482–485, 487 hyperbolas, 1143–1147 intercepts, 193–194, 197–199, 1118–1119 inverse functions, 339–340 linear inequalities, 140–142, 998–1003 linear programming, 1010–1015 lines, 191, 204–209 logarithmic functions, 499–502 nonlinear inequalities, 1164–1165 parabolas, 378–383, 1115–1116, 1118–1120 piecewise-defined functions, 296–299 polar coordinate system, 930–938, 1187–1190 polynomial functions, 396–397, 400–404, 430–432 bindsub.qxd 11/28/12 11:42 AM Page 1399 SUBJECT INDEX polynomial inequalities, 151 projectile motion, 1198–1199 quadratic functions, 378–383 rational functions, 454–459 scatterplots, 230–233, 256–257 secant function, 696–697, 702–704 shifts (horizontal/vertical), 308–312, 484–485, 501–502, 675–677, 704–707 sine function, 663–665 sinusoidal functions, 668–677 stretching and compression, 315–317, 668–671 sums of functions, 683–685 symmetry in, 194–199 systems of linear equations, 963–965 tangent function, 693–695, 699–702 transformations, 308–318, 397, 466 Greatest common factor (GCF), factoring with, 38–39 Greatest integer function, defined, 299 Grouping, in factoring polynomials, 44–45 Growth, modeling, 487–491, 527–528, 532–533, 536 H Half-angle identities, 773–780 finding exact values, 776–778 Half-life, in exponential decay, 488–489 Half-open intervals, and inequalities, 140–141 Harmonic motion, 678–683 Heaviside step function, defined, 299 Heron’s Formula, for area of SSS triangle, 874–876 Horizontal asymptotes, 450–452 defined, 450 Horizontal components, of vectors, 886 Horizontal lines characteristics of, 204–208 finding equation of, 212 Horizontal line test, for functions, 335–336, 793 Horizontal shifts defined, 309 graphing, 308–312, 484–485, 501–502, 675–677, 704–707 Hyperbolas applications of, 1147–1148 with center at (h, k), 1145–1147 with center at origin, 1141–1145 defined, 1110, 1140 finding equation of, 1141–1143, 1146 graphing, 1143–1147 vertices of, 1140, 1141–1147 Hypotenuse defined, 559 in Pythagorean theorem, 559–560 I Identities defined, 92, 730 reciprocal, 576–578 Identity functions defined, 288 graphing, 297–299 Identity properties, of real numbers, 10 Imaginary numbers, 73–78 See also Complex numbers Imaginary part, of complex number, 74, 906 Imaginary unit (i), 73–75 defined, 74 raising to powers, 77–78 Implicit domains, 278–280 defined, 278 Improper rational expressions, in partialfraction decomposition, 986–987 Improper rational functions, defined, 451 Inclination, angle of, 594–596 Inconsistent systems of linear equations defined, 956 solving, 958, 962, 965, 978, 1041–1043 Increasing functions defined, 292 determining, 292–293 Independent events, probability of, 1278–1279 Independent systems of linear equations defined, 956 solving, 958, 960, 964, 979 Independent variables as argument, 273 assigning, 270 defined, 230 Index, of root, 65 Index of summation, defined, 1222 Induction, mathematical, 1250–1253 Inequalities See also specific inequality type www.FreeEngineeringbooksPdf.com 1399 absolute value, 165–167 graphing, 140–142, 998–1003 linear, 140–146 nonlinear, 1164–1165 nonstrict, 1164–1165 polynomial, 151–155 properties of, 143 rational, 156–159 strict, 1164–1165 Infinite sequences, defined, 1218 Infinite series defined, 1222 evaluating, 1223–1224, 1242–1244 geometric, 1242–1244 Infinity, defined, 141 Initial point, of line segment, 881–882 Initial ray/side, of angle, 556 Inquiry-Based Learning Projects equivalent equations and extraneous solutions, 170–171 exponential and logarithmic functions, 544 linear relationships, 256–257 matrix multiplication, 1096 Pascal’s triangle and probability, 1283–1284 quadratic and square functions, 465 rules, patterns, and examples, 81–82 systems of linear inequalities, 1018 transformations of functions, 361–362, 466 Integer exponents, 18–25 complex numbers and, 77–78 properties of, 20–21 Integers, as numbers subset, 4–5 Intercepts defined, 193 as graphing aids, 197–199 of lines, 205–206 of rational functions, 454–459 and vertices, 1129–1133 Interest (finance) See also Applications Index defined, 106 solving problems, 106–108, 489–491 as variation, 354 Intermediate value theorem approximating zeros with, 429–430 defined, 399, 429 Intersections (sets), 141–142 in probability, 1276–1279 bindsub.qxd 1400 11/28/12 11:42 AM Page 1400 SUBJECT INDEX Interval notation, 140–141 and functions, 292–293 Intervals open and closed, 140–141 test, 151–152, 401–402 Inverse functions defined, 338 finding, 340–344, 544 graphing, 339–340, 499–502, 544 properties of, 337–339, 512–516 Inverse identities, and logarithms, 512–513 Inverse identities, trigonometric, 796, 799–800, 802, 805 Inverse of square matrix defined, 1069 finding, 1068–1073 Inverse properties of real numbers, 10 solving equations with, 521–522 Inverse variation defined, 352 modeling with, 352–353 Irrational numbers as coefficients, 1090–1091 defined, properties of, 4–5 Irreducible quadratic expressions, as factors, 424 Irreducible quadratic factors, in partialfraction decomposition, 991–993 Isosceles triangles, defined, 559 J Joint variation defined, 353 modeling with, 353–354 L Latus rectum, defined, 1115 Law of cosines and areas of triangles, 872–876 derivation of, 861–863 solving SAS triangles, 863–864 solving SSS triangles, 865–866 vectors and, 888–891 Law of sines applications of, 856 and areas of triangles, 872–876 derivation of, 848–849 solving two angles/side (AAS or ASA), 849–852 solving two sides/angle (SSA), 852–855 vectors and, 888–891 Least common denominator (LCD) defined, 14 and rational expressions, 55–58 in solving equations, 94 Least common multiple (LCM) defined, 14 and rational expressions, 55–58 Lemniscates, graphing, 936 Like terms, of polynomials, 29 Limaỗons, graphing, 934–936 Linear equations, 92–98, 102–111, 209–212 applications involving, 214–215 defined, 93 forms of, 209–212 solving algebraically, 92–95 solving with matrices See Matrix/matrices solving with modeling, 102–111 systems of See Systems of linear equations trigonometric, 819–820 Linear factors defined, 424 in partial-fraction decomposition, 987–991 Linear functions, 287–288 defined, 287 graphing, 298–299 Linear inequalities, 140–146 applications involving, 145–146 graphing, 140–142, 998–1000 interval notation for, 140–142 linear programming, 1010–1015 solving, 142–145 systems of See systems of linear inequalities in two variables Linearity, in scatterplots, 235–240, 256–257 Linear programming model, 1010–1015 Linear regression, 230–247 See also Scatterplots best fit line in, 240–245 in prediction, 245–247 Linear speed, 643–644 and angular speed, 645–646 defined, 643 Lines, 205–216 www.FreeEngineeringbooksPdf.com best fit, 240–247 defined, 556 equations of, 209–212 general form, 204 graphing, 191, 204–208 parallel, 212–213 perpendicular, 213–214 Line segment, defined, 556 Line segment, midpoint of, 185 Local (relative) maxima/minima, finding, 402–403 Logarithmic equations, 521–528 applications involving, 526–528 properties of, 521–522 solving, 524–528 Logarithmic functions, 496–518 applications involving, 503–506 common and natural, 498–499 defined, 496 domain of, 500 evaluating, 496–498 graphing, 499–502 modeling with, 532, 537–538 properties of, 512–516 Logarithmic models, 532, 537–538 Logarithmic scale, 505–506 defined, 505 Logarithms applications involving, 503–506 change-of-base formula, 517–518 common (base 10), 498–499 defined, 496 evaluating, 496–498 operations on, 514–516 properties of, 512–516 Logistic growth models, 532, 536 defined, 536 Long division, of polynomials, 410–414 M Magnitude of earthquake, 504–505 of vectors, 881–883 Main diagonal entries, of matrix, 1031 Major axis, defined, 1127 Mathematical induction, 1250–1253 Matrix algebra, 1053–1062 addition and subtraction, 1055–1056 equality of matrices, 1053–1054 matrix multiplication, 1058–1062 scalar multiplication, 1056–1057 bindsub.qxd 11/28/12 11:42 AM Page 1401 SUBJECT INDEX Matrix equations, 1067–1076 applications involving, 1076 defined, 1067 finding inverse of square matrix, 1068–1073 solving systems of linear equations, 1074–1076 Matrix/matrices, 1030–1046 applications involving, 1043–1046 augmented, 1032–1037, 1070–1073 components of, 1031–1032 in cryptography, 1028 defined, 1031 equality of, 1053–1054 finding order of, 1031–1032 Gaussian elimination with backsubstitution, 1035–1037 inverse of, 1068–1076 notation for, 1053 operations on See Matrix algebra row operations on, 1033–1034 solving linear equations with, 1192–1197 types of solutions, 1041–1043 Matrix multiplication, 1058–1062, 1096 defined, 1056, 1059 by inverse, 1068–1073 Members, of sets, Midpoint (line segment) defined, 185 midpoint formula, 185 Minor axis, defined, 1127 Minors, of square matrix, 1082–1083 Mixtures See also Applications Index defined, 108 solving problems, 108–109 Modeling See also Polynomial functions with absolute values, 172 dot products, 944 exponential decay, 487–491, 532, 534–535 with exponential functions, 487–491 exponential growth, 487–491, 532–533 functions and variation, 350–355 Gaussian (normal) distribution, 532, 535–536 inverse trigonometric functions, 833 linear equations, 102–111 linear programming, 1010–1015 logarithmic functions, 532, 537–538 logistic growth, 532, 536 polynomial functions in, 394–396 sinusoidal functions, 713 systems of linear equations, 979–981 systems of nonlinear equations, 1204 and word problems, 102–104 Modeling Our World absolute value equations and inequalities, 172 Climate Carbon Wedge model, 258–259 climate change case, 713, 833, 944, 1204 modeling functions with variation, 363 modeling with exponential and logarithmic functions, 545–546 modeling with matrices, 1097–1098 modeling with polynomial functions, 467–468 modeling with series, 1285 modeling with systems of linear equations, 1019–1020 Modulus, defined, 906 Monomials, defined, 28 Multiplication of binomials, 31–34 commutative property of, 10 of complex numbers, 76, 915–916 dot product, 896–901 exponents and, 20–21 of fractions, 13–14 of functions, 323–325 of matrices, 1056–1062 order of operations and, 7–8 of polynomials, 30–34 of radicals, 65–66 of rational expressions, 52–53 scalar, 885 of vectors, 885 zero in, 12–13 Multiplicative identity matrix, 1068–1069, 1096 defined, 1069 Multiplicative identity property defined, 10 and matrices, 1068–1069, 1096 Multiplicative inverse property, defined, 10 Multiplicity of zeros, 399–400 defined, 400 and graph of polynomial function, 402–403 www.FreeEngineeringbooksPdf.com 1401 Mutually exclusive events, probability of, 1276–1278 N Natural base e, 486–487 Natural exponential function, 486–487 defined, 486 Natural logarithmic function evaluating, 498–499, 517 properties of, 513–516 Natural numbers, 4–5 as exponents, 18 Navigation, applications of, 596 Negative-integer exponent property, 19 Negative numbers See also Complex numbers in exponents, 18–19, 69–70 properties of, 11–12, 19, 73–75 Nonacute angles, 611–612 evaluating trigonometric functions for, 626–630 Nondistinguishable permutations, 1268–1269 Nonlinear equations, 1153–1154 See also Systems of nonlinear equations Nonlinear inequalities in two variables, 1164–1165 Nonrigid transformations, defined, 315 Nonsingular matrices, 1070–1073 defined, 1070 Nonstrict inequalities, graphing, 1000, 1164–1165 Notation for absolute value, 162 arrow, 447–448 for binomial coefficient, 1256 for composite functions, 326 for determinants, 1081 factorial notation, 1220–1221, 1256 function, 273–277, 280 interval, 140–141, 292–293 for inverse functions, 337, 795 for matrices, 1053 for operations, for probability, 1275 scientific, 22–24 for sequences, 1218 set, 141–142 sigma/summation notation, 1222, 1241 for vectors, 881 bindsub.qxd 1402 11/28/12 11:42 AM Page 1402 SUBJECT INDEX nth partial sum of arithmetic sequence, 1232–1234 defined, 1222 of geometric sequence, 1241–1244 nth root theorem, for complex numbers, 919–921 Null set, defined, Number line, absolute value and, 12 Numbers complex, 73–78 complex numbers, 905–911 irrational, 4–5 natural, 4–5 negative, 11–12 rational, 4–6 subsets of, 4–5 whole, 4–5 Numerators, in rational numbers, n zeros theorem, defined, 436 O Objective function defined, 1010 in linear programming, 1010–1015 Oblique triangles area of, 872–876 defined, 846 solving using law of cosines, 861–866 solving using law of sines, 848–857 Obtuse angles, defined, 557 Obtuse triangles, 846 Odd functions, 657–658 sine function as, 665 Odd functions, determining, 290–291 One-to-one functions, defined, 334 One-to-one properties, for exponential equations, 521 Open intervals, and inequalities, 140–141 Operations with exponents, 18–21 on fractions, 13–14 on functions, 323–329 on logarithms, 514–516 order of, 7–8 with radicals, 65–68 on vectors, 885 Optimization defined, 1010 solving problems of, 712, 1010–1015 Order in combinations and permutations, 1265 of matrix, 1031–1032 Ordered pairs, in graphing, 182 Order of operations, 7–8 exponents in, 18–19 Ordinate, defined, 182 Orientation, along curve, 1195 Origin defined, 182 symmetry about, 195–196 Orthogonal vectors, 899 Outcomes, 1273–1279 defined, 1273 P Parabolas applications involving, 384–387 applications of, 1120–1121 defined, 1110, 1113 finding equation of, 1114, 1116–1118 finding equations of, 384 graphing, 378–383, 1115–1116, 1118–1120 as graphs of quadratic functions, 376–377 with vertex at (h, k), 1117–1120 with vertex at origin, 1114–1117 vertex/vertices of, 1113, 1114, 1117 Parallel lines, 212–213 defined, 212 Parameters, defined, 1194 Parametric equations, 1194–1196 applications of, 1197–1199 defined, 1194 Parametric representation, of line, 977 Partial-fraction decomposition, 986–995 with combinations of cases, 993–994 defined, 986 with distinct irreducible quadratic factors, 991–992 with distinct linear factors, 988–989 procedure for, 987 with repeated irreducible quadratic factors, 992–993 with repeated linear factors, 989–991 Partial fractions, defined, 986 Pascal’s triangle, and binomial expansion, 1258–1259 Perfect cubes, 33–34 Perfect squares, 32–34 factoring, 39–40 and quadratic equations, 120–121 www.FreeEngineeringbooksPdf.com square roots of, 64 Period, of sinusoidal function, 671–672 Periodic function, defined, 663 Permutations, 1265–1270 defined, 1265 finding, 1265–1267 with repetition, 1268–1269 Perpendicular lines, 213–214 defined, 213 Piecewise-defined functions, 296–299 inverse of, 344 Plane, in graphing, 974 Plane curves, defined, 1194 Point-plotting defined, 182 graphing equations by, 190–192 polar coordinates, 927–929, 1189 Point-slope form, of straight line equation, 210–212 Points of discontinuity, 298–299 Polar coordinate system, 927–929 and conics, 1183–1189 defined, 927 graphing, 930–938 Polar equations of conics, 1183–1189 converting to rectangular form, 937 graphing, 930–936 Polar form, of complex numbers, 905–911 Polynomial factoring, 37–46, 424–430 Descartes’s rule of signs in, 425–430 with factor theorem, 420–421 greatest common factor in, 38–39 by grouping, 44–45 rational zero theorem in, 422–425 of special forms, 39–41 strategy for, 45 trinomials, 40–44 upper and lower bound rules, 427–428 Polynomial functions complex zeros of, 435–441 defined, 376 graphing, 396–397, 400–404, 430–432 identifying, 394–396 quadratic See Quadratic functions real zeros of See Real zeros remainder and factor theorems, 419–422 Polynomial inequalities, 151–155 graphing, 151 solving, 151–155 bindsub.qxd 11/28/12 11:42 AM Page 1403 SUBJECT INDEX Polynomials See also Rational expressions adding and subtracting, 28–30 defined, 28 division of, 410–416 factoring See Polynomial factoring multiplying, 30–34 Position vector, 882–884 Power functions defined, 396 end behavior of, 403 Power reduction formulas, in calculus, 774 Powers See also Exponents variations with, 351–353 Prediction, using best fit line, 245–247 Predictor variables, defined, 230 Prime polynomials defined, 38 identifying, 44 Principal (finance), 106–108, 489–491 defined, 106 Principal nth roots, defined, 65 Principal square roots of complex numbers, 74 defined, 63 Probability, 1273–1279 calculating, 1274–1275 defined, 1274 of independent events, 1278–1279 of mutually exclusive events, 1276–1278 of non-occurrence of event, 1275–1276 sample space and, 1273 Producer surplus, in supply and demand, 1004–1005 Product functions, 323–325 Products of complex numbers, 76–77, 915–916 defined, dot product, 896–901, 944 in factorial notation, 1220–1221 special, 31–34 and sums of functions, 784 Product-to-sum identities, 784–786, 788–789 Proper rational expressions, in partialfraction decomposition, 986–987 Proper rational functions, defined, 451 Pythagorean identities, 654, 733–735 Pythagorean theorem, 559–560 as case of law of cosines, 863 Q Quadrantal angles calculating for, 614, 621 defined, 604–605 Quadrants algebraic signs of, 619–621 of Cartesian plane, 182, 604–608 Quadratic equations, 116–126 completing the square, 120–121 defined, 116 factoring, 116–118 quadratic formula use, 122–125 solving with square root method, 118–119 trigonometric, 819–820 Quadratic formula, 122–126 applications involving, 124–125 defined, 122 solving equations with, 122–124 Quadratic functions, 376–387 defined, 377 finding equation of parabola, 384 graphing in general form, 380–383 graphing in standard form, 378–380, 466 graphs of, 376–377 Quadratic inequalities See Polynomial inequalities Quadratic in form equations, solving, 134–135 Quotient functions, and domain restrictions, 323–325 Quotient identities, 732–733 Quotients of complex numbers, 77, 916–917 defined, 7, 410 R Radian measure, 636–637 arc length and, 640–641 area of circular sector and, 641–643 conversion to degree measure, 637–639 defined, 636 linear and angular speed, 643–646 in polar coordinate graphing, 927–928 Radians, defined, 635 Radical equations, solving, 130–133 Radicals operations with, 65–68 properties of, 65, 81–82 www.FreeEngineeringbooksPdf.com 1403 simplified form of, 68–69 Radical sign, defined, 63 Radicand, defined, 63 Radius (circle), 221–225 defined, 221 Range defined, 268 of exponential functions, 481–482 of functions, 268–272, 622–623 and inequalities, 142 of inverse functions, 338–339 of trigonometric functions, 622–623, 693–700, 793 Rational equations, 95–98 defined, 95 solving, 95–98 Rational exponents, 69–70 defined, 69 Rational expressions adding and subtracting, 53–56 complex, 56–58 defined, 48 domain restrictions of, 48–50, 156–158 multiplying and dividing, 52–53 simplifying, 50–52 Rational functions, 445–460 asymptotes and, 447–453 defined, 445 domain of, 445–446 graphing, 454–459 proper/improper, 451 Rational inequalities, 156–159 applications involving, 158 solving, 156–158 Rationalizing denominators, 67–69 Rational numbers defined, properties of, 4–5 Rational zero (root) theorem defined, 422 finding zeros using, 422–425 Ratios, in right triangles, 574–580 Ray, defined, 556 Real number line defined, test intervals, 151–152 Real numbers, 4–15 order of operations with, 7–8 properties of, 9–15 set of, 4–6 Real part, of complex number, 74 bindsub.qxd 1404 11/28/12 11:42 AM Page 1404 SUBJECT INDEX Real zeros, 419–432 approximating, 429–430 defined, 398 Descartes’s rule of signs and, 425–430 finding, 399–400, 411–412, 422–430 number of, 422 Reciprocal functions, defined, 290 Reciprocal identities, in trigonometric functions, 576–578, 730–732 Rectangular coordinate system and polar coordinates, 929–930 vectors in, 882–884 Rectangular coordinate system, defined, 182 Rectangular form, of complex numbers, 905–911 Recursion formulas, and sequences, 1221–1222 Reduced row-echelon form properties of, 1034–1035 solving with, 1038–1041 Reference angles, 623–626 defined, 624 Reference triangles, 623–626 defined for right triangle, 624 Reflection about the axes, graphing using, 312–314 Relations defined, 268 and functions, 268–270, 280 Remainders, defined, 410 Remainder theorem defined, 420 and polynomials, 419–422 Repetition, permutations with, 1268–1269 Resonance, 679, 683 Response variables, defined, 230 Richter scale, logarithmic application, 504–505 Right angle, defined, 557 Right triangles applications of, 594–596 defined, 559 ratios in, 574–580 solving, 559–563, 589–597 special, 561–564 Rigid transformations, defined, 315 Rising lines, characteristics of, 207–208 Roots of complex numbers, 919–923 cube root, 65 other (nth) roots, 65–69 square, 63–64, 74 Roots (solution) See also Zeros complex, 123, 124 defined, 92 double, 117, 124 repeated, 399–400 Rotation, angle of, 1175–1179 Rotation of axes, formulas for, 1172–1175 Rounding, of decimals, 6–7 Row-echelon form properties of, 1034–1035 solving with, 1035–1037 Row index, of matrix, 1031 Row matrix, defined, 1032 Rule of 4, for functions, 272 S Sample space, defined, 1273 SAS (side-angle-side) triangles, 847, 863–864, 873–874 Scalar multiplication defined, 1057 of matrices, 1056–1057 of vectors, 885 Scalars in dot products, 896 vs vectors, 881 Scatterplots, 230–240 defined, 230 drawing, 230–233 patterns in, 234–240, 256–257 Scientific notation, 22–24 and decimal conversion, 24 defined, 24 Secant function calculating as ratio, 578 defined, 575–576 graphing, 696–697, 702–704 inverse, 803–806 inverse identities, 805 Secant line, and average rate of change, 293–296 Second-degree equation, 1111–1112 See also Quadratic equations defined, 93 Sequences, 1218–1225 arithmetic, 1229–1235 defined, 1218 factorial notation and, 1220–1221 www.FreeEngineeringbooksPdf.com finding, 1219–1220 geometric, 1238–1246 recursion formulas and, 1221–1222 Series, 1222–1225 applications involving, 1224 arithmetic, 1232–1234 defined, 1222 evaluating, 1223–1224, 1242–1244 geometric, 1241–1246 Sets and probability, 1276–1279 union and intersection of, 141–142, 1276–1279 Shifts, horizontal/vertical, 308–312, 484–485, 501–502, 675–677, 704–707 Sigma notation, 1222–1223, 1241 Significant digits and accuracy, 589–590 defined, 589 least number of, 590 Similar triangles, 564–568 applications of, 567 defined, 565 solving, 565–566 Simple harmonic motion, 678, 679–681 Simple interest (finance), defined, 106 Simplification of complex rational expressions, 56–58 of exponential expressions, 21–22 and order of operations, 7–8 of radical expressions, 68–70, 74 of rational expressions, 50–52 of roots, 64, 66–70 Sine function calculating as ratio, 576–577 defined, 575–576 graphing, 663–665 inverse, 794–797 inverse identities, 796 sum and difference identities, 767–769 Singular matrices, 1070, 1072 defined, 1070 Sinusoidal functions amplitude of, 668–671 graphing, 668–677 and harmonic motion, 678–683 modeling, 713 period of, 671–672 shifted, 675–677 sums of functions, 683–685 bindsub.qxd 11/28/12 11:42 AM Page 1405 SUBJECT INDEX Slant asymptotes, defined, 453 Slope in average rate of change, 293–296 defined, 206 determining, 206–208 and straight line equations, 209–212 Slope-intercept form graphing using, 964 of straight line equation, 209–210 Smooth graphs, defined, 396 Solutions defined, 92 extraneous, 96, 130–133 feasible, 1010–1015 Solution sets defined, 92 for systems of nonlinear equations, 1153–1154 Special angles, evaluating trigonometric functions, 580–583, 605–607 Special products of polynomials, 31–34 applying, 34 factoring, 39–41 Special triangles, 559, 561–564 Speed linear and angular, 643–646 solving problems, 109–111 vs velocity, 643 Square functions defined, 288 graphing, 297–299 Square matrix/matrices defined, 1031 determinants and Cramer’s rule, 1081–1091 inverse of, 1067–1076 minors and cofactors of, 1082–1083 Square root functions defined, 289 inverse of, 342 Square root method, and quadratic equations, 118–121 Square root property defined, 118 solving equations with, 118–119 Square roots, 63–64, 74 defined, 63 Squares, in special products, 32–34, 39–41 SSA (side-side-angle) triangles, 847, 852–856 SSS (side-side-side) triangles, 847, 865–866, 874–876 Standard form of complex numbers, 74 of equation of circle, 222 of equation of ellipse, 1129, 1132 of equation of hyperbola, 1141–1142, 1146 of equation of parabola, 1114, 1117 of polynomials, 28–29 of quadratic equation, 116 of quadratic function, 378, 466 Standard position, defined, 604 Step functions, defined, 299 Straight angle, defined, 557 Stretching, graphing using, 315–317 Stretching, of graphs, 668–671 Strict inequalities defined, 140 graphing, 999, 1164–1165 Substitution method for evaluating functions, 274 for solving systems of linear equations, 957–959, 974–976, 1035–1037 for solving systems of nonlinear equations, 1158–1161 Substitution principle, evaluating with, Subtraction of complex numbers, 75 exponents and, 20–21 of fractions, 13–14 of functions, 323–325 of matrices, 1055–1056 order of operations and, 7–8 of polynomials, 28–30 of radicals, 66 of rational expressions, 53–56 of real numbers, 10 of vectors, 885 Sum and difference identities for cosine function, 752–756 and product-to-sum identities, 784–786 for sine function, 756–758 for tangent function, 759–761 Sum functions, 323–325 Summation notation, 1222–1223, 1241 Sums defined, of functions, graphing, 683–685 www.FreeEngineeringbooksPdf.com 1405 of matrices, 1055–1056 and series, 1222–1224 Sum-to-product identities, 784, 786–789 Supplementary angles, defined, 558 Symmetry about the origin, 195–196 in binomial expansions, 1255 and even/odd functions, 290–291 as graphing aids, 197–199 with respect to axes, 194–199, 1113 tests for, 195 Synthetic division defined, 414 of polynomials, 414–416 Systems of linear equations, 956–969, 973–981 solving with Cramer’s rule, 1081–1091 solving with matrix equations, 1067–1076 Systems of linear equations in three variables, 973–981 defined, 973 modeling with, 979–981 solving, 974–976, 1036–1041, 1088–1091 solving with Cramer’s rule, 1088–1091 types of solutions, 977–979 Systems of linear equations in two variables, 956–969 See also Matrix/matrices applications involving, 966–968 characteristics of, 956–957 choosing solution method, 966 defined, 956 solving by elimination, 959–963, 1035–1036 solving by graphing, 963–965 solving by substitution, 957–959 solving with Cramer’s rule, 1086–1088 Systems of linear inequalities in two variables, 998–1006, 1018 applications involving, 1004–1006 graphing, 1000–1003, 1018 Systems of nonlinear equations applications of, 1160–1161 modeling, 1204 solving by elimination, 1154–1157 solving by substitution, 1158–1161 Systems of nonlinear inequalities, solving by graphing, 1165–1169 bindsub.qxd 1406 11/28/12 11:42 AM Page 1406 SUBJECT INDEX T Tangent function calculating as ratio, 576–578 defined, 575–576 graphing, 693–695, 699–702 inverse, 801–803 inverse identities, 802 sum and difference identities, 759–761 Technology use for absolute values, 163, 166 for approximating exponential functions, 481 for approximating logarithms, 498–499, 523 for binomial expansions, 1256, 1260 for computing correlation coefficient, 237–238 and Cramer’s rule, 1087–1088 for evaluating series, 1223–1224, 1252 for factoring polynomials, 421, 424, 438 for finding best fit line, 242–244 for finding determinants, 1082, 1084 for finding factorials, 1220–1221, 1256 for finding permutations and combinations, 1267–1269 for finding real zeros, 399, 404, 426, 430 for graphing circles, 223, 225 for graphing equations, 192, 195–196, 209 for graphing exponential functions, 483, 485, 487, 536–537 for graphing functions, 275–276, 297–298, 310, 312, 314, 317, 385–386, 401 for graphing linear inequalities, 999–1003 for graphing logarithmic functions, 526 for graphing parabolas, 381–382 for graphing rational functions, 452–453 for inverse functions, 342–343 for matrices, 1045, 1055, 1057, 1059–1060, 1070–1071, 1073, 1075, 1082 for natural base e, 486–487 for partial-fraction decomposition, 988–989, 992 for scatterplots, 231–233 for sequences and series, 1231, 1233, 1246 for solving inequalities, 143–145, 152, 156 for solving linear equations, 93–98 for solving quadratic equations, 117–118, 121, 123, 125 for solving radical equations, 131, 133 for solving systems of linear equations, 963–965 using asymptotes, 453, 455–459 Terminal point, of line segment, 881–882 Terminal ray/side, of angle, 556, 604–605 Terms defined, 9, 28 like, 29 in sequences, 1218–1219 Test intervals, of real number line, 151–152 Third-degree equations, defined, 93 Three-dimensional coordinate system, graphing with, 973–974 TI calculators computing correlation coefficient with, 237–238 finding best fit line with, 242–243 for scatterplots, 231–232 Transcendental functions, defined, 480 Transformations, 308–317, 361–362, 466 of circular functions, 704–707, 1117 of exponential functions, 484–485 horizontal and vertical shifts, 308–312, 484–485, 501–502, 675–677, 704–707 of power functions, 396–397 reflection about axes, 312–314 stretching and compressing, 315–317, 668–671 summary table, 318 Transverse axis, of hyperbola, 1140 Triangles classification of, 559 congruent, 565 oblique, 846 properties of, 559 reference, 623–626 right, 559–563, 589–597 similar, 564–568 special, 559, 561–564 Trichotomy property, of real numbers, 140 Trigonometric equations www.FreeEngineeringbooksPdf.com applications of, 825–826 solving by inspection, 815–819 solving with algebraic techniques, 819–820 solving with inverse functions, 820–822 solving with trigonometric identities, 822–824 Trigonometric expressions with inverse functions, 806–809 simplifying with identities, 742–744 Trigonometric functions algebraic signs of, 619–621 and angle of rotation, 1175–1179 defined by ratios, 575 defined in Cartesian plane, 609–614 evaluating with calculator, 583, 767–769 inverse cosine, 798–800 inverse secant/cosecant/cotangent, 803–806 inverse sine, 794–797 inverse tangent, 801–803 modeling, 833 for nonacute angles, 626–630 range of, 622–623 right triangle ratios, 574–580 of unit circle, 654–658 Trigonometric identities double-angle, 765–769 even-odd, 742 half-angle, 773–780 inverse, 796, 799–800, 802, 805 product-to-sum, 784–786, 788–789 Pythagorean, 654, 733–735 quotient, 732–733 reciprocal, 730–732 in solving trigonometric equations, 822–824 sum and difference, 751–761 sum-to-product, 784, 786–789 verifying, 744–747 Trigonometry, uses of, 554 Trinomials defined, 28 factoring, 40–44 Truncation, of decimals, 6–7 U Unbounded graphs, 1164–1169 Union (sets), 141–142 in probability, 1276–1279 bindsub.qxd 11/28/12 11:42 AM Page 1407 SUBJECT INDEX Unit circle defined, 222 trigonometric functions and, 654–658 Unit vector defined, 886 finding, 886–887 Upper/lower bound rules, for real zeros, 427–428 u-substitution, solving equations with, 134–135 V Variables assigning in models, 102–104 defined, dependent and independent, 230, 270, 273 solving for, 92–95 Variation modeling with, 350–355, 363 in sign, 425 Vectors addition of, 882, 884 algebraic interpretation of, 882–884 applications of, 887–891 dot product, 896–901 equality of, 881, 884 geometric interpretation of, 881–882 magnitude of, 881–883 modeling, 944 operations on, 885 orthogonal, 899 Velocity vectors, 887–889 Vertex/vertices See also Quadratic functions of angles, 556 of ellipses, 1129–1133 of hyperbolas, 1140, 1141–1147 of inequalities graphs, 1004–1006 of parabolas, 377, 383, 1113–1114 Vertical asymptotes, 448–450, 693 defined, 448 Vertical components, of vectors, 886 Vertical lines characteristics of, 204–208 finding equation of, 212 Vertical line test, for functions, 271–272 Vertical shifts defined, 309 graphing, 308–312, 484–485, 501–502 graphing of, 675–677, 704–707 W Whole numbers, 4–5 Wiles, Andrew, proof of Fermat’s last theorem, 1250 Word problems examples of, 104–111 solving, 102–104 Work defined, 901 solving problems, 111 vectors in, 900–901 X x-axis defined, 182 www.FreeEngineeringbooksPdf.com 1407 symmetry about, 194–196 x-coordinates, defined, 182 x-intercepts See also Intercepts; Symmetry; Zeros finding, 193–194 of parabolas, 378–383, 1118–1119 Y y-axis defined, 182 symmetry about, 194–196 y-coordinates, defined, 182 y-intercepts See also Intercepts; Symmetry finding, 193–194 of parabolas, 378–383, 1118–1119 Z Zero, properties of, 13, 20 Zero-exponent property, 20 Zero matrix, defined, 1056 Zero product property defined, 13 and factorable equations, 136 in factoring quadratic equations, 116–118 Zeros complex See Complex zeros of polynomials, 151–155, 398–400, 411–412, 422 rational, 422–426 real See Real zeros Zero vector, 885 bindsub.qxd 11/28/12 11:42 AM Page 1408 www.FreeEngineeringbooksPdf.com FMEndpapers.qxd 11/27/12 G RAPHS 9:59 AM OF TH E Page B1 T R IGONOM ETR IC F U NCTIONS y y = cos x y y = sin x y y = tan x 1 ␲ –␲ x x –2␲ –2␲ 2␲ ␲ –␲ 2␲ –2␲ –␲ –1 x ␲ 2␲ –3 –1 –1 y y = sec x y y = csc x 3 –␲ –1 y y = cot x 5 –2␲ –5 ␲ x –2␲ 2␲ –␲ –1 x ␲ 2␲ –2␲ –␲ –1 –3 –3 –3 –5 –5 –5 A M PLITU DE , P E R IOD , AN D P HASE S H IF T x = r cos ␪ y = r sin ␪ y ϭ A sin(Bx + C ) ⎫ ⎬ ⎭ y ϭ A tan(Bx + C) Period = P OWE RS AN D ⎫ ⎬ ⎭ Phase shift ϭ C left B right R OOTS OF p B y ϭ A cot(Bx + C) if C/B if C/B C OM PLEX N U M B E RS zn = (x + iy)n = [r(cos ␪ + i sin ␪)]n = r n[cos (n ␪) + isin(n␪)] n = 1, 2, Á n 2z = (x + iy)1/n = [r (cos ␪ + i sin ␪)]1/n = r1/n c cos a ␪ + 2k␲ ␪ + 2k␲ b + i sin a bd n n k = 0, 1, 2, Á , n - www.FreeEngineeringbooksPdf.com 2␲ P OLAR C OOR DINATES y y ϭ A cos(Bx + C) 2p Amplitude ϭ 0A Period = B C left if C/B Phase shift = B right if C/B x ␲ (r, ␪) (x, y) r y x ␪ x r2 = x2 + y2 y tan ␪ = x Complex Numbers x ϩ iy ϭ Rectangular form r (cos␪ + isin ␪) Trigonometric (polar) form FMEndpapers.qxd 11/27/12 9:59 AM Page B2 T R IGONOM ETR IC F U NCTIONS IN TH E C ARTESIAN P LAN E R IG HT T R IANG LE T R IGONOM ETRY opp hyp sin ␪ = csc ␪ = hyp opp adj cos ␪ = hyp hyp sec ␪ = adj opp tan ␪ = adj adj cot ␪ = opp b Opposite ␪ a Adjacent E XACT VALU ES OF T R IGONOM ETR IC F U NCTIONS x radians sin x cos x tan x 0° p p p p 2 12 13 13 12 2 13 — 45° 60° 90° r y r sec ␪ = x x cot ␪ = y y csc ␪ = (x, y) r y ␪ x x A NG LE M EASU R E M E NT x degrees 30° y r x cos ␪ = r y tan ␪ = x sin ␪ = Hypotenuse c p radians = 180° A = 12 r 2␪ s = r␪ 1␪ in radians) s r ␪ To convert from degrees to p radians, multiply by 180° r To convert from radians to degrees, multiply by 13 180° p O B LIQU E T R IANG LE Law of Sines S PECIAL R IG HT T R IANG LES In any triangle, sin ␤ sin ␥ sin ␣ = = a c b 30º 45º 45º 60º C IR CU LAR F U NCTIONS ( COS⍜ , SIN⍜ ) y ( ) ) ( ( ) ( ( – , √3 2 (0, 1) ( ) 2 –√ , √ 2 90º ␲ 3␲ 45º 135º ( 2 –√ , –√ 2 ) 5␲ ( √2 , √ 2 ) x 0º 360º 2␲ (1, 0) 315º 7␲ √2 √ (–1, 0) ␲ 180º ␲ 225º 3␲ 270º (0, –1) c ␤ a2 = b2 + c2 - 2bc cos ␣ b2 = a2 + c2 - 2ac cos ␤ c2 = a2 + b2 - 2ab cos ␥ 1 ␣ a Law of Cosines √3 √2 ( ,– ) ) y (0, 1) ( √3 2, 90º ␲ ␲ 2␲ √3 , 3 120º –√ , 60º ␲ 2 5␲ 30º 6 150º x (–1, 0) 0º ␲ 180º 360º 2␲ (1, 0) 330º 11␲ 7␲ 210º √3 , – 240º 300º –√ , – 5␲ 4␲ 2 2 3␲ 3 270º ( – 1, – √ 2 b ␥ ) www.FreeEngineeringbooksPdf.com (0, –1) ( √3 2,– ) ) ) FMEndpapers.qxd 11/27/12 9:59 AM Page B3 I DE NTITIES Sum Identities Cofunction Identities sin(x + y) = sin x cos y + cos x sin y cos(x + y) = cos x cos y - sin x sin y tan x + tan y tan(x + y) = - tan x tan y (Replace p/2 with 90° if x is in degree measure.) sin a p - xb = cos x cos a p - xb = sin x Difference Identities tan a p - xb = cot x cot a p - xb = tan x p sec a - x b = csc x csc a p - x b = sec x sin(x - y) = sin x cos y - cos x sin y cos(x - y) = cos x cos y + sin x sin y tan x - tan y tan(x - y) = + tan x tan y Product–Sum Identities Double-Angle Identities sin x cos y = 12 [sin(x + y) + sin(x - y)] sin 2x ϭ 2sin x cos x cos x sin y = 12 [sin(x + y) - sin(x - y)] cos x - sin x cos 2x = - sin2 x L cos2 x - tan 2x ϭ sin x sin y = 12 [cos(x - y) - cos(x + y)] cos x cos y = 12 [cos(x + y) + cos(x - y)] 2tan x 2cot x ϭ ϭ 2 cot x Ϫ tan x Ϫ tan x cot x Ϫ Half-Angle Identities x - cos x sin a b = ; A Sign ( +/ -) is determined by quadrant in which x/2 lies x + cos x cos a b = ; A x - cos x sin x - cos x tan a b = = = ; sin x + cos x A + cos x Sum–Product Identities x + y x - y b cos a b 2 x + y x - y sin x - sin y = 2cos a b sin a b 2 x + y x - y cos x + cos y = 2cosa b cos a b 2 x + y x - y cos x - cos y = - 2sin a b sin a b 2 sin x + sin y = 2sin a Identities for Reducing Powers - cos 2x cos 2x tan2 x = + cos 2x sin2 x = cos2 x = + cos 2x B ASIC T R IGONOM ETR IC I DE NTITIES Reciprocal Identities csc x = sin x sec x = cos x Identities for Negatives cot x = tan x sin1 -x2 = - sin x tan1-x2 = -tan x Quotient Identities Pythagorean Identities sin x tan x = cos x sin2x + cos2x = cos x cot x = sin x + cot2x = csc2x www.FreeEngineeringbooksPdf.com cos 1- x2 = cosx tan2x + = sec2x bindsub.qxd 11/28/12 11:42 AM Page 1408 ~StormRG~ www.FreeEngineeringbooksPdf.com ... Algebra and Trigonometry Third Edition www.FreeEngineeringbooksPdf.com ffirs.qxd 11/14/12 10:09 AM Page iv www.FreeEngineeringbooksPdf.com ffirs.qxd 11/14/12 10:09 AM Page v Algebra and Trigonometry. .. of Mathematics at the University of Central Florida (UCF) and the author of College Algebra, Trigonometry, Algebra and Trigonometry, and Precalculus She holds a B.A degree in Secondary Mathematics... www.FreeEngineeringbooksPdf.com xi fprep.qxd xii 11/14/12 10:10 AM Page xii P R E FA C E New to the Third Edition The first edition was my book, the second edition was our book, and this third edition is our