,
, STUDENT SOLUTIONS
MANUAL
KEVIN BODDEN RANDY GALLAHER
LEWIS AND CLARK COMMUNITY COLLEGE
8
��e6ra & �ri8onometry
MICHAEL SULLIVAN
PEARSON
----
Prentice
Hall
Upper Saddle River, NJ 07458
,Vice President and Editorial Director, Mathematics: Christine Hoag
Senior Editor: Adam Jaworski
Editorial AssistantlPrint Supplements Editor: Christine Whitlock
Senior Managing Editor: Linda Behrens
Assistant Managing Editor: Lynn Savino Wendel
Production Editor: Ashley M. Booth
Supplement Cover Manager: Paul Gourhan
Supplement Cover Designer: Victoria Colotta
Manufacturing Buyer: Ilene Kahn
Associate Director of Operations Alexis Heydt-Long
PEARSON © 2008 Pearson Education, Inc.
----- Pearson Prentice Hall
Prentice Pearson Education, Inc.
Hall Upper Saddle River, NJ 07458
All rights reserved. No part of this book may be reproduced in any form or by any means, without permission in
writing from the publisher.
Pearson Prentice Hall™ is a trademark of Pearson Education, Inc.
The author and publisher of this book have used their best efforts in preparing this book. These efforts include the
development, research, and testing of the theories and programs to determine their effectiveness. The author and
publisher make no warranty of any kind, expressed or implied, with regard to these programs or the documentation
contained in this book. The author and publisher shall not be liable in any event for incidental or consequential
damages in connection with, or arising out of, the furnishing, performance, or use of these programs.
This work is protected by United States copyright laws and is provided solely for teaching courses and assessing
student learning. Dissemination or sale of any part of this work (including on the World Wide Web) will destroy
the integrity of the work and is not permitted. The work and materials from it should never be made available
except by instructors using the accompanying text in their classes. All recipients of this work are expected to
abide by these restrictions and to honor the intended pedagogical purposes and the needs of other instructors
who rely on these materials.
Printed in the United States of America
10 9 8 7 6 5 4 3 2
ISBN 13: 978-0-13-232124-2
ISBN 10: 0-13-232124-6
Pearson Education Ltd., London
Pearson Education Australia Pty. Ltd., Sydney
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Pearson Education Canada, Inc., Toronto
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Pearson Education Malaysia, Pte. Ltd.
, Table of Contents
Preface
Chapter R Review
R.l Real Numbers ............................................................................................................................... 1
R.2 Algebra Essentials . .... ... . . . .. . .
. . . . . .. .. . .
. . . . . . . . . . . . . . . .. .
. . . .. .
. . . . . 3
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
R.3 Geometry Essentials ..................................................................................................................... 6
R.4 Polynomials .................................................................................................................................. 8
R.5 Factoring Polynomials . .. . ..
. . .. . . .
. . . . . . . . . . . . . . .. . . . .
. . . . . . . . . . . . . .. II
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
R.6 Synthetic Division ...................................................................................................................... 14
R.7 Rational expressions .....................: ............................................................................................. 15
R.8 nth Roots; Rational Exponents ................................................................................................... 20
Chapter Review ..................................................................................................................................... 24
Chapter Test .......................................................................................................................................... 28
Chapter 1 Equations and Inequalities
1.1 Linear Equations ......................................................................................................................... 30
1.2 Quadratic Equations.................................................................................................................... 39
1.3 Complex Numbers; Quadratic Equations in the Complex Number System ............................... 48
1.4 Radical Equations; Equations Quadratic in Form; Factorable Equations ................................... 50
1.5 Solving Inequalities .................................................................................................................... 59
1.6 Equations and Inequalities Involving Absolute Value ................................................................ 65
1.7 Problem Solving: Interest, Mixture, Uniform Motion, and Constant Rate Job Applications ..... 69
Chapter Review ..................................................................................................................................... 73
Chapter Test .......................................................................................................................................... 80
Chapter 2 Graphs
2.1 The Distance and Midpoint Formulas......................................................................................... 82
2.2 Graphs of Equations in Two Variables; Intercepts; Symmetry .................................................. 87
2.3 Lines 93
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4 Circles ....................................................................................................................................... 101
2.5 Variation ................................................................................................................................... 106
Chapter Review ................................................................................................................................... 108
Chapter Test ........................................................................................................................................ 114
Cumulative Review ............................................................................................................................. 116
Chapter 3 Functions and Their Graphs
3.1 Functions................................................................................................................................... 118
3.2 The Graph of a Function ........................................................................................................... 125
3.3 Properties of Functions ............................................................................................................. 129
3.4 Library of Functions; Piecewise-defined Functions ................................................................. 136
3.5 Graphing Techniques: Transformations ................................................................................... 142
3.6 Mathematical Models: Building Functions ............................................................................... 150
Chapter Review................................................................................................................................... 153
Chapter Test ........................................................................................................................................ 159
Cumulative Review ............................................................................................................................. 163
© 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently
exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
, Chapter 4 Linear and Quadratic Functions
4.1 Linear Functions and Their Properties . . . 165
...... ............................. . .. ......................................... ....
4.2 Building Linear Functions from Data ....................................................................................... 170
4.3 Linear and Quadratic Functions . . . . . . 172
.. .... .................... ....................... ...... ............................... ....
4.4 Properties of Quadratic Functions ............................................................................................ 182
4.5 Inequalities Involving Quadratic Functions . . 185
...................................... ................................... ...
Chapter Review . .. . . .
........................... ........ . . 195
. .... .... ........ .................................................. .....................
Chapter Test 201
........................................................................................................................................
Cumulative Review ................................................................................................ :............................ 203
Chapter 5 Polynomial and Rational Functions
5.1 Polynomial Functions and Models . . :.................................................................. 205
.... ............. ......
5.2 Properties of Rational Functions . . . . . .
........... ........ ..... .......... . . ..
....... 215
.... ..... ...................... . .. . ... ......
5.3 The Graph of a Rational Function . . . . . .. .
..... .. .. .... ..... ................ 219
. . . ... ............................................
5.4 Polynomial and Rational Inequalities . . . . . . 242
... . . ....................... .... ...... ....... . ................ . .... .... ..........
5.5 The Real Zeros of a Polynomial Function . . . .
.................. ...... ........ . 249
............. . . . ....... ....... ........ .....
5.6 Complex Zeros; Fundamental Theorem of Algebra . . . . 270
......... ........ ...... ...................... . . . . ........ ....
Chapter Review . .
... . . : ................................................................... 273
............. ............................... . .. ... ... ...
Chapter Test . . . .. . .
........ .... ... ........ . . . 289
. .... .................. .............................................. .................... ....... .......
Cumulative Review . . . . . . . . 293
................... ......... ........... . ... ............................ .................. ....... .. .... ...............
Chapter 6 Exponential and Logarithmic Functions
6.1 Composite Functions ................................................................................................................ 296
6.2 One-to-One Functions; Inverse Functions . . . .. . . .
........... ...... .... . . 303
..... . ..... .................................... ..
6.3 Exponential Functions . . . . . .
.. .......... ............ . ........ ............... ....... . . 313
................... ................ ....... .....
6.4 Logarithmic Functions .............................................................................................................. 321
6.5 Properties of Logarithms .......................................................................................................... 329
6.6 Logarithmic and Exponential Equations . ... . .
................................ ............. ......... 333 .......... .... .........
6.7 Compound Interest . .
..... . . . . . . .
..... . .......... ...... ............ ............................. ... ....... 341 .. .... .......................
6.8 Exponential Growth and Decay Models; Newton's Law; Logistic Growth
and Decay Models . . . . . . 345
............. .................. . ... .... .................................. . .............. ...... ............. ...
6.9 Building Exponential, Logarithmic, and Logistic Models from Data....................................... 348
Chapter Review . .
.............................................................. .... . . .. 351
....................... . .......... ..... ....... . ............
Chapter Test . . . . 361
... .......................................... ..................... ............ .. . ............ . ... . .. ................................
Cumulative Review . . . . .
............. ............. .. ..... ...... . .. .
.................... ...... . . .... 364
.................. .. ................... . .....
Chapter 7 Trigonometric Functions
7.1 Angles and Their Measure . . . . . . .
.. ........ ......... ........................................... 367 . ... .. .............. ..... ..........
7.2 Right Triangle Trigonometry . . . . . 371
......... ......... ........ ........... ..... ..................... ................................
7.3 Computing the Values of Trigonometric Functions of Acute Angles . . . 378 ............. ...... ......... .......
7.4 Trigonometric Functions of General Angles ............................................................................ 384
7.5 Unit Circle Approach: Properties of the Trigonometric Functions . . . 391 .. . ............ .........................
7.6 Graphs of the Sine and Cosine Functions ... . ............. . ..
. . ........ . . .. . 394
..................... ... . . . ........... .. . .. ...
7.7 Graphs of the Tangent, Cotangent, Cosecant, and Secant Functions.. . .. . 404 .......... ............ . . ........ ..
7.8 Phase Shift; Sinusoidal Curve Fitting . . . . . 407
............................. .. . . . ............. ...... .................... .........
Chapter Review . . . . 413
......................... ................... .. . ........................... ........................................ ...... .......
Chapter Test . . . . . . . . . . . 421
. ...... ........................ .................. ........ ...... ......................... . . ............ . . ........ ...... .. . ....
Cumulative Review . . . . .. . . . .. . . . .
............... ........... .............. .... ................ .... . . . 425
.... .. .......... .. .... .... . ...... .. ... .....
© 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently
exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
, Chapter 8 Analytic Trigonometry
8.1 The Inverse Sine, Cosine, and Tangent Functions . .
. ......... . ...... .......... ............... ................. . . . . . 428..
8.2 The Inverse Trigonometric Functions (continued) . ..................... . . ....... . . . . . . .. . .. . . . . . . ................... 433
8.3 Trigonometric Identities .......... . . . . . .......................................................... . . . . ....... .. . . . .... . . . . . .. . ... . .. 439
8.4 Sum and Difference Formulas ........................ ........................................ . . ................................ 445
8.5 Double-Angle and Half-Angle Formulas .
......... . . . ....... ............... . . . . . . . . ..... . ... . ............................. 455
8.6 Product-to-Sum and Sum-to-Product Formulas ...... ........................ . .
. . . . . . ........... .................... . .. 466
8.7 Trigonometric Equations I . . . ............ ......................................................... ................................ 469
8.8 Trigonometric Equations 11. ................... .................................. ................................................ . 476
Chapter Review ................................................................ . ....................................... .................. ......... 481
Chapter Test ........................................................................................................................................ 494
Cumulative Review .
.................................................................... ...... .......... . ..... . .. ............ .... . ..... ...... . . . 499
Chapter 9 Applications of Trigonometric Functions
9.1 Applications Involving Right Triangles . ................. ......... . . . ............... . . . . . .... ......... . . .. . .. . . . ........ . . . 503
9.2 The Law of Sines . . . . . . .
...... . . . . .............. ............ ................... ..... ... .. .. ...... ... . .............................. . .. . 506
9.3 The Law of Cosines . . .. . . . ... ....... . . . ......... . . . . . . . .
. ... . . . . . . . . . . . . . .... . . .. . . . ................. . . . . ..... . . ......... ........ .. . . 513
9.4 Area of a Triangle .. . .. .
. . . ............... . .. . ... . . ....... . . . ... . .. . .
. . ....... . . ..... ......... . .. . . . ... . . . . .... . ................ .. .. . . . 517
9.5 Simple Harmonic Motion; Damped Motion; Combining Waves ... .. . . . ..
.... . . . . . . .. . . . ... . .... . . . . .. . .. . . . 521
Chapter Review . . . . . . ..
....... ...... ..... . ........... .. . ..... . .. .... ........ . . ... . . . . ... ..... . . . . ....... .. . .
. ..... . . . .. ... . . . ...... . . . . . . ...... . .. 524
Chapter Test .. ..
. .... . .. . . . .
.... .
........ . . . .... . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . ... . . . . . . . . . . . . . . . . . . ... . . . . . . . . . . . .. 529
Cumulative Review . . .
.. .......... . . . . ...... . . ...... ............... ............. ............ . . ...... . . . . . .
..... . . . . .
. ... . ... . .... . . . . . .... . .. ... 532
Chapter 10 Polar Coordinates; Vectors
10.1 Polar Coordinates . .
................................. . . ..... ......... ............... . . . . .............. . .. . . ....... . . .... . .............. . 535
10.2 Polar Equations and Graphs .
.................................................. ................................................... 538
10.3 The Complex Plane; DeMoivre's Theorem ......................... ............ . . ....................................... 550
10.4 Vectors ................. .............................. ..................... . . ............. . . ...... ............................... . . .......... 555
10.5 The Dot Product. ......... . . ............................... ............................................................. ................ 558
Chapter Review . .
........ . ......... ...... ........ . . .......................................................... ............................... ..... . 562
Chapter Test . ... .............................................................................................. . ....... . . ...... .............. ....... . 569
Cumulative Review .................................... . . . ........................................... . . .................. . . ............. ........ 572
Chapter 1 1 Analytic Geometry
11.2 The Parabola .
............................................................... .. . ............ . ........... . . . ................... ............ 574
11.3 The Ellipse . .... .............. .............. ................. ...... . . . ............ . . . ...... . . ........... . .
. . . . ......... 581 . .
. . . . ...... . . ......
11.4 The Hyperbola . . . .. . .
. .... ....... . . ..... ...... . . . .. ...... . . . ..... . . . .. .. . .... . . . . . .......... . . . .. .. . . . . .... . ... .. .
. . . 589
. . . . ... . . . ........ .
11.5 Rotation of Axes; General Form of a Conic . . ... . . . . . . . .... . ... .
. . .. ......... ... . ..... . . . .. . . . . . 597
......... . . . . . . .. . .. . ...
11.6 Polar Equations of Conics .. . . . . . . . . .. . . ...... . . . . ..... . . . . . ....... . .
. . . . . . . ........ . . . . ........... . . ...... . . . . . . ... .... . .. . . ..... . 603
11.7 Plane Curves and Parametric Equations . . ..... . . . . . .. . . .... . ....... ... . . ...... .. . ..
. . ..
... . ..... ..... . . . .......... 606 . . . .....
Chapter Review . .. . . ..
................ . . ............ . . ... . . . . . ..... . .. . . . . ...... . . . . .. . . . . ... . ............ . .. . . . . . . .... . . .. . .. . . . . 612 . . . . ........... .
Chapter Test .. . . . .
. .. . . ..
... . . . . . . .
. . . . .. . .. . . . . ... . . .. . . . .. . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .... . . . .. . .. . . . . . . . . .. . . . . . . . . 622
. . . .. . . . . . . . . . . . . . . .. .
Cumulative Review . .
................ .... . . .. . . . ............. . . . . . . ..... . ....... . ... . ............. . ........... . . . . . ..
......... 626 .. . ...... . .... . ...
© 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently
exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
, Chapter 12 Systems of Equations and Inequalities
12.1 Systems of Linear Equations: Substitution and Elimination ... . ...... . . . ...... .. . . . . . . .... . ..... . ........ 628 .......
12.2 Systems of Linear Equations: Matrices . . ........ ... . . . ... . . . . . .
. ... . .. . . . . ... . .. ... . . ... . . ...... . . . . .... . ....
. . 637 . . . . . . . .. . .
12.3 Systems of Linear Equations: Determinants . . . . . . .... . . . . . .. .. . . . . .. .. . . . . . . . .. . ... . . . . . . . ... 649
. . . . . . . . . . . . . . . . . . . . . . . . . ..
12.4 Matrix Algebra. . . . . .
............... ...... . . . . . ....... . . . . . . . . . . . . . . . . . . .. . . . . .. . .. ........... . ......... ........ . .
. . . . . . .. . . .. . 656 . . . . .. .. . .
12.5 Partial Fraction Decomposition . . .. . .. . . . .... . . . . . . . . . . . . . ... . . . . . . . . . .. .. . . . .. . . .. ... . . . . .. . .. . . . . . .... . .. .. .. . .. ... 663
... . . . . . . .
12.6 Systems of Nonlinear Equations ..... . . . . . . ... . . . ... . . . . ............... . . .. . . . . .
.. . . ... . .. . ............ ..... .. . . ..... . . . . . . . . . . 669
12.7 Systems of Inequalities .
........ . ................................. ............... . .. . . ......... . . . . . . . . . . . .. 682
. . ...... . ... . .. . . .......
12.8 Linear Programming . ... . . . ..... . . . . . . .. . . . . ... . . . . . . . .. . . . ... . . . .. . .... . . . . . . .. . . . . . . ......... . . . . . . . .
. . . . .. . . .. . . . . . .. 689
. ..... . .. . . . .
Chapter Review .. ..... . ............ .. .
....................... . ..... ................ . . .. . .. . . .. . . .................................... . . .... . ...... . 695
Chapter Test ........................................................................................................................................ 709
Cumulative Review ..... ............ ........... . . ..... ....................... ............ ............................ ........................... 717
Chapter 13 Sequences; Induction; the Binomial Theorem
13.1 Sequences ............. .. ............... . . . . .................... .......................................... . .
. . . . . ... . . ..... . . .....720 . ........
13.2 Arithmetic Sequences ........ . . . . ........... . . . . ........................ . .. . ....................................... . . . . ............. 723
13.3 Geometric Sequences; Geometric Series ................................................... . . . . . 726
..........................
13.4 Mathematical Induction ..... . ..
. ... . ................... . ......... . ... . . ... . .. . .. . . .... ..................... . . .. . .
.. 731
.... . . . .. . . . .. . .
13.5 The Binomial Theorem ........................ .. .
.. . . .. ................. . . ......... ................ . ...... ... . . 734
... . ........... . ....
Chapter Review .
...... . . .......... . ............. .
.. . ....... . ... . . .
.. . . . . ........ . ... .......... . . .
. .... ................. .. .... . ................. . . . 737
Chapter Test . ..... . ... . . . ....... .. . . .... . . .
... . . . ....... ........ . . . .
. ... . . ....... . .... . . . . . . . . . . . ........ . .. . . . 741
. . . . . . . . . ... . .. . . . . ...... .... . ... ....
Cumulative Review . . . .. . . .. . . ....... . .. . . ..... . . . . . . . . .
....... . . . . . .. . ... .... . . ....... .... ... . ...... . ... . . . . . ..... . . ... . ... . . .. 744 . . . . . . . . . . .... . .
Chapter 14 Counting and Probability
14.1 Sets and Counting .. . . .... . .. . .......... . ... . ... . . . . .... . .. . . . . . . . . ... . . .. . . . . ...... . .
...... ..... . . .... . . . . . ... . . 745
. . . . . .. . . . . . . . . . . . . . . .
14.2 Permutations and Combinations . . ........ ......... . . . . .. . .. . . .
. . .. . . . . . ....... ....... . .. . . ... . . . . . . .
.. . 746
. .......... . . . ... . . . . . .
14.3 Probability . . . . . . . .. .
. . . .
. . .. . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . .. ..
. . . . 747
... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Chapter Review . . . ... . . . .. . . .
. . . . ... ...... . . .... .. .. .. ... . . . . . . .... . . . ..... . . . . . . ..... . . .. ... . . . . . . . . . . . .. . . .... . . . . . . . . . . ...... . . . . . . . 750
. . .. . . . ....
Chapter Test .. .... . . . .
. . . ......... . . ........... . . .... . .. . . ................ ............... . . . ............ . . . ..
..... . . . ........... . . .. . 751
. .. . . . . . . . . . . . .
Cumulative Review . .. . ...... .............. ............. . . . . .. . . . . . . ............ . ........ .......... . ...... ...... . . .. . .... . . .. . . .. 753
. . . . . .... . . . . . .
Appendix Graphing Utilities
Section I The Viewing Rectangle . .. . . . . . ... .................................... . . . .... ... . ......... . . . .. . . ... ........... . .. . .. . .. . . . .. 755
Section 2 Using a Graphing Utility to Graph Equations .. . ........................... . . . .. . . ....... . . ........... ......... . . 755
Section 3 Using a Graphing Utility to Locate Intercepts and Check for Symmetry .
... . .... ............. . ... 759
Section 5 Square Screens ................. . .. . .......................... . . . ............... . . . ........ ....................................... 760
© 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently
exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
, Preface
This solution manual accompanies Algebra & Trigonometry, 8e by Michael Sullivan.
The Instructor Solutions Manual (ISM) contains detailed solutions to all exercises in the
text and the chapter projects (both in the text and those posted on the internet). The
Student Solutions Manual (SSM) contains detailed solutions to all odd exercises in the
text and all solutions to chapter tests. In both manuals, some TI-84 Plus graphing
calculator screenshots have been included to demonstrate how technology can be used to
solve problems and check solutions. A concerted effort has been made to make this
manual as user-friendly and error free as possible. Please feel free to send us any
suggestions or corrections.
We would like to extend our thanks to Dawn Murrin, Christine Whitlock and Bob
Walters from Prentice Hall for all their help with manuscript pages and logistics. Thanks
for everything!
We would also like to thank our wives (Angie and Karen) and our children (Annie, Ben,
Ethan,Logan, Payton, and Shawn) for their patient support and for enduring many late
evemngs.
Kevin Bodden and Randy Gallaher
Department of Mathematics
Lewis and Clark Community College
5800 Godfrey Road
Godfrey,IL 62035
,
, STUDENT SOLUTIONS
MANUAL
KEVIN BODDEN RANDY GALLAHER
LEWIS AND CLARK COMMUNITY COLLEGE
8
��e6ra & �ri8onometry
MICHAEL SULLIVAN
PEARSON
----
Prentice
Hall
Upper Saddle River, NJ 07458
,Vice President and Editorial Director, Mathematics: Christine Hoag
Senior Editor: Adam Jaworski
Editorial AssistantlPrint Supplements Editor: Christine Whitlock
Senior Managing Editor: Linda Behrens
Assistant Managing Editor: Lynn Savino Wendel
Production Editor: Ashley M. Booth
Supplement Cover Manager: Paul Gourhan
Supplement Cover Designer: Victoria Colotta
Manufacturing Buyer: Ilene Kahn
Associate Director of Operations Alexis Heydt-Long
PEARSON © 2008 Pearson Education, Inc.
----- Pearson Prentice Hall
Prentice Pearson Education, Inc.
Hall Upper Saddle River, NJ 07458
All rights reserved. No part of this book may be reproduced in any form or by any means, without permission in
writing from the publisher.
Pearson Prentice Hall™ is a trademark of Pearson Education, Inc.
The author and publisher of this book have used their best efforts in preparing this book. These efforts include the
development, research, and testing of the theories and programs to determine their effectiveness. The author and
publisher make no warranty of any kind, expressed or implied, with regard to these programs or the documentation
contained in this book. The author and publisher shall not be liable in any event for incidental or consequential
damages in connection with, or arising out of, the furnishing, performance, or use of these programs.
This work is protected by United States copyright laws and is provided solely for teaching courses and assessing
student learning. Dissemination or sale of any part of this work (including on the World Wide Web) will destroy
the integrity of the work and is not permitted. The work and materials from it should never be made available
except by instructors using the accompanying text in their classes. All recipients of this work are expected to
abide by these restrictions and to honor the intended pedagogical purposes and the needs of other instructors
who rely on these materials.
Printed in the United States of America
10 9 8 7 6 5 4 3 2
ISBN 13: 978-0-13-232124-2
ISBN 10: 0-13-232124-6
Pearson Education Ltd., London
Pearson Education Australia Pty. Ltd., Sydney
Pearson Education Singapore, Pte. Ltd.
Pearson Education North Asia Ltd., Hong Kong
Pearson Education Canada, Inc., Toronto
Pearson Educaci6n de Mexico, S.A. de C.V.
Pearson Education-Japan, Tokyo
Pearson Education Malaysia, Pte. Ltd.
, Table of Contents
Preface
Chapter R Review
R.l Real Numbers ............................................................................................................................... 1
R.2 Algebra Essentials . .... ... . . . .. . .
. . . . . .. .. . .
. . . . . . . . . . . . . . . .. .
. . . .. .
. . . . . 3
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
R.3 Geometry Essentials ..................................................................................................................... 6
R.4 Polynomials .................................................................................................................................. 8
R.5 Factoring Polynomials . .. . ..
. . .. . . .
. . . . . . . . . . . . . . .. . . . .
. . . . . . . . . . . . . .. II
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
R.6 Synthetic Division ...................................................................................................................... 14
R.7 Rational expressions .....................: ............................................................................................. 15
R.8 nth Roots; Rational Exponents ................................................................................................... 20
Chapter Review ..................................................................................................................................... 24
Chapter Test .......................................................................................................................................... 28
Chapter 1 Equations and Inequalities
1.1 Linear Equations ......................................................................................................................... 30
1.2 Quadratic Equations.................................................................................................................... 39
1.3 Complex Numbers; Quadratic Equations in the Complex Number System ............................... 48
1.4 Radical Equations; Equations Quadratic in Form; Factorable Equations ................................... 50
1.5 Solving Inequalities .................................................................................................................... 59
1.6 Equations and Inequalities Involving Absolute Value ................................................................ 65
1.7 Problem Solving: Interest, Mixture, Uniform Motion, and Constant Rate Job Applications ..... 69
Chapter Review ..................................................................................................................................... 73
Chapter Test .......................................................................................................................................... 80
Chapter 2 Graphs
2.1 The Distance and Midpoint Formulas......................................................................................... 82
2.2 Graphs of Equations in Two Variables; Intercepts; Symmetry .................................................. 87
2.3 Lines 93
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4 Circles ....................................................................................................................................... 101
2.5 Variation ................................................................................................................................... 106
Chapter Review ................................................................................................................................... 108
Chapter Test ........................................................................................................................................ 114
Cumulative Review ............................................................................................................................. 116
Chapter 3 Functions and Their Graphs
3.1 Functions................................................................................................................................... 118
3.2 The Graph of a Function ........................................................................................................... 125
3.3 Properties of Functions ............................................................................................................. 129
3.4 Library of Functions; Piecewise-defined Functions ................................................................. 136
3.5 Graphing Techniques: Transformations ................................................................................... 142
3.6 Mathematical Models: Building Functions ............................................................................... 150
Chapter Review................................................................................................................................... 153
Chapter Test ........................................................................................................................................ 159
Cumulative Review ............................................................................................................................. 163
© 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently
exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
, Chapter 4 Linear and Quadratic Functions
4.1 Linear Functions and Their Properties . . . 165
...... ............................. . .. ......................................... ....
4.2 Building Linear Functions from Data ....................................................................................... 170
4.3 Linear and Quadratic Functions . . . . . . 172
.. .... .................... ....................... ...... ............................... ....
4.4 Properties of Quadratic Functions ............................................................................................ 182
4.5 Inequalities Involving Quadratic Functions . . 185
...................................... ................................... ...
Chapter Review . .. . . .
........................... ........ . . 195
. .... .... ........ .................................................. .....................
Chapter Test 201
........................................................................................................................................
Cumulative Review ................................................................................................ :............................ 203
Chapter 5 Polynomial and Rational Functions
5.1 Polynomial Functions and Models . . :.................................................................. 205
.... ............. ......
5.2 Properties of Rational Functions . . . . . .
........... ........ ..... .......... . . ..
....... 215
.... ..... ...................... . .. . ... ......
5.3 The Graph of a Rational Function . . . . . .. .
..... .. .. .... ..... ................ 219
. . . ... ............................................
5.4 Polynomial and Rational Inequalities . . . . . . 242
... . . ....................... .... ...... ....... . ................ . .... .... ..........
5.5 The Real Zeros of a Polynomial Function . . . .
.................. ...... ........ . 249
............. . . . ....... ....... ........ .....
5.6 Complex Zeros; Fundamental Theorem of Algebra . . . . 270
......... ........ ...... ...................... . . . . ........ ....
Chapter Review . .
... . . : ................................................................... 273
............. ............................... . .. ... ... ...
Chapter Test . . . .. . .
........ .... ... ........ . . . 289
. .... .................. .............................................. .................... ....... .......
Cumulative Review . . . . . . . . 293
................... ......... ........... . ... ............................ .................. ....... .. .... ...............
Chapter 6 Exponential and Logarithmic Functions
6.1 Composite Functions ................................................................................................................ 296
6.2 One-to-One Functions; Inverse Functions . . . .. . . .
........... ...... .... . . 303
..... . ..... .................................... ..
6.3 Exponential Functions . . . . . .
.. .......... ............ . ........ ............... ....... . . 313
................... ................ ....... .....
6.4 Logarithmic Functions .............................................................................................................. 321
6.5 Properties of Logarithms .......................................................................................................... 329
6.6 Logarithmic and Exponential Equations . ... . .
................................ ............. ......... 333 .......... .... .........
6.7 Compound Interest . .
..... . . . . . . .
..... . .......... ...... ............ ............................. ... ....... 341 .. .... .......................
6.8 Exponential Growth and Decay Models; Newton's Law; Logistic Growth
and Decay Models . . . . . . 345
............. .................. . ... .... .................................. . .............. ...... ............. ...
6.9 Building Exponential, Logarithmic, and Logistic Models from Data....................................... 348
Chapter Review . .
.............................................................. .... . . .. 351
....................... . .......... ..... ....... . ............
Chapter Test . . . . 361
... .......................................... ..................... ............ .. . ............ . ... . .. ................................
Cumulative Review . . . . .
............. ............. .. ..... ...... . .. .
.................... ...... . . .... 364
.................. .. ................... . .....
Chapter 7 Trigonometric Functions
7.1 Angles and Their Measure . . . . . . .
.. ........ ......... ........................................... 367 . ... .. .............. ..... ..........
7.2 Right Triangle Trigonometry . . . . . 371
......... ......... ........ ........... ..... ..................... ................................
7.3 Computing the Values of Trigonometric Functions of Acute Angles . . . 378 ............. ...... ......... .......
7.4 Trigonometric Functions of General Angles ............................................................................ 384
7.5 Unit Circle Approach: Properties of the Trigonometric Functions . . . 391 .. . ............ .........................
7.6 Graphs of the Sine and Cosine Functions ... . ............. . ..
. . ........ . . .. . 394
..................... ... . . . ........... .. . .. ...
7.7 Graphs of the Tangent, Cotangent, Cosecant, and Secant Functions.. . .. . 404 .......... ............ . . ........ ..
7.8 Phase Shift; Sinusoidal Curve Fitting . . . . . 407
............................. .. . . . ............. ...... .................... .........
Chapter Review . . . . 413
......................... ................... .. . ........................... ........................................ ...... .......
Chapter Test . . . . . . . . . . . 421
. ...... ........................ .................. ........ ...... ......................... . . ............ . . ........ ...... .. . ....
Cumulative Review . . . . .. . . . .. . . . .
............... ........... .............. .... ................ .... . . . 425
.... .. .......... .. .... .... . ...... .. ... .....
© 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently
exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
, Chapter 8 Analytic Trigonometry
8.1 The Inverse Sine, Cosine, and Tangent Functions . .
. ......... . ...... .......... ............... ................. . . . . . 428..
8.2 The Inverse Trigonometric Functions (continued) . ..................... . . ....... . . . . . . .. . .. . . . . . . ................... 433
8.3 Trigonometric Identities .......... . . . . . .......................................................... . . . . ....... .. . . . .... . . . . . .. . ... . .. 439
8.4 Sum and Difference Formulas ........................ ........................................ . . ................................ 445
8.5 Double-Angle and Half-Angle Formulas .
......... . . . ....... ............... . . . . . . . . ..... . ... . ............................. 455
8.6 Product-to-Sum and Sum-to-Product Formulas ...... ........................ . .
. . . . . . ........... .................... . .. 466
8.7 Trigonometric Equations I . . . ............ ......................................................... ................................ 469
8.8 Trigonometric Equations 11. ................... .................................. ................................................ . 476
Chapter Review ................................................................ . ....................................... .................. ......... 481
Chapter Test ........................................................................................................................................ 494
Cumulative Review .
.................................................................... ...... .......... . ..... . .. ............ .... . ..... ...... . . . 499
Chapter 9 Applications of Trigonometric Functions
9.1 Applications Involving Right Triangles . ................. ......... . . . ............... . . . . . .... ......... . . .. . .. . . . ........ . . . 503
9.2 The Law of Sines . . . . . . .
...... . . . . .............. ............ ................... ..... ... .. .. ...... ... . .............................. . .. . 506
9.3 The Law of Cosines . . .. . . . ... ....... . . . ......... . . . . . . . .
. ... . . . . . . . . . . . . . .... . . .. . . . ................. . . . . ..... . . ......... ........ .. . . 513
9.4 Area of a Triangle .. . .. .
. . . ............... . .. . ... . . ....... . . . ... . .. . .
. . ....... . . ..... ......... . .. . . . ... . . . . .... . ................ .. .. . . . 517
9.5 Simple Harmonic Motion; Damped Motion; Combining Waves ... .. . . . ..
.... . . . . . . .. . . . ... . .... . . . . .. . .. . . . 521
Chapter Review . . . . . . ..
....... ...... ..... . ........... .. . ..... . .. .... ........ . . ... . . . . ... ..... . . . . ....... .. . .
. ..... . . . .. ... . . . ...... . . . . . . ...... . .. 524
Chapter Test .. ..
. .... . .. . . . .
.... .
........ . . . .... . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . ... . . . . . . . . . . . . . . . . . . ... . . . . . . . . . . . .. 529
Cumulative Review . . .
.. .......... . . . . ...... . . ...... ............... ............. ............ . . ...... . . . . . .
..... . . . . .
. ... . ... . .... . . . . . .... . .. ... 532
Chapter 10 Polar Coordinates; Vectors
10.1 Polar Coordinates . .
................................. . . ..... ......... ............... . . . . .............. . .. . . ....... . . .... . .............. . 535
10.2 Polar Equations and Graphs .
.................................................. ................................................... 538
10.3 The Complex Plane; DeMoivre's Theorem ......................... ............ . . ....................................... 550
10.4 Vectors ................. .............................. ..................... . . ............. . . ...... ............................... . . .......... 555
10.5 The Dot Product. ......... . . ............................... ............................................................. ................ 558
Chapter Review . .
........ . ......... ...... ........ . . .......................................................... ............................... ..... . 562
Chapter Test . ... .............................................................................................. . ....... . . ...... .............. ....... . 569
Cumulative Review .................................... . . . ........................................... . . .................. . . ............. ........ 572
Chapter 1 1 Analytic Geometry
11.2 The Parabola .
............................................................... .. . ............ . ........... . . . ................... ............ 574
11.3 The Ellipse . .... .............. .............. ................. ...... . . . ............ . . . ...... . . ........... . .
. . . . ......... 581 . .
. . . . ...... . . ......
11.4 The Hyperbola . . . .. . .
. .... ....... . . ..... ...... . . . .. ...... . . . ..... . . . .. .. . .... . . . . . .......... . . . .. .. . . . . .... . ... .. .
. . . 589
. . . . ... . . . ........ .
11.5 Rotation of Axes; General Form of a Conic . . ... . . . . . . . .... . ... .
. . .. ......... ... . ..... . . . .. . . . . . 597
......... . . . . . . .. . .. . ...
11.6 Polar Equations of Conics .. . . . . . . . . .. . . ...... . . . . ..... . . . . . ....... . .
. . . . . . . ........ . . . . ........... . . ...... . . . . . . ... .... . .. . . ..... . 603
11.7 Plane Curves and Parametric Equations . . ..... . . . . . .. . . .... . ....... ... . . ...... .. . ..
. . ..
... . ..... ..... . . . .......... 606 . . . .....
Chapter Review . .. . . ..
................ . . ............ . . ... . . . . . ..... . .. . . . . ...... . . . . .. . . . . ... . ............ . .. . . . . . . .... . . .. . .. . . . . 612 . . . . ........... .
Chapter Test .. . . . .
. .. . . ..
... . . . . . . .
. . . . .. . .. . . . . ... . . .. . . . .. . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .... . . . .. . .. . . . . . . . . .. . . . . . . . . 622
. . . .. . . . . . . . . . . . . . . .. .
Cumulative Review . .
................ .... . . .. . . . ............. . . . . . . ..... . ....... . ... . ............. . ........... . . . . . ..
......... 626 .. . ...... . .... . ...
© 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently
exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
, Chapter 12 Systems of Equations and Inequalities
12.1 Systems of Linear Equations: Substitution and Elimination ... . ...... . . . ...... .. . . . . . . .... . ..... . ........ 628 .......
12.2 Systems of Linear Equations: Matrices . . ........ ... . . . ... . . . . . .
. ... . .. . . . . ... . .. ... . . ... . . ...... . . . . .... . ....
. . 637 . . . . . . . .. . .
12.3 Systems of Linear Equations: Determinants . . . . . . .... . . . . . .. .. . . . . .. .. . . . . . . . .. . ... . . . . . . . ... 649
. . . . . . . . . . . . . . . . . . . . . . . . . ..
12.4 Matrix Algebra. . . . . .
............... ...... . . . . . ....... . . . . . . . . . . . . . . . . . . .. . . . . .. . .. ........... . ......... ........ . .
. . . . . . .. . . .. . 656 . . . . .. .. . .
12.5 Partial Fraction Decomposition . . .. . .. . . . .... . . . . . . . . . . . . . ... . . . . . . . . . .. .. . . . .. . . .. ... . . . . .. . .. . . . . . .... . .. .. .. . .. ... 663
... . . . . . . .
12.6 Systems of Nonlinear Equations ..... . . . . . . ... . . . ... . . . . ............... . . .. . . . . .
.. . . ... . .. . ............ ..... .. . . ..... . . . . . . . . . . 669
12.7 Systems of Inequalities .
........ . ................................. ............... . .. . . ......... . . . . . . . . . . . .. 682
. . ...... . ... . .. . . .......
12.8 Linear Programming . ... . . . ..... . . . . . . .. . . . . ... . . . . . . . .. . . . ... . . . .. . .... . . . . . . .. . . . . . . ......... . . . . . . . .
. . . . .. . . .. . . . . . .. 689
. ..... . .. . . . .
Chapter Review .. ..... . ............ .. .
....................... . ..... ................ . . .. . .. . . .. . . .................................... . . .... . ...... . 695
Chapter Test ........................................................................................................................................ 709
Cumulative Review ..... ............ ........... . . ..... ....................... ............ ............................ ........................... 717
Chapter 13 Sequences; Induction; the Binomial Theorem
13.1 Sequences ............. .. ............... . . . . .................... .......................................... . .
. . . . . ... . . ..... . . .....720 . ........
13.2 Arithmetic Sequences ........ . . . . ........... . . . . ........................ . .. . ....................................... . . . . ............. 723
13.3 Geometric Sequences; Geometric Series ................................................... . . . . . 726
..........................
13.4 Mathematical Induction ..... . ..
. ... . ................... . ......... . ... . . ... . .. . .. . . .... ..................... . . .. . .
.. 731
.... . . . .. . . . .. . .
13.5 The Binomial Theorem ........................ .. .
.. . . .. ................. . . ......... ................ . ...... ... . . 734
... . ........... . ....
Chapter Review .
...... . . .......... . ............. .
.. . ....... . ... . . .
.. . . . . ........ . ... .......... . . .
. .... ................. .. .... . ................. . . . 737
Chapter Test . ..... . ... . . . ....... .. . . .... . . .
... . . . ....... ........ . . . .
. ... . . ....... . .... . . . . . . . . . . . ........ . .. . . . 741
. . . . . . . . . ... . .. . . . . ...... .... . ... ....
Cumulative Review . . . .. . . .. . . ....... . .. . . ..... . . . . . . . . .
....... . . . . . .. . ... .... . . ....... .... ... . ...... . ... . . . . . ..... . . ... . ... . . .. 744 . . . . . . . . . . .... . .
Chapter 14 Counting and Probability
14.1 Sets and Counting .. . . .... . .. . .......... . ... . ... . . . . .... . .. . . . . . . . . ... . . .. . . . . ...... . .
...... ..... . . .... . . . . . ... . . 745
. . . . . .. . . . . . . . . . . . . . . .
14.2 Permutations and Combinations . . ........ ......... . . . . .. . .. . . .
. . .. . . . . . ....... ....... . .. . . ... . . . . . . .
.. . 746
. .......... . . . ... . . . . . .
14.3 Probability . . . . . . . .. .
. . . .
. . .. . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . .. ..
. . . . 747
... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Chapter Review . . . ... . . . .. . . .
. . . . ... ...... . . .... .. .. .. ... . . . . . . .... . . . ..... . . . . . . ..... . . .. ... . . . . . . . . . . . .. . . .... . . . . . . . . . . ...... . . . . . . . 750
. . .. . . . ....
Chapter Test .. .... . . . .
. . . ......... . . ........... . . .... . .. . . ................ ............... . . . ............ . . . ..
..... . . . ........... . . .. . 751
. .. . . . . . . . . . . . .
Cumulative Review . .. . ...... .............. ............. . . . . .. . . . . . . ............ . ........ .......... . ...... ...... . . .. . .... . . .. . . .. 753
. . . . . .... . . . . . .
Appendix Graphing Utilities
Section I The Viewing Rectangle . .. . . . . . ... .................................... . . . .... ... . ......... . . . .. . . ... ........... . .. . .. . .. . . . .. 755
Section 2 Using a Graphing Utility to Graph Equations .. . ........................... . . . .. . . ....... . . ........... ......... . . 755
Section 3 Using a Graphing Utility to Locate Intercepts and Check for Symmetry .
... . .... ............. . ... 759
Section 5 Square Screens ................. . .. . .......................... . . . ............... . . . ........ ....................................... 760
© 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently
exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
, Preface
This solution manual accompanies Algebra & Trigonometry, 8e by Michael Sullivan.
The Instructor Solutions Manual (ISM) contains detailed solutions to all exercises in the
text and the chapter projects (both in the text and those posted on the internet). The
Student Solutions Manual (SSM) contains detailed solutions to all odd exercises in the
text and all solutions to chapter tests. In both manuals, some TI-84 Plus graphing
calculator screenshots have been included to demonstrate how technology can be used to
solve problems and check solutions. A concerted effort has been made to make this
manual as user-friendly and error free as possible. Please feel free to send us any
suggestions or corrections.
We would like to extend our thanks to Dawn Murrin, Christine Whitlock and Bob
Walters from Prentice Hall for all their help with manuscript pages and logistics. Thanks
for everything!
We would also like to thank our wives (Angie and Karen) and our children (Annie, Ben,
Ethan,Logan, Payton, and Shawn) for their patient support and for enduring many late
evemngs.
Kevin Bodden and Randy Gallaher
Department of Mathematics
Lewis and Clark Community College
5800 Godfrey Road
Godfrey,IL 62035
,
