Birthday polynomial project 2000

GEOMETRIC FORMULAS

Formulas for area A, perimeter P, circumference C, volume V :

Rectangle Box

A � l„ V � l„ h

P � 2l � 2„

Triangle Pyramid

A � �1 2

� bh V � �1 3

�ha2

Circle Sphere

A � �r 2 V � �4 3

��r 3

C � 2�r A � 4�r 2

Cylinder Cone

V � �r 2h V � �1 3

��r 2h

HERON’S FORMULA

Area � �s 1�s��� a�2 1�s��� b�2 1�s��� c�2� where s � �

a � 2 b � c �

hh

r

r

r r

h

b a a

h

„

l

h

l „

EXPONENTS AND RADICALS

xmxn � xm�n � x x

m

n� � x m�n

1xm2n � xm n x�n � � x 1

n�

1xy2n � xnyn a�xy�b n

� � x y

n

n�

x1�n � �n x� xm�n � �n xm�� � Q�n x�Rm

�n xy� � �n x� �n y� �n xy��� � �

n x�

�n y� ��

�m �n��x� ��n �m��x� � �mnx�

SPECIAL PRODUCTS

1x � y22 � x 2 � 2xy � y 2 1x � y22 � x 2 � 2xy � y 2 1x � y23 � x 3 � 3x 2y � 3xy 2 � y 3 1x � y23 � x 3 � 3x 2y � 3xy 2 � y 3

FACTORING FORMULAS

x 2 � y 2 � 1x � y2 1x � y2 x 2 � 2xy � y 2 � 1x � y22 x 2 � 2xy � y 2 � 1x � y22 x 3 � y 3 � 1x � y2 1x 2 � xy � y 22 x 3 � y 3 � 1x � y2 1x 2 � xy � y 22

QUADRATIC FORMULA

If ax 2 � bx � c � 0, then

x �

INEQUALITIES AND ABSOLUTE VALUE

If a � b and b � c, then a � c.

If a � b, then a � c � b � c.

If a � b and c � 0, then ca � cb.

If a � b and c � 0, then ca � cb.

If a � 0, then

⏐x⏐ � a means x � a or x � �a.

⏐x⏐ � a means �a � x � a.

⏐x⏐ � a means x � a or x � �a.

�b �b�2��� 4�a�c� ��

2a

b

B

CA

ac

DISTANCE AND MIDPOINT FORMULAS

Distance between P11x1, y12 and P21x2, y2 2 : d � � 1x�2��� x�12�2��� 1�y2� �� y�12�2�

Midpoint of P1P2: a�x1 � 2

x2. �, �

y1 �

2

y2. �b

LINES

Slope of line through P11x1, y12 and P21x2, y2 2 Point-slope equation of line y � y1 � m 1x � x12 through P11x1, y12 with slope m Slope-intercept equation of y � mx � b line with slope m and y-intercept b

Two-intercept equation of line with x-intercept a and y-intercept b

LOGARITHMS

y � loga x means a y � x

loga a x � x a loga x � x

loga 1 � 0 loga a � 1

log x � log10 x ln x � loge x

loga xy � loga x � loga y loga a�xy�b � loga x � loga y loga x

b � b loga x logb x �

EXPONENTIAL AND LOGARITHMIC FUNCTIONS

0

1

y=a˛ 0<a<1

0

1

y=a˛ a>1

1

y=loga x a>1

0

y=loga x 0<a<1

10

y

x

y

x

y

x

y

x

loga x

log a b

� a x

� � � b y

� � 1

m � � x

y2

2

�

�

y

x 1

1 �

GRAPHS OF FUNCTIONS

Linear functions: f 1x2 � mx � b

Power functions: f 1x2 � xn

Root functions: f 1x2 � �n x�

Reciprocal functions: f 1x2 � 1/xn

Absolute value function Greatest integer function

Ï=“x ‘

1

1

x

y

Ï=|x |

x

y

Ï= 1≈

x

y

Ï= 1x

x

y

Ï=£œ∑x

x

y

Ï=œ∑x

x

y

Ï=x£

x

y

Ï=≈ x

y

Ï=mx+b

b

x

y

Ï=b

b

x

y

COMPLEX NUMBERS

For the complex number z � a � bi the conjugate is

the modulus is ⏐z⏐ � �a2 � b2���� the argument is , where tan � b/a

Polar form of a complex number

For z � a � bi, the polar form is

z � r 1cos � i sin 2 where r � ⏐z⏐ is the modulus of z and is the argument of z

De Moivre’s Theorem

z n � �r 1cos � i sin 2 n � rn 1cos n � i sin n 2 �n z� � �r 1cos � i sin 2 1�n

� r 1�n acos � � n 2k� � � i sin �

�

n 2k� �b

where k � 0, 1, 2, . . . , n � 1

ROTATION OF AXES

0

P(x, y) P(X, Y)

Y

X

ƒ x

y

Re

Im

bi

0

| z| a+bi

¨ a

z � a � bi

Angle-of-rotation formula for conic sections

To eliminate the xy-term in the equation

Ax2 � Bxy � Cy2 � Dx � Ey � F � 0

rotate the axis by the angle � that satisfies

cot 2� � � A �

B C

�

POLAR COORDINATES

x � r cos

y � r sin

r2 � x2 � y 2

tan � � y

x �

x

y

0

r

¨ x

y

P (x, y) P (r, ¨)

Rotation of axes formulas

x � X cos � � Y sin �

y � X sin � � Y cos �

CONIC SECTIONS

Circles

1x � h2 2 � 1y � k2 2 � r 2

Parabolas x 2 � 4py y 2 � 4px

Focus 10, p2 , directrix y � �p Focus 1p, 02 , directrix x � �p

y � a 1x � h22 � k, y � a 1x � h2 2 � k, a � 0, h � 0, k � 0 a � 0, h � 0, k � 0

Ellipses

� a

x2

2� � �b

y2

2� � 1 x2

b2 �� �

y2

a2 �� � 1

Foci 1 c, 02 , c2 � a2 � b2 Foci 10, c2 , c2 � a2 � b2

Hyperbolas

� a

x2

2� � �b

y2

2� � 1 � x2

b2 �� �

y2

a2 �� � 1

Foci 1 c, 02 , c2 � a2 � b2 Foci 10, c2 , c2 � a2 � b2

a

b

_a

_b

b

a

_b

_a

_c c

c

_c

x

y

x

y

a>b

b

a

_b

_a

c

_c

a>b

a

b

_a

_b

c_c x

y

x

y

0

y

x

(h, k)

0

y

x

(h, k)

y

x

p>0

p<0

y

x

p>0p<0

p

p

0

C(h, k)

r

x

y

This page intentionally left blank

This is an electronic version of the print textbook. Due to electronic rights restrictions, some third party content may be suppressed. Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. The publisher reserves the right to remove content from this title at any time if subsequent rights restrictions require it. For valuable information on pricing, previous editions, changes to current editions, and alternate formats, please visit www.cengage.com/highered to search by ISBN#, author, title, or keyword for materials in your areas of interest.

www.cengage.com/highered

T H I R D E D I T I O N

ALGEBRA AND TRIGONOMETRY

A B O U T T H E A U T H O R S

JAMES STEWART received his MS from Stanford University and his PhD

from the University of Toronto. He did

research at the University of London

and was influenced by the famous

mathematician George Polya at Stan-

ford University. Stewart is Professor

Emeritus at McMaster University and is

currently Professor of Mathematics at

the University of Toronto. His research

field is harmonic analysis and the con-

nections between mathematics and

music. James Stewart is the author of a

bestselling calculus textbook series

published by Brooks/Cole, Cengage

Learning, including Calculus, Calculus:

Early Transcendentals, and Calculus:

Concepts and Contexts; a series of pre-

calculus texts; and a series of high-

school mathematics textbooks.

LOTHAR REDLIN grew up on Van- couver Island, received a Bachelor of

Science degree from the University of

Victoria, and received a PhD from

McMaster University in 1978. He sub-

sequently did research and taught at

the University of Washington, the Uni-

versity of Waterloo, and California

State University, Long Beach. He is

currently Professor of Mathematics at

The Pennsylvania State University,

Abington Campus. His research field is

topology.

SALEEM WATSON received his Bachelor of Science degree from

Andrews University in Michigan. He

did graduate studies at Dalhousie

University and McMaster University,

where he received his PhD in 1978.

He subsequently did research at the

Mathematics Institute of the University

of Warsaw in Poland. He also taught at

The Pennsylvania State University. He

is currently Professor of Mathematics

at California State University, Long

Beach. His research field is functional

analysis.

Stewart, Redlin, and Watson have also published Precalculus: Mathematics for Calculus, College Algebra, Trigonometry, and

(with Phyllis Panman) College Algebra: Concepts and Contexts.

The cover photograph shows the Science Museum in the City of Arts and Sciences in Valencia, Spain. Built from 1991 to 1996, it was designed by Santiago Calatrava, a Spanish architect. Calatrava has always been very interested in how mathematics can help him realize the buildings he imagines. As a young student, he taught himself descriptive geometry from books in order to represent

three-dimensional objects in two dimensions. Trained as both an engineer and an architect, he wrote a doctoral thesis in 1981 entitled “On the Foldability of Space Frames,” which is filled with mathematics, especially geometric transformations. His strength as an engineer enables him to be daring in his architecture.

ABOUT THE COVER

ALGEBRA AND TRIGONOMETRY JAMES STEWART M C M A S T E R U N I V E R S I T Y A N D U N I V E R S I T Y O F TO R O N TO

LOTHAR REDLIN T H E P E N N S Y LVA N I A S TAT E U N I V E R S I T Y

SALEEM WATSON C A L I F O R N I A S TAT E U N I V E R S I T Y, LO N G B E A C H

Australia • Brazil • Japan • Korea • Mexico • Singapore • Spain • United Kingdom • United States

T H I R D E D I T I O N

Algebra and Trigonometry, Third Edition James Stewart, Lothar Redlin, Saleem Watson

Acquisitions Editor: Gary Whalen

Developmental Editor: Stacy Green

Assistant Editor: Cynthia Ashton

Editorial Assistant: Sabrina Black

Media Editor: Lynh Pham

Marketing Manager: Myriah Fitzgibbon

Marketing Assistant: Shannon Myers

Marketing Communications Manager: Darlene Macanan

Content Project Manager: Jennifer Risden

Design Director: Rob Hugel

Art Director: Vernon Boes

Print Buyer: Karen Hunt

Rights Acquisitions Specialist: Dean Dauphinais

Production Service: Martha Emry

Text Designer: Lisa Henry

Photo Researcher: Bill Smith Group

Copy Editor: Barbara Willette

Illustrator: Matrix Art Services, Precision Graphics

Cover Designer: Lisa Henry

Cover Image: © Jose Fuste Raga/ photolibrary.com

Compositor: Graphic World, Inc.

© 2012, 2007 Brooks/Cole, Cengage Learning

ALL RIGHTS RESERVED. No part of this work covered by the copyright herein may be reproduced, transmitted, stored, or used in any form or by any means graphic, electronic, or mechanical, including but not limited to photocopying, recording, scanning, digitizing, taping, Web distribution, information networks, or info