Algebra and Trigonometry , Third Edition

1049 Pages·2011·31.53 MB·English

by  James Stewart

EXPONENTS AND RADICALS GEOMETRIC FORMULAS
xmxn(cid:2)xm(cid:3)n (cid:4)x(cid:4)m (cid:2)xm(cid:6)n Formulas for area A,perimeter P,circumference C,volume V:
xn
xmn (cid:2)xmn x(cid:6)n (cid:2)(cid:4)1(cid:4) Rectangle Box
1 2 xn
xyn (cid:2)xnyn (cid:4)x(cid:4) n (cid:2)(cid:4)x(cid:4)n A(cid:2)l„ V(cid:2)l„h
1 2 ayb yn P(cid:2)2l (cid:3)2„
x1(cid:4)n(cid:2)(cid:2)nx(cid:3) xm(cid:4)n(cid:2)(cid:2)nx(cid:3)(cid:3)m(cid:2) (cid:2)nx(cid:3) m
Q R
(cid:2)nx(cid:3)y (cid:2)(cid:2)nx(cid:3)(cid:2)ny(cid:3) (cid:5)n(cid:4)(cid:6)x(cid:4) (cid:2)(cid:4)(cid:2)n(cid:4)x(cid:3) „ h
(cid:2)m(cid:2)(cid:3)n(cid:3)x(cid:3)(cid:2)(cid:2)n(cid:2)(cid:3)m(cid:3)x(cid:3)(cid:2)(cid:2)mnx(cid:3) y (cid:2)n(cid:3)y
„
l l
SPECIAL PRODUCTS
Triangle Pyramid
1x (cid:3)y22(cid:2)x2 (cid:3)2xy (cid:3)y2 A(cid:2)(cid:4)1(cid:4)bh V(cid:2)(cid:4)1(cid:4)ha2
2 3
x (cid:6)y2(cid:2)x2 (cid:6)2xy (cid:3)y2
1 2
x (cid:3)y3(cid:2)x3 (cid:3)3x2y (cid:3)3xy2 (cid:3)y3
1 2
x (cid:6)y3(cid:2)x3 (cid:6)3x2y (cid:3)3xy2 (cid:6)y3 h
1 2
h
a
FACTORING FORMULAS b a
x2 (cid:6)y2(cid:2) x (cid:3)y x (cid:6)y
1 21 2 Circle Sphere
x2 (cid:3)2xy (cid:3)y2(cid:2)1x (cid:3)y22 A(cid:2)(cid:5)r2 V(cid:2)(cid:4)4(cid:4)(cid:5)r3
3
x2 (cid:6)2xy (cid:3)y2(cid:2) x (cid:6)y2 C(cid:2)2(cid:5)r A(cid:2)4(cid:5)r2
1 2
x3 (cid:3)y3(cid:2) x (cid:3)y x2 (cid:6)xy (cid:3)y2
1 21 2
x3 (cid:6)y3(cid:2) x (cid:6)y x2 (cid:3)xy (cid:3)y2 r r
1 21 2
QUADRATIC FORMULA
If ax2 (cid:3)bx (cid:3)c(cid:2)0,then Cylinder Cone
(cid:6)b(cid:9)(cid:2)b(cid:3)(cid:3)2(cid:6)(cid:3)(cid:3)4a(cid:3)c(cid:3) V(cid:2)(cid:5)r2h V(cid:2)(cid:4)1(cid:4)(cid:5)r2h
x(cid:2)(cid:4)(cid:4) 3
2a
r
INEQUALITIES AND ABSOLUTE VALUE
h h
If a(cid:7)band b(cid:7)c,then a(cid:7)c. r
If a(cid:7)b,then a(cid:3)c(cid:7)b(cid:3)c.
If a(cid:7)band c(cid:8)0,then ca(cid:7)cb.
If a(cid:7)band c(cid:7)0,then ca(cid:8)cb. HERON’S FORMULA
If a(cid:8)0,then
B
⏐x⏐(cid:2)a means x(cid:2)a or x(cid:2)(cid:6)a. Area(cid:2)(cid:2)(cid:3)s s(cid:3)(cid:3)(cid:6)(cid:3)a(cid:3)s(cid:3)(cid:6)(cid:3)(cid:3)b(cid:3)s(cid:3)(cid:6)(cid:3)(cid:3)c(cid:3)
1 21 21 2 c a
⏐x⏐(cid:7)a means (cid:6)a(cid:7)x(cid:7)a. a(cid:3)b(cid:3)c
where s(cid:2)(cid:4)(cid:4)
⏐x⏐(cid:8)a means x(cid:8)a or x(cid:7)(cid:6)a. 2 A C
b
DISTANCE AND MIDPOINT FORMULAS GRAPHS OF FUNCTIONS
Distancebetween P x ,y and P x ,y : Linear functions: f x (cid:2)mx(cid:3)b
11 1 12 21 2 22 1 2
d(cid:2)(cid:2)(cid:3)1x(cid:3)2(cid:6)(cid:3)(cid:3)x(cid:3)12(cid:3)2(cid:3)(cid:3)(cid:3)1(cid:3)y2(cid:3)(cid:6)(cid:3)y(cid:3)12(cid:3)2 y y
x (cid:3)x. y (cid:3)y.
Midpointof P1P2: (cid:4)1 (cid:4)2, (cid:4)1 (cid:4)2 b
a 2 2 b
b
LINES
x x
y (cid:6)y
Slope of linethrough m(cid:2)(cid:4)2 (cid:4)1 Ï=b Ï=mx+b
x (cid:6)x
P x ,y and P x ,y 2 1
11 1 12 21 2 22 Power functions: f x (cid:2)xn
1 2
Point-slope equationof line y(cid:6)y (cid:2)m x(cid:6)x
1 1 12 y y
through P x ,y with slopem
11 1 12
Slope-intercept equationof y(cid:2)mx(cid:3)b
line with slopemand y-intercept b
x
Two-intercept equationof line x y
(cid:4)(cid:4)(cid:3)(cid:4)(cid:4)(cid:2)1 x
with x-intercept aand y-intercept b a b
Ï=≈ Ï=x£
LOGARITHMS Root functions: f x (cid:2)(cid:2)nx(cid:7)
1 2
y(cid:2)log x means ay(cid:2)x y y
a
log ax(cid:2)x alogax(cid:2)x
a
log 1(cid:2)0 log a(cid:2)1
a a x x
logx(cid:2)log x lnx(cid:2)log x
10 e
log xy(cid:2)log x (cid:3)log y log (cid:4)x(cid:4) (cid:2)log x (cid:6)log y Ï=œx∑ Ï=£œx∑
a a a aayb a a
log xb(cid:2)blog x log x(cid:2) logax Reciprocal functions: f x (cid:2)1/xn
a a b log b 1 2
a
y y
EXPONENTIAL AND LOGARITHMIC FUNCTIONS
y y
x x
y=a˛ y=a˛
a>1 0<a<1
1 1
Ï= Ï=
1 1 x ≈
0 x 0 x Absolute value function Greatest integer function
y y
y y=loga x y y=loga x
a>1 0<a<1
1
1 x
0 x 0 x x
1 1
Ï=|x| Ï=“x‘
COMPLEX NUMBERS CONIC SECTIONS
y
For the complex number z(cid:2)a(cid:3)bi Circles
r
the conjugateis z(cid:2)a(cid:6)bi x (cid:6)h 2(cid:3) y (cid:6)k 2 (cid:2)r2
the modulusis ⏐z⏐(cid:2)(cid:2)a(cid:3)2(cid:3)(cid:3)(cid:3)b(cid:7)2 1 2 1 2 C(h, k)
the argumentis (cid:10),where tan (cid:10)(cid:2)b/a 0 x
Im
Parabolas
bi a+bi x2 (cid:2)4py y2 (cid:2)4px
|z|
y y
¨ p>0 p<0 p>0
0 a Re p
x p x
Polar form of a complex number p<0
For z(cid:2)a(cid:3)bi,the polar formis
Focus 0, p ,directrixy (cid:2)(cid:6)p Focus p, 0 ,directrixx (cid:2)(cid:6)p
z(cid:2)r cos(cid:10)(cid:3)isin(cid:10) 1 2 1 2
1 2
y y
where r(cid:2)⏐z⏐is the modulus of zand (cid:10)is the argument of z (h, k)
De Moivre’s Theorem
0 x
zn(cid:2)(cid:8)r cos(cid:10)(cid:3)isin(cid:10)(cid:9)n(cid:2)rn cosn(cid:10)(cid:3)isinn(cid:10) (h, k)
1 2 1 2
0 x
(cid:2)nz(cid:3)(cid:2)(cid:8)r cos(cid:10)(cid:3)isin(cid:10)(cid:9)1(cid:4)n
1 2 y (cid:2)a x (cid:6)h 2(cid:3)k, y (cid:2)a x (cid:6)h 2(cid:3)k,
(cid:2)r1(cid:4)n cos(cid:4)(cid:10)(cid:3)(cid:4)2k(cid:5)(cid:3)isin(cid:4)(cid:10)(cid:3)(cid:4)2k(cid:5) a (cid:7)0,1 h (cid:8)20, k (cid:8)0 a (cid:8)0,1 h (cid:8)20, k (cid:8)0
a n n b
where k(cid:2)0,1,2,...,n(cid:6)1 Ellipses
x2 y2 x2 y2
ROTATION OF AXES (cid:4)a(cid:4)2 (cid:3)(cid:4)b(cid:4)2 (cid:2)1 (cid:4)b(cid:4)2 (cid:3)(cid:4)a(cid:4)2 (cid:2)1
y y
y
Y P(x, y) Rotation of axes a a>b
P(X, Y) formulas b a>b c
x(cid:2)Xcos(cid:11)(cid:6)Ysin(cid:11)
X
y(cid:2)Xsin(cid:11)(cid:3)Ycos(cid:11) _a _c c a x _b b x
_c
_b
ƒ _a
0 x
Foci (cid:9)c, 0 ,c2 (cid:2)a2(cid:6)b2 Foci 0, (cid:9)c ,c2 (cid:2)a2(cid:6)b2
1 2 1 2
Angle-of-rotation formula for conic sections Hyperbolas
x2 y2 x2 y2
To eliminate the xy-term in the equation (cid:4)(cid:4)(cid:6)(cid:4)(cid:4) (cid:2)1 (cid:6)(cid:4)(cid:4) (cid:3)(cid:4)(cid:4)(cid:2)1
a2 b2 b2 a2
Ax2(cid:3)Bxy(cid:3)Cy2(cid:3)Dx(cid:3)Ey(cid:3)F(cid:2)0
y y
rotate the axis by the angle (cid:11)that satisfies a c
b
A(cid:6)C
cot2(cid:11)(cid:2)(cid:4)(cid:4) _c c
B
_a a x _b b x
POLAR COORDINATES _b
_c
_a
y x(cid:2)rcos(cid:10) Foci (cid:9)c, 0 ,c2 (cid:2)a2(cid:3)b2 Foci 0, (cid:9)c ,c2 (cid:2)a2(cid:3)b2
1 2 1 2
P(x, y)
P(r, ¨) y(cid:2)rsin(cid:10)
r y r2(cid:2)x2(cid:3)y2
y
¨ tan(cid:10)(cid:2)(cid:4)(cid:4)
x
0 x x
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THIRD EDITION
A
LGEBRA AND
T
RIGONOMETRY
ABOUT THE AUTHORS
JAMES STEWART received his MS LOTHAR REDLIN grew up on Van- SALEEM WATSON received his
from Stanford University and his PhD couver Island,received a Bachelor of Bachelor of Science degree from
from the University of Toronto.He did Science degree from the University of Andrews University in Michigan.He
research at the University of London Victoria,and received a PhD from did graduate studies at Dalhousie
and was influenced by the famous McMaster University in 1978.He sub- University and McMaster University,
mathematician George Polya at Stan- sequently did research and taught at where he received his PhD in 1978.
ford University.Stewart is Professor the University of Washington,the Uni- He subsequently did research at the
Emeritus at McMaster University and is versity of Waterloo,and California Mathematics Institute of the University
currently Professor of Mathematics at State University,Long Beach.He is of Warsaw in Poland.He also taught at
the University of Toronto.His research currently Professor of Mathematics at The Pennsylvania State University.He
field is harmonic analysis and the con- The Pennsylvania State University, is currently Professor of Mathematics
nections between mathematics and Abington Campus.His research field is at California State University,Long
music.James Stewart is the author of a topology. Beach.His research field is functional
bestselling calculus textbook series analysis.
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.
Stewart,Redlin,and Watson have also published Precalculus:Mathematics for Calculus,College Algebra,Trigonometry,and
(with Phyllis Panman) College Algebra:Concepts and Contexts.
ABOUT THE COVER
The cover photograph shows the Science Museum in the City of three-dimensional objects in two dimensions.Trained as both an
Arts and Sciences in Valencia,Spain.Built from 1991 to 1996,it was engineer and an architect,he wrote a doctoral thesis in 1981
designed by Santiago Calatrava,a Spanish architect.Calatrava has entitled “On the Foldability of Space Frames,”which is filled with
always been very interested in how mathematics can help him mathematics,especially geometric transformations.His strength
realize the buildings he imagines.As a young student,he taught as an engineer enables him to be daring in his architecture.
himself descriptive geometry from books in order to represent
THIRD EDITION
A
LGEBRA AND
T
RIGONOMETRY
J S
AMES TEWART
MCMASTER UNIVERSITY AND UNIVERSITY OF TORONTO
L R
OTHAR EDLIN
THE PENNSYLVANIA STATE UNIVERSITY
S W
ALEEM ATSON
CALIFORNIA STATE UNIVERSITY, LONG BEACH
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Algebra and Trigonometry,Third Edition © 2012, 2007Brooks/Cole, Cengage Learning
James Stewart, Lothar Redlin, Saleem Watson
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