Precalculus with Limits: A Graphing Approach

1148 Pages·2008·60.06 MB·English

Triangle

Laws of Cosines:

Circular Ring

w � width of ring�
�p � average radius,
� 2�pw
Area � ��R2 � r2�

R
p
w
r

c 2 � a2 � b2 � 2ab cos �

Area � 1

2bh

a

b

h
c

θ

h � a sin �

FORMULAS FROM GEOMETRY

Right Triangle

Pythagorean Theorem:
Ellipse

Circumference � 2��
a2 � b2

2

Area � �ab
a
b

c2 � a2 � b2

a

b

c

Equilateral Triangle
Cone

(
area of base)

Volume � Ah

3

A �
h

A
Area � �3s2

4

h � �3s

2

s

s

h
s

Parallelogram
Right Circular Cone

Lateral Surface Area � �r�r2 � h2

Volume � �r2h

3
r

h
Area � bh

b

h

Sector of Circle
Sphere

Surface Area � 4�r 2

Volume � 4

3�r3
r

�� in radians�
s � r�

Area � �r2

2

r

s

θ

Circle

Circumference

Right Circular Cylinder

Volume
Lateral Surface Area � 2�rh
� �r2h

r

h
� 2�r
Area � �r 2
r

Trapezoid
Frustum of Right Circular Cone

Volume

Lateral Surface Area � �s�R � r�
h
R

r

s

� ��r 2 � rR � R2�h

3
Area � h

2�a � b�

a

h

b

a
b

h

333351_FES.qxp 1/23/07 10:33 AM Page 1

Definition of the Six Trigonometric Functions

Right triangle definitions, where

Circular function definitions, where
is any angle

Reciprocal Identities

Quotient Identities

Pythagorean Identities

Cofunction Identities

Even/Odd Identities

Sum and Difference Formulas

Double-Angle Formulas

Power-Reducing Formulas

Sum-to-Product Formulas

Product-to-Sum Formulas

cos u sin v � 1

2�sin�u � v� � sin�u � v��

sin u cos v � 1

2�sin�u � v� � sin�u � v��

cos u cos v � 1

2�cos�u � v� � cos�u � v��

sin u sin v � 1

2�cos�u � v� � cos�u � v��

cos u � cos v � �2 sin�
u � v

2 � sin�
u � v

2 �

cos u � cos v � 2 cos�
u � v

2 � cos�
u � v

2 �

sin u � sin v � 2 cos�
u � v

2 � sin�
u � v

2 �

sin u � sin v � 2 sin�
u � v

2 � cos�
u � v

2 �

tan2 u � 1 � cos 2u

1 � cos 2u

cos2 u � 1 � cos 2u

2

sin2 u � 1 � cos 2u

2

tan 2u �
2 tan u

1 � tan2 u

� 2 cos2 u � 1 � 1 � 2 sin2 u
cos 2u � cos2 u � sin2 u

sin 2u � 2 sin u cos u

x

y

(1, 0)

240°

90°

0°
360°

330°

30°
150°

210°
315°

45°
135°

225°
300°

60°
120°

270°

180°

(0, 1)

1
2

1
2
1
2

1
2

1
2

1
2
1
2

1
2

2

2
2

2

2

2
2

2
2

2
2

2

2

2
2

2

3

3
3

3

2

2
2

2
2

2
2

2

3

3
3

3

(

(
(

(

(

(
(

(
(

(
(

(

)

)
)

)

)

)
)

)
)

)
)

)

,

,
,

,

,

,
,

,

,

,
,

,
−

−

−
−

−

−

−
−

−

−

−
−

( 1, 0)
−

(0,
1)
−

0
π
π
2

6

4
3
3

2
3
3

4

4
4

6

6
6

2

3
4
5

3

5
7

5

7
11

2

π

π
π
π

π
π
π

π

π
π

π

π
π

π

tan�u ± v� � tan u ± tan v

1 � tan u tan v

cos�u ± v� � cos u cos v � sin u sin v
sin�u ± v� � sin u cos v ± cos u sin v

csc��u� � �csc u
tan��u� � �tan u
sec��u� � sec u
cos��u� � cos u
cot��u� � �cot u
sin��u� � �sin u

csc�
�
2 � u� � sec u
tan�
�
2 � u� � cot u

sec�
�
2 � u� � csc u
cos�
�
2 � u� � sin u

cot�
�
2 � u� � tan u
sin�
�
2 � u� � cos u

1 � cot2 u � csc2 u
1 � tan2 u � sec2 u
sin2 u � cos2 u � 1

cot u � cos u

sin u
tan u � sin u

cos u

cot u �
1

tan u
sec u �
1

cos u
csc u �
1

sin u

tan u �
1

cot u
cos u �
1

sec u
sin u �
1

csc u

cot � � x

y
tan � � y

x

sec � � r

x
cos � � x

r

csc � � r

y
sin � � y

r

�

cot � � adj.

opp.
tan � � opp.

adj.

sec � � hyp.

adj.
cos � � adj.

hyp.

csc � � hyp.

opp.
sin � � opp.

hyp.

0 < � < ��2

Opposite

Adjacent

Hypotenuse

θ

θ

x

y

r
x
y
=
+
2
2
( , )
x y

y

x

r

333351_FES.qxp 1/23/07 10:33 AM Page 2

Precalculus with Limits
A Graphing Approach

Ron Larson
The Pennsylvania State University
The Behrend College
Robert Hostetler
The Pennsylvania State University
The Behrend College
Bruce H. Edwards
University of Florida

With the assistance of David C. Falvo
The Pennsylvania State University
The Behrend College

F I F T H E D I T I O N

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

0618851526_AP_FM.qxp 8/10/09 2:10 PM Page i

Precalculus with Limits
A Graphing Approach

Ron Larson
Robert Hostetler
Bruce H. Edwards

Publisher: Richard Stratton

Sponsoring Editor: Cathy Cantin

Senior Marketing Manager: Jennifer Jones

Development Editor: Lisa Collette

Supervising Editor: Karen Carter

Senior Project Editor: Patty Bergin

Art and Design Manager: Gary Crespo

Cover Design Manager: Anne S. Katzeff

Photo Editor: Jennifer Meyer Dare

Composition Buyer: Chuck Dutton

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Editorial Assistant: Jill Clark

Composition and Art: Larson Texts, Inc.

Cover photograph
© Rosemary Calvert/Getty Images

© 2008 Brooks/Cole, Cengage Learning

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