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AMAZINGTIPS
Copyright 2021 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
, REFERENCE page 1
ALGEBRA GEOMETRY
Arithmetic Operations Geometric Formulas
Cut here and keep for reference
a c ad 1 bc
asb 1 cd − ab 1 ac 1 − Formulas for area A, circumference C, and volume V:
b d bd
a Triangle Circle Sector of Circle
a1c a c b a d ad A − 12 bh A − r 2 A − 12 r 2
− 1 − 3 −
b b b c b c bc
d − 12 ab sin
C − 2 r s − r
s
in radiansd
Exponents and Radicals
a
xm h r s
x m x n − x m1n − x m2n
xn ¨ r
¨
1 b
sx mdn − x m n x2n − n r
x
sxydn − x n y n SD x
y
n
−
xn
yn
Sphere Cylinder Cone
x − (s x)
m
x 1yn − s
n
x x myn − s
n m n
V− 4 3
V − r h 2
V − 13 r 2h
3 r
Î sx n
x A − 4 r 2 A − rsr 2 1 h 2
s xy − s x s y
n n n n − n
y sy
U
Factoring Special Polynomials
U K
x 2 2 y 2 − sx 1 ydsx 2 yd K
x 3 1 y 3 − sx 1 ydsx 2 2 xy 1 y 2d r
x 3 2 y 3 − sx 2 ydsx 2 1 xy 1 y 2d
Binomial Theorem Distance and Midpoint Formulas
sx 1 yd2 − x 2 1 2xy 1 y 2 sx 2 yd2 − x 2 2 2xy 1 y 2 Distance between P1sx1, y1d and P2sx 2, y2d:
sx 1 yd3 − x 3 1 3x 2 y 1 3xy 2 1 y 3
d − ssx 2 2 x1d2 1 s y2 2 y1d2
sx 2 yd3 − x 3 2 3x 2 y 1 3xy 2 2 y 3
sx 1 ydn − x n 1 nx n21y 1
nsn 2 1d n22 2
2
x y
Midpoint of P1 P2: S x1 1 x 2 y1 1 y2
, D
SD
2 2
n n2k k …
1 … 1 x y 1 1 nxy n21 1 y n
k
where SD n
k
−
nsn 2 1d … sn 2 k 1 1d
1?2?3?…?k
Lines
Slope of line through P1sx1, y1d and P2sx 2, y2d:
Quadratic Formula m−
y2 2 y1
2b 6 sb 2 2 4ac x 2 2 x1
If ax 2 1 bx 1 c − 0, then x − .
2a
Point-slope equation of line through P1sx1, y1d with slope m:
Inequalities and Absolute Value
y 2 y1 − msx 2 x1d
If a , b and b , c, then a , c.
If a , b, then a 1 c , b 1 c. Slope-intercept equation of line with slope m and y-intercept b:
If a , b and c . 0, then ca , cb.
y − mx 1 b
If a , b and c , 0, then ca . cb.
If a . 0, then Circles
| |
x − a means x − a or x − 2a
Equation of the circle with center sh, kd and radius r:
| |
x , a means 2a , x , a
| x | . a means x . a or x , 2a sx 2 hd2 1 s y 2 kd2 − r 2
AMAZINGTIPS
Copyright 2021 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
, REFERENCE page 2
TRIGONOMETRY
Angle Measurement Fundamental Identities
radians − 1808 1 1
csc
− sec
−
s sin
cos
180° r
18 − rad 1 rad − sin
cos
180 ¨
tan
− cot
−
r cos
sin
s − r
s
in radiansd 1
cot
− sin2
1 cos2
− 1
tan
Right Angle Trigonometry 1 1 tan2
− sec 2
1 1 cot 2
− csc 2
opp hyp
sin
− csc
− sins2
d − 2sin
coss2
d − cos
hyp opp
S D
hyp
adj hyp opp
cos
− sec
− tans2
d − 2tan
sin 2
− cos
hyp adj ¨ 2
S D S D
adj
opp adj
tan
− cot
− cos 2
− sin
tan 2
− cot
adj opp 2 2
Trigonometric Functions
The Law of Sines B
y r y
sin
− csc
− sin A sin B sin C
r y − −
a
(x, y) a b c
x r r
cos
− sec
− C
r x c
y x ¨
tan
− cot
−
x
The Law of Cosines
x y b
a 2 − b 2 1 c 2 2 2bc cos A
Graphs of Trigonometric Functions b 2 − a 2 1 c 2 2 2ac cos B
y y y y=tan x c 2 − a 2 1 b 2 2 2ab cos C A
y=sin x y=cos x
1 1
π 2π Addition and Subtraction Formulas
2π
x π 2π x π x sinsx 1 yd − sin x cos y 1 cos x sin y
_1 _1 sinsx 2 yd − sin x cos y 2 cos x sin y
cossx 1 yd − cos x cos y 2 sin x sin y
y y=csc x y y=sec x y y=cot x cossx 2 yd − cos x cos y 1 sin x sin y
tan x 1 tan y
1 1 tansx 1 yd −
1 2 tan x tan y
π 2π x π 2π x π 2π x tan x 2 tan y
tansx 2 yd −
_1 _1 1 1 tan x tan y
Double-Angle Formulas
sin 2x − 2 sin x cos x
Trigonometric Functions of Important Angles
cos 2x − cos 2x 2 sin 2x − 2 cos 2x 2 1 − 1 2 2 sin 2x
radians sin
cos
tan
2 tan x
08 0 0 1 0 tan 2x −
1 2 tan2x
308 y6 1y2 s3y2 s3y3
458 y4 s2y2 s2y2 1 Half-Angle Formulas
608 y3 s3y2 1y2 s3
1 2 cos 2x 1 1 cos 2x
908 y2 1 0 — sin 2x − cos 2x −
2 2
AMAZINGTIPS
Copyright 2021 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
,Study Smarter.
Ever wonder if you studied enough? WebAssign from
Cengage can help.
WebAssign is an online learning platform for your math,
statistics, physical sciences and engineering courses.
It helps you practice, focus your study time and absorb
what you learn. When class comes—you’re way
more confident.
With WebAssign you will:
Get instant feedback Know how well you
and grading understand concepts
Watch videos and tutorials Perform better on
when you’re stuck in-class assignments
Ask your instructor today how you can get
access to WebAssign!
cengage.com/webassign
AMAZINGTIPS
Copyright 2021 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
, REFERENCE page 1
ALGEBRA GEOMETRY
Arithmetic Operations Geometric Formulas
Cut here and keep for reference
a c ad 1 bc
asb 1 cd − ab 1 ac 1 − Formulas for area A, circumference C, and volume V:
b d bd
a Triangle Circle Sector of Circle
a1c a c b a d ad A − 12 bh A − r 2 A − 12 r 2
− 1 − 3 −
b b b c b c bc
d − 12 ab sin
C − 2 r s − r
s
in radiansd
Exponents and Radicals
a
xm h r s
x m x n − x m1n − x m2n
xn ¨ r
¨
1 b
sx mdn − x m n x2n − n r
x
sxydn − x n y n SD x
y
n
−
xn
yn
Sphere Cylinder Cone
x − (s x)
m
x 1yn − s
n
x x myn − s
n m n
V− 4 3
V − r h 2
V − 13 r 2h
3 r
Î sx n
x A − 4 r 2 A − rsr 2 1 h 2
s xy − s x s y
n n n n − n
y sy
U
Factoring Special Polynomials
U K
x 2 2 y 2 − sx 1 ydsx 2 yd K
x 3 1 y 3 − sx 1 ydsx 2 2 xy 1 y 2d r
x 3 2 y 3 − sx 2 ydsx 2 1 xy 1 y 2d
Binomial Theorem Distance and Midpoint Formulas
sx 1 yd2 − x 2 1 2xy 1 y 2 sx 2 yd2 − x 2 2 2xy 1 y 2 Distance between P1sx1, y1d and P2sx 2, y2d:
sx 1 yd3 − x 3 1 3x 2 y 1 3xy 2 1 y 3
d − ssx 2 2 x1d2 1 s y2 2 y1d2
sx 2 yd3 − x 3 2 3x 2 y 1 3xy 2 2 y 3
sx 1 ydn − x n 1 nx n21y 1
nsn 2 1d n22 2
2
x y
Midpoint of P1 P2: S x1 1 x 2 y1 1 y2
, D
SD
2 2
n n2k k …
1 … 1 x y 1 1 nxy n21 1 y n
k
where SD n
k
−
nsn 2 1d … sn 2 k 1 1d
1?2?3?…?k
Lines
Slope of line through P1sx1, y1d and P2sx 2, y2d:
Quadratic Formula m−
y2 2 y1
2b 6 sb 2 2 4ac x 2 2 x1
If ax 2 1 bx 1 c − 0, then x − .
2a
Point-slope equation of line through P1sx1, y1d with slope m:
Inequalities and Absolute Value
y 2 y1 − msx 2 x1d
If a , b and b , c, then a , c.
If a , b, then a 1 c , b 1 c. Slope-intercept equation of line with slope m and y-intercept b:
If a , b and c . 0, then ca , cb.
y − mx 1 b
If a , b and c , 0, then ca . cb.
If a . 0, then Circles
| |
x − a means x − a or x − 2a
Equation of the circle with center sh, kd and radius r:
| |
x , a means 2a , x , a
| x | . a means x . a or x , 2a sx 2 hd2 1 s y 2 kd2 − r 2
AMAZINGTIPS
Copyright 2021 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
, REFERENCE page 2
TRIGONOMETRY
Angle Measurement Fundamental Identities
radians − 1808 1 1
csc
− sec
−
s sin
cos
180° r
18 − rad 1 rad − sin
cos
180 ¨
tan
− cot
−
r cos
sin
s − r
s
in radiansd 1
cot
− sin2
1 cos2
− 1
tan
Right Angle Trigonometry 1 1 tan2
− sec 2
1 1 cot 2
− csc 2
opp hyp
sin
− csc
− sins2
d − 2sin
coss2
d − cos
hyp opp
S D
hyp
adj hyp opp
cos
− sec
− tans2
d − 2tan
sin 2
− cos
hyp adj ¨ 2
S D S D
adj
opp adj
tan
− cot
− cos 2
− sin
tan 2
− cot
adj opp 2 2
Trigonometric Functions
The Law of Sines B
y r y
sin
− csc
− sin A sin B sin C
r y − −
a
(x, y) a b c
x r r
cos
− sec
− C
r x c
y x ¨
tan
− cot
−
x
The Law of Cosines
x y b
a 2 − b 2 1 c 2 2 2bc cos A
Graphs of Trigonometric Functions b 2 − a 2 1 c 2 2 2ac cos B
y y y y=tan x c 2 − a 2 1 b 2 2 2ab cos C A
y=sin x y=cos x
1 1
π 2π Addition and Subtraction Formulas
2π
x π 2π x π x sinsx 1 yd − sin x cos y 1 cos x sin y
_1 _1 sinsx 2 yd − sin x cos y 2 cos x sin y
cossx 1 yd − cos x cos y 2 sin x sin y
y y=csc x y y=sec x y y=cot x cossx 2 yd − cos x cos y 1 sin x sin y
tan x 1 tan y
1 1 tansx 1 yd −
1 2 tan x tan y
π 2π x π 2π x π 2π x tan x 2 tan y
tansx 2 yd −
_1 _1 1 1 tan x tan y
Double-Angle Formulas
sin 2x − 2 sin x cos x
Trigonometric Functions of Important Angles
cos 2x − cos 2x 2 sin 2x − 2 cos 2x 2 1 − 1 2 2 sin 2x
radians sin
cos
tan
2 tan x
08 0 0 1 0 tan 2x −
1 2 tan2x
308 y6 1y2 s3y2 s3y3
458 y4 s2y2 s2y2 1 Half-Angle Formulas
608 y3 s3y2 1y2 s3
1 2 cos 2x 1 1 cos 2x
908 y2 1 0 — sin 2x − cos 2x −
2 2
AMAZINGTIPS
Copyright 2021 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
