SAT II Math Formula Reference
MATH LEVEL IIC
1.3ver
CHAPTER 1
INTRODUCTION TO FUNCTIONS
(f+g)(x)=f(x)+g(x)
(f·g)(x)=f(x)·g(x)
(f/g)(x)=f(x)/g(x)
(f
g)(x)=f(x)
g(x)=f(g(x))
CHAPTER 2
POLYNOMIAL FUNCTIONS
Linear Functions
Distance=
2
2
1
2
1
2
(x -x ) +(y -y )
Distance=
1
1
2
2
Ax
By
C
A
B
Tan
=
1
2
1
2
1
m
m
m m
（
mis the slope of l.）
2
4
2
b
b
ac
x
a
Sumof zeros (roots)=
b
a
Product of zeros (roots)=
c
a
CHAPTER 3
TRIGONOMETRIC FUNCTIONS
Graphs:
(
)
is the amplitude
is the period of the graph
C
is the phase shift
B
y
A f Bx C
A
f
B
sin
csc
1
cos
sec
1
tan
cot
1
sin
tan
cos
cos
cot
sin
Quadrant
I
II
III
IV
Function:
sin,csc
+
+
-
-
cos,sec
+
-
-
+
tan,cot
+
-
+
-
Arcs and Angles
2
1
2
s r
A
r
Special Angles
0
2
3
2
2
sine
0
1
0
-1
0
cosine
1
0
-1
0
1
tangent
0
und
0
und
0
cotangent
und
0
und
0
und
secant
1
und
-1
und
1
cosecant
und
1
und
-1
und
*und: means that the function is undefined because the definition of the function necessitates division by
zero.
30
6
or
45
4
or
60
3
or
sine
1
2
2
2
3
2
cosine
3
2
2
2
1
2
tangent
3
3
1
3
cotangent
3
1
3
2
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1000+ College Admission Essay Samples:
secant
2 3
3
2
2
cosecant
2
2
2 3
3
Formulas:
2
2
2
2
2
2
1.sin
cos
1
2.tan
1 sec
3.cot
1 csc
4.sin(
) sin
cos
cos
sin
5.sin(
) sin
cos
cos
sin
6.cos(
) cos
cos
sin
sin
7.cos(
) cos
cos
sin
sin
tan
tan
8.tan(
)
1 tan
tan
9.ta
x
x
x
x
x
x
A B
A
B
A
B
A B
A
B
A
B
A B
A
B
A
B
A B
A
B
A
B
A
B
A B
A
B
tan
tan
n(
)
1 tan
tan
A
B
A B
A
B
2
2
2
2
2
10.sin 2
2sin cos
11.cos 2
cos
sin
12.cos 2
2cos
1
13.cos 2
1 2sin
2 tan
14.tan 2
1 tan
1
1 cos
15.sin
2
2
1
1 cos
16.cos
2
2
1
1 cos
17.tan
2
1 cos
1 cos
18.
sin
sin
19.
1 cos
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
*The correct sign for Formulas 15 through 17is determined by the quadrant in which angle
1
2
A
lies.
Triangles
Law of sines:
sin
sin
sin
A
B
C
a
b
c
Law of cosines:
2
2
2
2
2
2
2
2
2
2 cos
2 cos
2
cos
a
b
c
bc
A
b
a
c
ac
B
c
a
b
ab
C
Area of a
:
1
sin
2
1
sin
2
1
sin
2
Area
bc
A
Area
ac
B
Area
ab
C
CHAPTER 4
MISCELLANEOUS RELATIONS AND FUNCTIONS
The general quadratic equation
2
2
0
Ax
Bxy Cy
Dx Ey F
If
2
4
0
B
AC
and
A C
, the graph is a circle.
If
2
4
0
B
AC
and
A C
, the graph is an ellipse.
If
2
4
0
B
AC
, the graph is a parabola.
If
2
4
0
B
AC
, the graph is a hyperbola.
Circle:
2
2
2
(
)
(
)
x h
y k
r
Ellipse:
if C>A,
2
2
2
2
(
)
(
)
1
x h
y k
a
b
, transverse axis horizontal
if C<A,
2
2
2
2
(
)
(
)
1
x h
y k
b
a
, transverse axis vertical, where
2
2
2
a
b
c
Vertices:
a
units along major axis fromcenter
Foci:
c
units along major axis fromcenter
Length=2b
Eccentricity=
c
a
<1
Length of latus rectum=
2
2b
a
Parabola:
if C=0,
2
(
)
4 (
)
x h
p y k
opens up and down---axis of symmetry is vertical
if A=0,
2
(
)
4 (
)
y k
p x h
opens to the side---axis of symmetry is horizontal
Equation of axis of symmetry:
x=h if vertical
y=k if horizontal
Focus: p units along the axis of symmetry from vertex
Equation of directrix:
y=-p if axis of symmetry is vertical
x=-p if axis of symmetry is horizontal
Eccentricity=
c
a
=1
Length of latus rectum=4p
Hyperbola:
2
2
2
2
(
)
(
)
1
x h
y k
a
b
, transverse axis horizontal
2
2
2
2
(
)
(
)
1
y k
x h
a
b
, transverse axis vertical, where
2
2
2
c
a
b
Vertices:
a
units along the transverse axis fromcenter
Foci:
c
units along the transverse fromcenter
Length of latus rectum=
2
2b
a
Eccentricity=
c
a
>1
the slopes of the asymptotes are
a
b
(vertical)or
b
a
(horizontal).
Exponential and Logarithmic Functions
0
log
1
1
( )
( )
log (
) log
log
log 1 0
log
log
log
log
1
log (
)
log
log
log
log
b
a
b
a b
a
a b
b
a
a
a b
ab
a
a
a
b
b
b
b
p
b
b
b
b
x
b
b
a
b
a
x
x
x
x
x
x
x
x
x
x
x
x
y
xy
p q
p
q
b
p
p
p
q
q
b
p
x
p
p
p
b
Greatest Integer Functions:
, where i is an interger and
1
x
i
i x i
Polar Coordinates:
2
2
2
cos
sin
x r
y r
x
y
r
De Moivre’s Throrem:
If
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
2
1
2
1
2
1
2
1
2
1
1
1
2
1
2
2
2
2
2
1/
1/
(cos
sin )
(cos
sin )
:
1.
[cos(
)
sin(
)]
2.
[cos(
)
sin(
)]
3.
(cos
sin
)
2
4.
(cos
n
n
n
n
n
z
x
y i r
i
r cis
and
z
x
y i r
i
r cis
z z
r r
i
z
r
i
z
r
z
r
n
i
n
r cisn
k
z
r
1/
2
2
sin
)
where k is an integer taking on values from 0 to n-1.
n
k
k
i
r cis
n
n
n
CHAPTER 5
MISCELLANEOUS TOPICS
!
(
1)(
2)...3 2 1
n
n n
n
Permutations:
Circular permutation (e.g., around a table) of n elements=
(
1)!
n
Circular permutation (e.g., beads on a bracelet) of n elements=
(
1)!
2
n
Permutations of n elements with a repetitions and with b repetitions=
!
! !
n
a b
!
!
n r
n
P
n r
the product of the largest r factors of n!
!
!
n r
n
P
r
r
r
The number of combinations of n things taken r at a time is denoted by
n
r
C
or C(n,r) or
n
r
.
n
n
r
n r
Binomial Theorem:
1
n r r
r
n
r
T
C a b
Probability:
Independent events:
(
)
( )
( )
P A B
P A
P B
Mutually exclusive events:
(
) 0
(
)
( )
( )
P A B
and P A B
P A
P B
Sequences and Series
In general, an arithmetic sequence is denoted by
1 1
1
1
1
,
,
2 ,
3 ......
(
1)
t t
d t
d t
d
t
n
d
1
1
(
)
2
[2
(
1) ]
2
n
n
n
n
S
t
t
or
n
S
t
n
d
In general, a geometric sequence is denoted by
2
3
1
1
1
1
1
1
, ,
,
,...,
n
t t r t r t r
t r
1
(1
)
1
n
n
t
r
S
r
1
lim
1
n
n
t
S
r
Geometry and Vectors
If
1
2
( , )
V v v
and
1
2
( , )
U u u
,
1
1
2
2
(
,
)
U V
u
v u
v
2
2
1
2
( )
( )
V
v
v
1 1
2 2
V U v u
v u
Two vectors are perpendicular if and only if
0
V U
Logic:
(
)
(
)
(
),
,
'
If
is true, then '
' is also true.
conjunction
A B
disjunction
A B
implication
A
B negation A B
A
B
B
A
Determinates:
a c
ad bc
b d
,
ax by c
If
dx ey
f
c b
a c
f e
d f
x
y
a b
a b
d e
d e
Geometry:
Distance between two points with coordinates
1
1
1
2
2
2
2
2
2
1
2
1
2
1
2
( , , )
, ,
(
)
(
)
(
)
x y z and x y z
x
x
y
y
z
z
The distance between a point and a plane:
Distance=
1
1
1
2
2
2
Ax
By
Cz
D
A
B
C
Triange:
Heron’s formular:
A=
(
)(
)(
); , , are the three sides of the triangle,
1
S= (
)
2
s s a s b s c a b c
a b c
Rhombus:
Area=bh=
1 2
1
;
,
,
2
d d b base h height d
diagonal
Cylinder
Volume=
2
r h
Later surface area=
2 rh
Total surface area=
2
2
2
rh
r
Cone:
The volume of the cone:
2
1
3
V
r h
Later surface area
2
2
r r
h
1
2
cl
Total surface area
2
2
2
r r
h
r
Sphere
Volume=
3
4
3
r
Surface area=
2
4 r
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