1. Geometry
Solid Geometry
Rectangular Solid
Volume: V = lwh
Cube
Volume: V = s3
Right Circular Cylinder
Volume: V = pr2
h
Lateral surface area:
L = 2prh
Total surface area:
S = 2prh + 2pr2
Right Circular Cone
Volume: V = 1
3pr2
h
Lateral surface area:
L = prs
Total surface area:
S = pr2
+ prs
Slant height:
s = 2r2
+ h2
Sphere
Volume: V = 4
3pr3
Surface area: S = 4pr2
Plane Geometry
Rectangle
Area: A = lw
Perimeter: P = 2l + 2w
Square
Area: A = s2
Perimeter: P = 4s
Triangle
Area: A = 1
2bh
Sum of Angle Measures
A + B + C = 180Њ
Right Triangle
Pythagorean theorem
(equation):
a2
+ b2
= c2
Parallelogram
Area: A = bh
Trapezoid
Area: A = 1
2h1a + b2
Circle
Area: A = pr2
Circumference:
C = pd = 2pr
w
l
s
s
h
b
A
B
C
a
b
c
h
b
h
b
a
r
d
l w
h
s
s
s
r
r
h
h
s
r
r

2. Algebra
Properties of Real Numbers
Commutative: a + b = b + a; ab = ba
Associative: a + 1b + c2 = 1a + b2 + c ;
a1bc2 = 1ab2c
Additive Identity: a + 0 = 0 + a = a
Additive Inverse: -a + a = a + 1-a2 = 0
Multiplicative Identity: a и 1 = 1 и a = a
Multiplicative Inverse: a и
1
a
=
1
a
и a = 1, a ϶ 0
Distributive: a1b + c2 = ab + ac
Exponents and Radicals
am
и an
= am+n am
an = am-n
1am
2n
= amn
1ab2m
= am
bm
a
a
b
b
m
=
am
bm a-n
=
1
an
If n is even, 2
n
an
= ͉ a͉.
If n is odd, 2
n
an
= a.
2
n
a и 2
n
b = 2
n
ab, a, b Ú 0
A
n a
b
=
2
n
a
2
n
b
2
n
am
= 12
n
a2m
= am>n
Special-Product Formulas
1a + b21a - b2 = a2
- b2
1a + b22
= a2
+ 2ab + b2
1a - b22
= a2
- 2ab + b2
1a + b23
= a3
+ 3a2
b + 3ab2
+ b3
1a - b23
= a3
- 3a2
b + 3ab2
- b3
1a + b2n
= a
n
k=0
a
n
k
ban-k
bk
, where
a
n
k
b =
n!
k! 1n - k2!
=
n1n - 121n - 22 g 3n - 1k - 124
k!
Factoring Formulas
a2
- b2
= 1a + b21a - b2
a2
+ 2ab + b2
= 1a + b22
a2
- 2ab + b2
= 1a - b22
a3
+ b3
= 1a + b21a2
- ab + b2
2
a3
- b3
= 1a - b21a2
+ ab + b2
2
Interval Notation
1a, b2 = 5x ͉a 6 x 6 b6
3a, b4 = 5x ͉a … x … b6
1a, b4 = 5x ͉a 6 x … b6
3a, b2 = 5x ͉a … x 6 b6
1- ϱ, a2 = 5x ͉x 6 a6
1a, ϱ2 = 5x ͉x 7 a6
1- ϱ, a4 = 5x ͉x … a6
3a, ϱ2 = 5x ͉x Ú a6
Absolute Value
͉ a͉ Ú 0
For a 7 0,
͉ X ͉ = a S X = -a or X = a,
͉ X ͉ 6 a S -a 6 X 6 a,
͉ X ͉ 7 a S X 6 -a or X 7 a.
Equation-Solving Principles
a = b S a + c = b + c
a = b S ac = bc
a = b S an
= bn
ab = 0 4 a = 0 or b = 0
x2
= k S x = 2k or x = - 2k
Inequality-Solving Principles
a 6 b S a + c 6 b + c
a 6 b and c 7 0 S ac 6 bc
a 6 b and c 6 0 S ac 7 bc
(Algebra continued)

3. The Distance Formula
The distance from 1x1, y12 to 1x2, y22 is given by
d = 21x2 - x122
+ 1y2 - y122
.
The Midpoint Formula
The midpoint of the line segment from 1x1, y12 to
1x2, y22 is given by
a
x1 + x2
2
,
y1 + y2
2
b.
Formulas Involving Lines
The slope of the line containing points 1x1, y12 and
1x2, y22 is given by
m =
y2 - y1
x2 - x1
.
Slope–intercept equation: y = f 1x2 = mx + b
Horizontal line: y = b or f 1x2 = b
Vertical line: x = a
Point–slope equation: y - y1 = m1x - x12
The Quadratic Formula
The solutions of ax2
+ bx + c = 0, a ϶ 0, are given by
x =
-b { 2b2
- 4ac
2a
.
Compound Interest Formulas
Compounded n times per year: A = P a1 +
i
n
b
nt
Compounded continuously: P1t2 = P0ekt
Properties of Exponential and
Logarithmic Functions
loga x = y 4 x = ay
ax
= ay
4 x = y
loga MN = loga M + loga N loga Mp
= p loga M
loga
M
N
= loga M - loga N
logb M =
loga M
loga b
loga a = 1 loga 1 = 0
loga ax
= x aloga x
= x
Conic Sections
Circle: 1x - h22
+ 1y - k22
= r2
Ellipse:
1x - h22
a2 +
1y - k22
b2 = 1,
1x - h22
b2 +
1y - k22
a2 = 1
Parabola: 1x - h22
= 4p1y - k2,
1y - k22
= 4p1x - h2
Hyperbola:
1x - h22
a2 -
1y - k22
b2 = 1,
1y - k22
a2 -
1x - h22
b2 = 1
Arithmetic Sequences and Series
a1, a1 + d, a1 + 2d, a1 + 3d, c
an+1 = an + d an = a1 + 1n - 12d
Sn =
n
2
1a1 + an2
Geometric Sequences and Series
a1, a1r, a1r2
, a1r3
, c
an+1 = anr an = a1rn-1
Sn =
a111 - rn
2
1 - r
Sϱ =
a1
1 - r
, ͉ r͉ 6 1
Algebra (continued)

4. Trigonometry
Trigonometric Functions
Acute Angles Any Angle Real Numbers
sin u =
opp
hyp
, csc u =
hyp
opp
,
cos u =
adj
hyp
, sec u =
hyp
adj
,
tan u =
opp
adj
, cot u =
adj
opp
sin u =
y
r
, csc u =
r
y
,
cos u =
x
r
, sec u =
r
x
,
tan u =
y
x
, cot u =
x
y
sin s = y, csc s =
1
y
,
cos s = x, sec s =
1
x
,
tan s =
y
x
, cot s =
x
y
Basic Trigonometric Identities
sin 1-x2 = -sin x,
cos 1-x2 = cos x,
tan 1-x2 = -tan x,
tan x =
sin x
cos x
,
cot x =
cos x
sin x
,
csc x =
1
sin x
,
sec x =
1
cos x
,
cot x =
1
tan x
Pythagorean Identities
sin2
x + cos2
x = 1,
1 + cot2
x = csc2
x,
1 + tan2
x = sec2
x
Identities Involving P,2
sin 1p>2 - x2 = cos x,
cos 1p>2 - x2 = sin x, sin 1x Ϯ p>22 = {cos x,
tan 1p>2 - x2 = cot x, cos 1x Ϯ p>22 = ϯsin x
Sum and Difference Identities
sin 1u Ϯ v2 = sin u cos v Ϯ cos u sin v,
cos 1u Ϯ v2 = cos u cos v ϯ sin u sin v,
tan 1u Ϯ v2 =
tan u Ϯ tan v
1 ϯ tan u tan v
Double-Angle Identities
sin 2x = 2 sin x cos x,
cos 2x = cos2
x - sin2
x
= 1 - 2 sin2
x
= 2 cos2
x - 1,
tan 2x =
2 tan x
1 - tan2
x
Half-Angle Identities
sin
x
2
= {
A
1 - cos x
2
, cos
x
2
= {
A
1 + cos x
2
,
tan
x
2
= {
A
1 - cos x
1 + cos x
=
sin x
1 + cos x
=
1 - cos x
sin x
(Trigonometry continued)
hyp
adj
opp
u
r
(x, y)
x
y
u
s
(x, y)
1
x
y

5. Trigonometry (continued)
The Law of Sines
In any ᭝ABC,
a
sin A
=
b
sin B
=
c
sin C
.
The Law of Cosines
In any ᭝ABC,
a2
= b2
+ c2
- 2bc cos A,
b2
= a2
+ c2
- 2ac cos B,
c2
= a2
+ b2
- 2ab cos C.
Graphs of Trigonometric Functions
Trigonometric Function Values
of Special Angles
CA
B
a
b
c
The sine function: f1x2 = sin x The cosecant function: f1x2 = csc x
The secant function: f1x2 = sec x
The cotangent function: f1x2 = cot x
The cosine function: f1x2 = cos x
The tangent function: f1x2 = tan x
͙2
2
͙2
2 ΃΂ ,
͙3
2 ΃΂q,
͙3
2
, q΃΂
(1, 0)
(0, 1)
(Ϫ1, 0)
(0, Ϫ1)
q
w
u
d
A
2pp
0
x
y
͙2
2
͙2
2 ΃΂Ϫ ,
͙3
2 ΃΂q, Ϫ
͙3
2
, Ϫq΃΂
͙2
2
͙2
2 ΃΂Ϫ ,Ϫ
͙2
2
͙2
2 ΃΂ ,Ϫ
f
135Њ
90Њ
120Њ
150Њ
180Њ
210Њ
225Њ
240Њ
270Њ
300Њ
315Њ
330Њ
45Њ
30Њ
0Њ
360Њ
60Њ
i
S
F
h
o p
j
G
͙3
2 ΃΂Ϫq, Ϫ
͙3
2 ΃΂Ϫq,
͙3
2
, Ϫq΃΂Ϫ
͙3
2
, q΃΂Ϫ
1
2
Ϫ1
Ϫ2
x
y
Ϫ2p Ϫw Ϫp Ϫq q p w 2p
1
Ϫ1
2
x
y
Ϫ2p Ϫw Ϫp Ϫq q p w 2p
2
Ϫ1
Ϫ2
x
y
Ϫ2p Ϫw Ϫp Ϫq q p w 2p
Ϫ2
Ϫw Ϫq q w
Ϫ1
2
x
y
Ϫ2p Ϫp p 2p
Ϫ2
Ϫw Ϫq q w
1
Ϫ1
2
x
y
Ϫ2p Ϫp p 2p
1
Ϫ1
2
x
y
Ϫ2p Ϫw Ϫp Ϫq q p w 2p

6. A Library of Functions
Linear function Linear function Constant function Absolute-value function
y
x
2
4
Ϫ2
Ϫ4
Ϫ2Ϫ4 42
f(x) ϭ 3x ϩ 2
y
x
2
4
Ϫ2
Ϫ4
Ϫ2Ϫ4 42
f(x) ϭ Ϫ x Ϫ 1
1
2
y
x
2
4
Ϫ2
Ϫ4
Ϫ2Ϫ4 42
f(x) ϭ Ϫ3
y
x
2
4
Ϫ2
Ϫ4
Ϫ2Ϫ4 42
f(x) ϭ |x|
Squaring function Quadratic function Quadratic function Square-root function
y
x
2
4
Ϫ2
Ϫ4
Ϫ2Ϫ4 42
f(x) ϭ x2
y
x
2
4
Ϫ4
Ϫ2Ϫ4 42
f(x) ϭ x2 Ϫ 2x Ϫ 3
y
x
2
Ϫ2
Ϫ4
Ϫ2Ϫ4 42Ϫ6
f(x) ϭ Ϫ x2 Ϫ x Ϫ 2
1
2
5
2
y
x
2
4
Ϫ2
Ϫ4
Ϫ2Ϫ4 42
f(x) ϭ ͙x
Cubing function Cube root function Greatest integer Rational function
function
y
x
2
4
Ϫ2
Ϫ4
Ϫ2Ϫ4 42
f(x) ϭ x3
y
x
2
4
Ϫ2
Ϫ4
Ϫ2Ϫ4 42
f(x) ϭ ͙x
3
y
x
2
4
Ϫ4
Ϫ2Ϫ4 42
f(x) ϭ [[x]]
f(x) ϭ
2
x
y
x
2
4
Ϫ2
Ϫ4
Ϫ2 42
Exponential function Exponential function Logarithmic function Logistic function
y
x
2
4
Ϫ2
Ϫ4
Ϫ2Ϫ4 42
f(x) ϭ ex
y
x
2
4
Ϫ2
Ϫ4
Ϫ2Ϫ4 42
f(x) ϭ 2Ϫx
y
x
2
4
Ϫ2
Ϫ4
Ϫ2Ϫ4 42
f(x) ϭ log x
y
x
2000
4000
Ϫ2000
Ϫ10Ϫ20 2010
3000
1 ϩ 5eϪ0.4xf(x) ϭ