Algebra Geometry

REFERENCE PAGES
Algebra
Geometry
Arithmetic Operations
Geometric Formulas
a
c
ad bc
b
d
bd
a
d
ad
a
b
c
b
c
bc
d
ab c ab ac
ac
a
c
b
b
b
Formulas for area A, circumference C, and volume V:
Triangle
A 12 bh
12 ab sin a
Exponents and Radicals
Circle
A r 2
C 2 r
Sector of Circle
A 12 r 2
s r in radians
r
h
¨
xm
x mn
xn
1
xn n
x
x m x n x mn
x mn x m n
x
y
xyn x n y n
n
n
n
n
xy s
xs
y
s
n
s
¨
b
r
Sphere
V 43 r 3
xn
yn
Cylinder
V r 2h
Cone
V 13 r 2h
A 4 r 2
n
n
x mn s
x m (s
x )m
n
x 1n s
x
r
A rsr 2 h 2
r
n
x
x
s
n
y
sy
r
Factoring Special Polynomials
r
x y x yx y
2
2
x 3 y 3 x yx 2 xy y 2
x 3 y 3 x yx 2 xy y 2
Distance and Midpoint Formulas
Binomial Theorem
Distance between P1x1, y1 and P2x 2, y2:
x y2 x 2 2xy y 2
x y2 x 2 2xy y 2
d sx 2 x12 y2 y12
x y3 x 3 3x 2 y 3xy 2 y 3
x y3 x 3 3x 2 y 3xy 2 y 3
x yn x n nx n1y nn 1 n2 2
x y
2
Midpoint of P1 P2 :
n nk k
x y nxy n1 y n
k
x1 x 2 y1 y2
,
2
2
Lines
n
nn 1 n k 1
where
k
1 2 3 k
Slope of line through P1x1, y1 and P2x 2, y2:
m
Quadratic Formula
If ax 2 bx c 0, then x b sb 2 4ac
.
2a
y2 y1
x 2 x1
Point-slope equation of line through P1x1, y1 with slope m:
y y1 mx x1
Inequalities and Absolute Value
If a b and b c, then a c.
Slope-intercept equation of line with slope m and y-intercept b:
If a b, then a c b c.
y mx b
If a b and c 0, then ca cb.
If a b and c 0, then ca cb.
If a 0, then
x a
x a
x a
means
xa
or
Circles
x a
Equation of the circle with center h, k and radius r:
means a x a
means
h
h
xa
or
x h2 y k2 r 2
x a
❙❙❙❙
1
❙❙❙❙
REFERENCE PAGES
Trigonometry
Angle Measurement
Fundamental Identities
radians 180
1 rad
180
1 rad s
r
180
r
in radians
Right Angle Trigonometry
cos tan hyp
csc opp
adj
hyp
sec opp
adj
cot 1
sin sec 1
cos tan sin cos cot cos sin cot 1
tan sin 2 cos 2 1
¨
s r
opp
sin hyp
csc hyp
hyp
adj
opp
¨
adj
adj
opp
1 tan 2 sec 2
1 cot 2 csc 2
sin sin cos cos tan tan sin
cos 2
tan
cot 2
cos
sin 2
Trigonometric Functions
y
sin r
r
csc y
cos x
r
sec r
x
tan y
x
cot x
y
r
sin A
sin B
sin C
a
b
c
(x, y)
C
¨
The Law of Cosines
x
b 2 a 2 c 2 2ac cos B
y
A
c 2 a 2 b 2 2ab cos C
y=tan x
y=cos x
1
1
π
b
a 2 b 2 c 2 2bc cos A
y
y=sin x
a
c
Graphs of Trigonometric Functions
y
B
The Law of Sines
y
2π
Addition and Subtraction Formulas
2π
x
_1
π
2π x
sinx y sin x cos y cos x sin y
x
π
_1
sinx y sin x cos y cos x sin y
y
cosx y cos x cos y sin x sin y
cosx y cos x cos y sin x sin y
y
y=csc x
1
y=sec x
y
y=cot x
1
π
2π x
_1
π
2π x
2π x
π
tanx y tan x tan y
1 tan x tan y
tanx y tan x tan y
1 tan x tan y
_1
Double-Angle Formulas
sin 2x 2 sin x cos x
Trigonometric Functions of Important Angles
cos 2x cos 2x sin 2x 2 cos 2x 1 1 2 sin 2x
radians
sin cos tan 0
30
45
60
90
0
6
4
3
2
0
12
s22
s32
1
1
s32
s22
12
0
0
s33
1
s3
—
tan 2x 2 tan x
1 tan2x
Half-Angle Formulas
sin 2x ❙❙❙❙
2
❙❙❙❙
1 cos 2x
2
cos 2x 1 cos 2x
2
REFERENCE PAGES
Differentiation Rules
General Formulas
1.
d
c 0
dx
2.
d
cf x c f x
dx
3.
d
f x tx f x tx
dx
4.
d
f x tx f x tx
dx
5.
d
f xtx f xtx txf x (Product Rule)
dx
6.
d
dx
7.
d
f tx f txtx (Chain Rule)
dx
8.
d
x n nx n1 (Power Rule)
dx
f x
tx
txf x f xtx
tx 2
(Quotient Rule)
Exponential and Logarithmic Functions
9.
d
e x e x
dx
10.
d
a x a x ln a
dx
11.
d
1
ln x dx
x
12.
d
1
log a x dx
x ln a
Trigonometric Functions
13.
d
sin x cos x
dx
14.
d
cos x sin x
dx
15.
d
tan x sec 2x
dx
16.
d
csc x csc x cot x
dx
17.
d
sec x sec x tan x
dx
18.
d
cot x csc 2x
dx
Inverse Trigonometric Functions
19.
d
1
sin1x dx
s1 x 2
20.
d
1
cos1x dx
s1 x 2
21.
d
1
tan1x dx
1 x2
22.
d
1
csc1x dx
x sx 2 1
23.
d
1
sec1x dx
x sx 2 1
24.
d
1
cot1x dx
1 x2
Hyperbolic Functions
25.
d
sinh x cosh x
dx
26.
d
cosh x sinh x
dx
27.
d
tanh x sech 2x
dx
28.
d
csch x csch x coth x
dx
29.
d
sech x sech x tanh x
dx
30.
d
coth x csch 2x
dx
Inverse Hyperbolic Functions
31.
d
1
sinh1x dx
s1 x 2
32.
d
1
cosh1x dx
sx 2 1
33.
d
1
tanh1x dx
1 x2
34.
d
1
csch1x dx
x sx 2 1
35.
1
d
sech1x dx
x s1 x 2
36.
d
1
coth1x dx
1 x2
❙❙❙❙
3
❙❙❙❙
REFERENCE PAGES
Table of Integrals
Basic Forms
1.
y u dv uv y v du
2.
yu
3.
y
4.
ye
u
du e u C
5.
ya
u
du 6.
n
du y csc u cot u du csc u C
12. y tan u du ln sec u C
13. y cot u du ln sin u C
14. y sec u du ln sec u tan u C
15. y csc u du ln csc u cot u C
11.
u n1
C, n 1
n1
du
ln u C
u
au
C
ln a
y sa
17.
ya
18.
y u su
y sin u du cos u C
2
7.
y cos u du sin u C
8.
y sec u du tan u C
9.
y csc u du cot u C
19.
ya
2
y sec u tan u du sec u C
20.
yu
2
10.
2
2
Forms Involving sa 2 u 2 , a 0
u
a2
ln(u sa 2 u 2 ) C
sa 2 u 2 2
2
y
sa 2 u 2 du 22.
y
u 2 sa 2 u 2 du 23.
y
a sa 2 u 2
sa 2 u 2
du sa 2 u 2 a ln
u
u
24.
y
sa 2 u 2
sa 2 u 2
du ln(u sa 2 u 2 ) C
2
u
u
25.
y sa
21.
u 2
a4
a 2u 2 sa 2 u 2 ln(u sa 2 u 2 ) C
8
8
du
u2
2
u 2 du
26.
y sa
27.
y u sa
28.
y u sa
29.
y a
u2
2
u2
du
2
2
2
C
ln(u sa 2 u 2 ) C
du
2
u2
u
a2
ln(u sa 2 u 2 ) C
sa 2 u 2 2
2
1
sa 2 u 2 a
ln
a
u
C
sa 2 u 2
C
a 2u
du
u
2
C
u 2 32
a sa 2 u 2
❙❙❙❙
4
du
16.
❙❙❙❙
2
u
2
sin1
u
C
a
du
1
u
tan1 C
u2
a
a
du
2
a2
1
u
sec1 C
a
a
du
1
ua
ln
u2
2a
ua
du
1
ua
ln
a2
2a
ua
C
C