Trigonometry Formulas
y
1. Definitions and Fundamental Identities
Sine:
sinO
T
_I_
esc f)
x
r
1
sec f)
y
1
eolO
cos ()
Cosine:
x
Ian 0
Tangent:
-sin 0,
sin2 0 + eos2 0
sin20
1,
0
eos (-0)
eos 0
see2 0
1 + lan2 0,
2sinOeosO,
2n_1+eos20
COSu2
'
Ian (A - E)
lanA - lanE
1 + lanA lanE
Sin(A - ; )
-eosA,
Sin(A + ; )
eosA,
sin2 0
ese2 0
smA cos B + cosA sinE
sin (A - E)
smA cos B - cosA sinE
eos(A + ; )
-sinA
sin A sinE
±eos (A - E) - ±eos (A + E)
eosA easE
1
1
2eos (A - E) + 2eos (A + E)
sin A + sin E
2 sin ± (A + E) eos ± (A - E)
sinA - sinE
2 eos± (A + E) sin ± (A - E)
eos A + eos E
1
1
2 eos 2 (A + E) eos 2 (A - E)
eosA - easE
-2sin±(A + E)sin±(A - E)
1 + eofO
1 - eos 20
2
sin (A + E)
sinA
COS(A - ; )
x
x
cos 2fJ = cos2 f) - sin2 f)
eos (A + E) = cosA cos B
lanA + lanE
1 - lanA lanE
P(x, y)
2. Identities
sin (-0)
Ian (A + E)
sin A sinB
eos (A - E) = cosA cos B + sin A sinB
Trigonometric Functions
Radian Measure
Degrees
Radians
L n
5
V2
45
,
,
90
2
4
'f,
Domain: (_00, 00)
Range: [-1,1]
y
, 2
Domain: (_00, 00)
Range: [-1,1]
y
y=tanx
Domain: All real numbers except odd
integer multiples of 7T12
Range: (_00,00)
or
180 0 = 1T radians.
() =
s
y'
y
The angles of two common triangles, in
degrees and radians.
,U
-
(\
"2
h
II
Domain: x *- 0, ±7T, ± 27T,.
RaIge: (_00, -1] U [1, co)
Domain: All real numbers except odd
integer multiples of 7T12
Range: (_00, -1] U [1, 00)
y
y=cscx
y=secx
y=cotx
x
Domain: x *- 0, ±7T, ±27T, .
Range: (_00,00)
SERIES
Tests for Convergence of Infinite Series
5. Series with some negative terms: Does LI an Iconverge? If
yes, so does Lan since absolute convergence implies convergence.
1. The nth-Term Test: Unless an
0, the series diverges.
n
2. Geometric series: Lar converges if Irl < 1; otherwise it
diverges.
6. Alternating series: Lan converges ifthe series satisfies the
conditions of the Alternating Series Test.
3. p-series: 2: IjnI' converges if p > I; otherwise it diverges.
4. Series with nonnegative terms: Try the Integral Test, Ratio
Test, or Root Test. Try comparing to a known series with the
Comparison Test or the Limit Comparison Test.
Taylor Series
00
-x
I +x
+ (-x)" + ...
- x + x2 -
xn
+-+
n!
x2
=I+x+2T+
X
Ixl < I
n=O
00
I
e
1 + x + x 2 + ... + x n + ...
00
n
Ixl < 00
n=O n.
x2n+ 1
.
x3
x5
smx = x - - + - 3!
5!
00
+ (- I)" -c(2"'n-+----I),--! + ...
x 2n
00
+ (-I)" (2n)! + ...
In(1 + x)
x -
1+X
I - x
tan
-1
x
2
x3
( _ 1 )n 2n+ 1
x
"'(2,--n"'+----I)-!
(-I )"x 2n
n
3
3
5
3
5
( x+-+-+'"
x
x
X
5
X
2n + 1
-1 <x:S;
+
:3 + 5 - ... + (-I)" 2n + I +
Ixl < 00
Ixl < 00
(2n)!
x
x
x
2
+ :3
- ... + (-I) n-1 n
+
-1
x -
Ixl < I
n=O
)
.
00
00
=
(-I)"x2n+!
-'--,2'--n'--+"---;IC-
2n+1
+ l'
Ixl < I
Ixl s I
Binomial Series
m(m - l)x 2
+ mx + ---'-2"'!C-'--- +
m(m - I)(m - 2)x 3
m(m - I)(m - 2) .. · (m - k + I)x k
----'--- + ... +
k!
+
Ixl < I,
where
m(m - I)
2!
m(m - I) .. · (m - k + I)
k!
for k '" 3.
BASIC ALGEBRA FORMULAS
Arithmetic Operations
a(b + c)
a c
ab + ac,
b'd
ac
bd
alb
cld
b' c
!!:..+!:..=ad+bc
b
d
bd
a d
Laws of Signs
-(-a)
Zero
-a
b
a,
a
a
b-b
Division by zero is not defined.
* 0: 7ia 0, aO
If a
I,
a
on
For any nwnber a: a· 0 = O· a = 0
Laws of Exponents
If a
* 0,
am _
m-n
-;;n-a
,
a
-m _
1
- am'
For any positive integer n,
The Binomial Theorem
(a + b)n = an + nan-1b +
+
n(n - I)
1.2
n(n - I)(n - 2)
1.2.3
a
a n- 2 b 2
n-3 3
b
+ ... +nab n - I +b n .
For instance,
(a + bf
a 2 + 2ab + b 2 ,
(a - b f
a 2 - 2ab + b 2
(a + b)'
a 3 + 3a 2b + 3ab 2 + b 3,
(a - b)3
a 3 - 3a 2 b + 3ab 2 - b 3
Factoring the Difference of Like Integer Powers, n > 1
an - b n = (a - b)(a n- l
+ a n- 2 b + a n- 3 b 2 + ... + ab n- 2 + b n- l )
For instance,
a2 - b2
(a- b)(a + b),
a3 - b3
(a - b )(a 2 + ab + b 2 ),
a4 - b4
(a - b)(a 3 + a 2 b + ab 2 + b 3).
Completing the Square
If a
ax 2 + bx + c
The Quadratic Formula
* 0,
au 2 + C
If a
(u
x + (bI2a), C
* a and ax + bx + c
2
-b ± Vb 2 - 4ac
2a
0, then
c - ::)
GEOMETRY FORMULAS
A = area, B = area of base, C = circumference, S = lateral area or surface area,
V
volume
Triangle
Similar Triangles
Pythagorean Theorem
a
a'
a
b'
c'
Z;=c
Trapezoid
Circle
b
A
+ b)h
Any Cylinder or Prism with Parallel Bases
Right Circular Cylinder
v = 1Tr2h
S = 21Trh = Area of side
Any Cone or Pyramid
Right Circular Cone
v = !1TTlh
3
S = 1Trs = Area of side
Sphere
LIMITS
General Laws
Specific Formulas
If P(x) = anx n + an_IX n- 1 + ... + ao, then
If L, M, c, and k are real numbers and
lim f(x)
L
lim g(x)
and
M,
then
Sum Rule:
lim(j(x) + g(x))
L +M
Difference Rule:
lim(j(x) - g(x))
L - M
Product Rule:
lim (j(x) , g(x))
Constant Multiple Rule:
lim(k' f(x))
lim P(x)
P(c)
ancn + an_1Cn-1 + ... + aO.
If P(x) and Q(x) are polynomials and Q(c) i' 0, then
.
L, M
P(x)
Q(x)
P(c)
Q(c)'
k, L
M i' 0
Quotient Rule:
If f(x) is continuous at x
The Sandwich Theorem
c, then
lim f(x)
f(c).
If g(x) s; f(x) s; h(x) in an open interval containing c, except
possibly at x
c, and if
then
f(x)
lim g(x)
lim hex)
x----+c
x----+c
L,
lim smx =
x---+O
L.
x
and
lim 1 - cosx = 0
x---+O
X
Inequalities
L'Hopital's Rule
If f(x) s; g(x) in an open interval containing c, except possibly
If f(a)
g(a)
0, both l' and g' exist in an open interval I
containing a, and g' (x) i' 0 on I if x i' a, then
at x = c, and both limits exist, then
lim f(x) s; lim g(x) .
x----+c
x----+c
Continuity
asswning the limit on the right side exists.
If g is continuous at L and
limg(f(x))
f(x)
L, then
gel).
DIFFERENTIATION RULES
General Formulas
Inverse Trigonometric Functions
Asswne u and v are differentiable flUlctions ofx.
d
dx (e)
Constant:
d
Sum:
du
d
Difference:
du
dv
dv
1
A.. (COS-I x)
A.. (tan-I x)
_1_
1 + x2
A.. (sec-I x)
_ _1_
1 + x2
A.. (esc-I x)
A..(cu)
dx
e du
dx
Product:
A.. (uv)
u dv + v du
dx
dx
dx
dv
du
d ()
vd; - ud;
Power:
.!:!....x n = nx n- 1
Chain Rule:
:x (j(g(x))
dx
A.. (eoCI x)
dx
Constant Multiple:
Quotient:
A.. (sin-I x)
dx
0
v2
_
1
1
dx
1
_
dx
Hyperbolic Functions
:x (sinh x) eoshx
:x (cosh x) sinhx
dx (tanh x)
d
seeh2 x
dx (seehx)
-seehxtanhx
d
-eseh2 x
d
dx (esehx)
-esehxeothx
dx (eothx)
dx
dx
d
f'(g(x))' g'(x)
Inverse Hyperbolic Functions
Trigonometric Functions
:x (sin x)
eosx
:x (cos x)
-sinx
d
dx(tarlX)
see 2 x
d
dx(seex)
seextanx
d
dx(eotx)
-ese2 x
d
dx(esex)
-esexeotx
Exponential and Logarithmic Functions
.!:!....e X = eX
d
1
-lnx = -
.!!....ax=axlna
dx
d
-d (loga x )
x
dx
dx
1
-In
x a
_
1
A.. (eseh-I x)
_
1
dx
A.. (eoth-I x)
dx
_1_
1 - x2
dx
Parametric Equations
If x
x
A.. (seeh-I x)
jet) and y
y'
dy
dx
get) are differentiable, then
dy/dt
dxjdt
and
d 2y
dx 2
dy'/dt
dxjdt'
INTEGRATION RULES
General Formulas
a
[f(X) dx
Zero:
l
Order a/Integration:
af
(x) dx
- [f(x) dx
k1
l\f(X) dx
Constant Multiples:
bf
(x) dx
(Any number
k)
-[f(X)dX
1
1
b
(j(X) ± g(x)) dx
Sums and Differences:
Additivity:
bf
[f(x) dx + [f(X) dx
(x) dx ±
1
b
g(x) dx
[f(X) dx
Max-Min Inequality: If max f and min f are the maximum and minimum values of f on [a, b], then
1
1 1
1
bf
minf'(b - a) S;
Domination:
f(x) '" g(x)
on
f(x) '" a on
[a, b]
[a, b]
bf
implies
implies
bf
(x)dx S; maxf'(b - a).
(x) dx '"
bg
(x) dx
(x) dx '" a
The Fundamental Theorem of Calculus
Part 1 If f is continuous on [a, b], then F(x)
f:f(t) dt is continuous on
[a, b] and differentiable on (a, b) and its derivative is f(x);
fx[f(t) dt
F'(x)
f(x).
Part 2 Iff is continuous at every point of [a, b] and F is any antiderivative of f
on [a, b], then
1
bf
(x) dx
F(b) - F(a).
Substitution in Oefinite Integrals
b
1
f(g(x))' g'(x) dx
a
Integration by Parts
19(b)
g{a)
feu) du
[f(x)g'(x) dx
f(x)g(x)
J: - [j'(x)g(x) dx