SET OF FORMULAS TO BE USED DURING CLASSES AND EXAMS
GEOMETRY
TRIGONOMETRY
Area of a parallelogram
Arc Length
A = b•h
l = θr
Area of a triangle
𝑨=
Area of a triangle
A=
b•h
2
x=
Quadratic Equations
∑ fi •Ci
n
Axis of symmetry: if
f ( x ) = ax 2 + bx + c
( a + b) • h
2
sin (α + β ) = sin α cos β + cos α sin β
sin (α − β ) = sin α cos β − cos α sin β
A = π r2
Law of Cosines
Circumference
𝑷 = 𝟐𝝅𝒓
c 2 = a 2 + b 2 − 2ab cosC
2
Volume of a rectangular prism
V = l •w•h
2
a +b −c
cosC =
2ab
2
Volume of a cylinder
V = π r 2h
Law of Sines
Volume of a cone
a
b
c
=
=
sin A sin B sinC
2
V = πr h
1
3
Volume of a sphere
V = 43 π r 3
S = 4π r
tan θ =
2
Distance between two points
2
( x2 − x1 ) + ( y2 − y1 )
2
2
sin θ
cosθ
2
sin θ + cos θ = 1
1+ tan 2 θ = sec 2 θ
Midpoint
! x2 + x1
,
#
" 2
𝒔𝟐 =
y2 + y1 $
&
2 %
1+ cot 2 θ = csc 2 θ
b
2a
∑ 𝒇(𝒙𝒊 3𝒙)𝟐
𝒏3𝟏
cos (α − β ) = cos α cos β + sin α sin β
tan α + tan β
tan (α + β ) =
1− tan α tan β
tan (α − β ) =
tan α − tan β
1+ tan α tan β
Standard Deviation
∑ 𝒇(𝒙𝒊 − 𝒙)𝟐
𝒔=5
𝒏−𝟏
cos 2θ = cos2 θ − sin 2 θ = 2 cos2 θ −1 = 1− 2sin 2 θ
tan 2θ =
−b ± b 2 − 4ac
2a
,
where a ≠ 0
Discriminant b − 4ac
Variance
𝒔𝟐 =
∑(𝒙𝒊 − 𝒙)𝟐
𝒏−𝟏
Exponentials and logarithms
a x = b ⇔ x = log a b
Double angle formulas
sin 2θ = 2sin θ cosθ
x=
2
Fundamental Identities
Surface area of a sphere
x=−
Variance
Quadratic formula
cos (α + β ) = cos α cos β − sin α sin β
,
then the axis of symmetry is
Sum and difference formulas
θr2
A=
2
Area of a circle
d=
Mean (Average)
ALGEBRA
Area of a circular sector
Area of a trapezoid
A=
𝟏
𝒂𝒃 𝒔𝒊𝒏𝑪
𝟐
STATISTICS
Standard Deviation
∑(𝒙𝒊 3𝒙)𝟐
𝒔=7
a x = e x ln a
log a a x = x = a loga x
𝒏3𝟏
2 tan θ
1− tan 2 θ
DO NOT WRITE ON THIS DOCUMENT
log a x =
log b x
log b a
CALCULUS
MATHEMATICS FOR DECISION MAKING
Addition rule for probabilities
Derivative
% f ( x + h) − f ( x ) (
dy
y = f ( x) ⇒
= f " ( x ) = Lim '
*
h→0
dx
h
&
)
𝑓(𝑥)
𝑓′(𝑥)
f ! ( x ) = nx n−1
f ( x) = x n
y = un
y = uv
y=
u
v
y = ln u
vu! − uv!
v2
y! =
u!
u
y = sin (u)
y = cos (u)
y! = −u! sin (u)
y = tan (u)
y! = u! sec 2 (u)
y=e
y = cot (u)
y! =− u'csc (u)
∫ sec (u) du = ln sec (u) + tan (u) +C
y = sec (u)
y! = u'sec (u) tan (u)
∫ csc (u) du = ln csc (u) − cot (u) + C
y = csc (u)
2
y! =− u'csc (u) cot (u)
s=
1+ "# f ! ( x )$% dx
a
Average y value
1
b−a
A=
Permutations
n!
,
(n − r)!
Combinations
n
Cr =
n!
r!(n − r)!
:
:
𝑉𝑎𝑟(𝑥) = 𝜎 T = K(𝑥M − 𝑥̅ )T ∗ 𝑃(𝑋 = 𝑥)
𝐸(𝑋) = 𝜇 = K 𝑥M 𝑃(𝑋 = 𝑥)
MOA
MOA
= 𝐸(𝑥 T ) − 𝜇T
𝑆𝐷(𝑋) = 𝜎 = Y𝑉𝑎𝑟(𝑋)
Binomial distribution
#n&
or P(X = k) = % ( p k ⋅ (1− p)n−k
$k '
P(X = x) = n Cx ⋅ p x ⋅ (1− p)n−x
E(X) = np , σ 2 = Var(X) = npq = np(1− p) , σ = SD(X) = npq
Standard z-score (Normal distribution)
𝑧=
Contrast value or t statistic for Tscore, Z-score and Paired sample T-test
c
[̅ 3\
[̅ 3\
b
𝑡=_
𝑧=]
𝑡 = de
[̅ 3\
]
a
√:
a
√:
∫ f ( x ) dx
E = Zα /2
σ
n
_
so that
_
E = tα /2
x− E < µ < x+ E
s
n
_
_
so that x − E < µ < x+ E
i
𝑉𝑎𝑟(𝑥) = 𝜎 T = g (𝑥 − 𝜇)T 𝑓(𝑥)𝑑𝑥 = 𝐸(𝑥 T ) − 𝜇T
𝐸(𝑋) = 𝜇 = g 𝑥𝑓(𝑥)𝑑𝑥
j
i
j
j
Simple Interest
Compound Interest
A = P(1+ rt)
! r$
A = P #1+ &
" n%
a
∫
f
√:
Confidence intervals, where E is the margin of error
𝑉𝑎𝑟(𝑥 ) = g 𝑥 T 𝑓(𝑥)𝑑𝑥 − 𝜇T 𝑆𝐷(𝑋) = 𝜎 = Y𝑉𝑎𝑟(𝑋)
f ( x ) dx
b
,
I = Prt
nt
Annuities
Ordinary
2
V = π ∫ !" f ( x )#$ dx
a
Integration by parts
Pr =
Expected value, variance and standard deviation of a discrete random variable
a
Volume
n
b
b
Area under the curve
Counting Techniques
Expected value, variance and standard deviation of a continuous random variable
2
∫
P(A ∩ B)
P(B)
P(A B) =
i
b
Arc Length
Conditional Probability
2
2
y! = u!eu
y! = u! cos (u)
u
𝑛 ≠ −1
∫ sin (u) du = −cos (u) + C
∫ cos (u) du = sin (u) + C
∫ sec (u) du = tan (u) + C
∫ csc (u) du = −cot (u) + C
∫ sec (u) tan (u) du = sec (u) + C
∫ csc (u) cot (u) du = −csc (u) + C
∫ tan (u) du = − ln cos (u) + C
∫ cot (u) du = ln sin (u) + C
𝑦´ = 𝑛 ∙ 𝑢´ ∙ 𝑢:3A
y! = vu! + uv!
y! =
<=>?
∫ 𝑢: 𝑑𝑢 = :@A + 𝐶,
1
∫ u du = ln u + C
∫ eu du = eu + C
P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
∫ u dv = uv − ∫ v du
A=
R ((1+ i)n −1)
i
Anticipated A = R(1+ i)
,
P=
R (1− (1+ i)−n )
i
((1+ i)n −1)
i
R (1− (1+ i)
−n
Deferred:
Ordinary
P=
(1− (1+ i) )
−n
, P = R(1+ i)
i
) (1+ i)
−K
i
0
