This document contains numerous formulas and theorems from topics including sequences and series, systems of linear equations, quadratic equations, geometry, trigonometry, and calculus. Key formulas include the quadratic formula, distance formulas, slope formula, midpoint formula, Pythagorean theorem, sine and cosine laws, and compound interest formula. Theorems cover properties of angles, circles, and trigonometric identities.

1. 1
Chapter 8: Sequences and Series
Arithmetic Sequence (common difference)
an = a1 + (n – 1)d
Arithmetic Mean
(a + b)/2
Arithmetic Series
Sn = n(a1 + an)/2
Sn = n[2a1 + (n – 1)d]/2
Geometric Sequence (common ratio)
an = a1rn – 1
Geometric Series
Sn =
( )
r-1
r-1a n
1
Infinite Geometric series
S =
r-1
a
Geometric Mean
G = ab
Harmonic Mean
H =
ab2
ba +
FORMULAS IN INTERMEDIATE ALGEBRA
Chapter 1: Systems of Linear Equations
- Consistent and Independent
f
c
e
b
d
a
≠≠
- Inconsistent
f
c
e
b
d
a
≠=
- Dependent
f
c
e
b
d
a
==
Chapter 4: Quadratic Equations
Quadratic Formula
a2
4ac-bb-
x
2
±
=
Discriminant = b2 – 4ac
Nature of the Roots
b2 – 4ac  0 two real roots
b2 – 4ac > 0 two unequal real roots
b2 – 4ac = 0 two equal real roots
b2 – 4ac < 0 two unequal imaginary
roots
Sum of the Roots = - b/a
Product of the Roots = c/a
Chapter 5: Quadratic Function
The domain of the QF is set of real numbers.
The range is …y 
a4
b-ac4 2
if a is positive.
y 
a4
b-ac4 2
if a is negative.
Vertex (h, k)   a4/)b-(4acb/2a,- 2
Line of symmetry = - b/2a
The QF has a minimum value if a is positive
and is given by the formula y = (4ac – b2)/4a.
The QF has a maximum value if a is negative
and is given by the formula y = (4ac – b2)/4a.
*** The minimum/maximum value occurs at
x = - b/2a

2. 2
FORMULAS IN GEOMETRY
No. of line segment = n(n – 1)/2
where n is the no. of points
No. of Rays = 2(n – 1)
where n is the no. of points
No. of angles = n(n – 1)/2
Where n is the no. of rays
No. of Diagonals = n(n – 3)/2
Sum of interior angles of a polygon =
1800(n – 2) where n is the no. of sides
Sum of exterior angles = 3600
Sum of each exterior angle of a regular polygon =
3600/n, where n is the number of sides
Perimeterand Circumference
Square P = 4s
Rectangle P = 2l + 2w
Circumference of a circle C = 2  r or  d
Area
Circle A =  r2
Parallelogram A = bh
Rectangle A = lw or A = bh
Regular Polygon A =
2
Pa
Rhombus A =
2
dd 21
or bh
Sector of a circle A =
360
2
rN
Square A = s2
Trapezoid A =
2
)bb(h 21 +
Triangle A =
2
bh
Surface Area
Cube SA = 6s2
Rectangular Prism SA = 2(lw + lh + wh)
Sphere SA = 4r2
Cylinder SA = 2r2 + 2rh
Volume
Cube V = s3
Rectangular Prism V = lwh
Sphere V = 4/3r3
Cylinder V = r2h
Cone V = 1/3r2h
Pythagorean Theorem
In a right triangle, for side lengths a, b, and c,
a2 + b2 = c2.
Distance
 Between two points on coordinate plane
d = 2
12
2
12 )()( yyxx 
 Between two points A and B on a number
line
d = ba 
 Arc length r
180
d)AB(L 0
0 
= , r – radius
 From a point (x1, y1) to a line with equation
Ax + By +C = 0
22
11
BA
CByAx
d



 Distance Between 2 Parallel Lines
d =
22
12
ba
cc


Midpoint
Between two points A and B on a number line
2
ba 
Between two points on a coordinate plane
2
yy
,
2
xx 2121 ++
Slope
Slope of a line
12
12
xx
y-y
m
-
=
Equations for Figures on a coordinate Plane
Slope – intercept form of a line y = mx + b
Point - slope form of a line y = m(x- x1) + y1
Standard form of a line -------- Ax + By = C
General Form ------- Ax+ By + C = 0
Equation of a Circle (Standard Form)
(x – h)2 + (y – k)2 = r2, Center (h, k) Radius = r

3. 3
Theorems for Angles and Circles
Theorems:
1. The radius drawn to the point of tangency is perpendicular to the tangent.
2. An angle inscribed in a semi – circle is a right angle.
3. Inscribed angles subtended by the same arc are equal.
4. Opposite angles of an inscribed quadrilaterals are supplementary.
x = 2x
(It means that the measure of an
inscribed angle is half the measure of the
intercepted arc.)
ABC =
2
1
(arc AB)
2
1 = 2 =
2
1
(arc AB + arc DE)
1 =
2
1
(arc JK – arc LM)
2 =
2
1
(arc SR – arc QR)
3 =
2
1
(arc BJH – arc BH)

4. 4
POWER THEOREMS
1. Two – Secant Power Theorem
(PA)(PB) = (PC)(PD)
2. Secant – Tangent Power Theorem
(PT)2 = (PA)(PB)
3. Two – Chord Power Theorem
(AR)(RB) = (CR)(CD)
P
B
A
D
C
P
T
A
B
A
B
C
D
R

5. 5
FORMULAS IN TRIGONOMETRY
1. Arc Length L = rd
180
0
0

2. Graphs of Functions
Amplitude Period
y = a cos bx  a  2/b
y = a sin bx  a  2/b
y = a tan bx no amplitude /b
3. Trigonometric Identities
sin2 x + cos2 x = 1 tan2 x + 1 = sec2 x
cot2 x + 1 = csc2 x
4. Sum and Difference Formulas
cos(A + B) = cosAcosB – sinAsinB
cos(A – B) = cosA cosB + sinAsinB
sin(A + B) = sinAcosB + cosAsinB
sin(A – B) = sinAcosB – cosAsinB
tan(A + B) =
BtanAtan1
BtanAtan


tan(A – B) =
BtanAtan1
BtanAtan


5. Double Angle Formulas
sin 2A = 2sinAcosA
cos 2A = cos2A – sin2A
= 2cos2A – 1
= 1 – 2sin2A
6. Half – Angle Formulas
cos
2
Acos1
2
A 
 sin
2
Acos1
2
A 


6. 6
7. Right Triangle Trigonometry
sin  =
h
o
cos  =
h
a
tan  =
a
o
8. Sine / Cosine Law
c
Csin
b
Bsin
a
Asin

a2 = b2 + c2 – 2bccosA
b2 = a2 + c2 – 2accosB
c2 = a2 + b2 – 2abcosC
COMPOUND INTEREST
A = P(1 + r/n)nt where P – principal invested
r – rate of interest
n – the number of times the amount is compounded
t – number of years
A
B C

Opposite (o)
Adjacent (a)
Hypotenuse (h)
CB
A
a
bc

7. 7
Pythagorean Theorem
In a right triangle, for side lengths a, b, and c,
a2 + b2 = c2.
Distance
 Between two points on coordinate plane
d = 2
12
2
12 )()( yyxx 
 Between two points A and B on a number line
d = ba 
 From a point (x1, y1) to a line with equation
Ax + By +C = 0
22
11
BA
CByAx
d



 Distance Between 2 Parallel Lines
d =
22
21
ba
cc


Midpoint
Between two points A and B on a number line
2
ba 
Between two points on a coordinate plane
2
yy
,
2
xx 2121 ++
Slope
Slope of a line
12
12
xx
y-y
m
-
=
Equations for Figures on a coordinate Plane
Slope – intercept form of a line y = mx + b
Point - slope form of a line y = m(x- x1) + y1
Standard form of a line -------- Ax + By = C
General Form ------- Ax+ By + C = 0
Equation of a Circle (Standard Form)
(x – h)2 + (y – k)2 = r2, Center (h, k) Radius = r