Formula Sheet (O level Mathematics)

Trigonometry
1. If the triangle is a right triangle.
i. Pythagoras Theorem

( Hypotenuse )2=( Perpendicular )2 + ( Base )2

ii. Trigonometric Ratios

Adjacent Base
cos θ= ∨
Hypotenuse Hypotenuse
Opposite Perpendicular
sin θ= ∨
Hypotenuse Hypotenuse
Opposite Perpendicular
tanθ= ∨
Adjacent Base
iii. Area of the Triangle
1
Areaof the Triangle= (base )×(height )
2

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2. If the trinagle is not a right triangle.

 A, B and C represents Angled and a,b,c are the corresponding sides of the triangle.
i. Sine Rule (If complete side-angle pair is given)
sin A sin B sin C
= =
a b c
ii. Cosine Rule (If complete side-angle pair is not given)
2 2 2
a =b +c −2 bc cos ⁡( A )
2 2 2
b =c + a −2 ca cos ⁡( B)
2 2 2
c =a + b −2 ab cos ⁡(C )

iii. Area of the Triangle

1
Areaof the Triangle= ( Neibhouring Sides ) × Sinθ
2

Bearing
i. Make a grid.
ii. Split Angles.
iii. Find Z-Angles.
iv. Mark Bearing.

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The angles are measured clockwise from the north line.
 The bearing of A from P is 45 o.
 The bearing of B from P is 260o .

Quadratic Equation
There are three methods to solve the quadratic equation.
i. Factorizing
 Ensure the quadratic is in standard form: a x 2 +bx +c=0 .
 Multiply a and c .
 Find two numbers that multiply to give a ⋅ c and add to give b .
 Break the middle term into two parts.
 Factor out the common binomial and solve for x .
ii. Quadratic Formula

−b ± √ b2−4 ac
x=
2a
iii. Completing Square
 Start with the quadratic equation in standard form: a x 2 +bx +c=0
 If a ≠ 1, divide the entire equation by a to make the coefficient of x 2 equal to 1.
 Move the constant term to the other side of the equation.
 Add and subtract the square of half the coefficient of x on the left-hand side to complete
the square.
 Rewrite the left side as a binomial square.
 Solve for x by taking the square root of both sides.
Basic Algebraic Identities:

( a+ b )2=a2+ 2 ab+b 2

( a−b )2=a2−2 ab +b2

2 2
a −b =(a+ b)(a−b)
3 3 2 2
a + b =(a+ b)(a −ab+b )
3 3 2 2
a −b =( a−b)(a +ab+ b )

( a+ b )3=a3 +b3 +3 ab (a+ b)

( a−b )3 =a3 −b3−3 ab(a−b)

Shoaib Akhtar | MPhil Mathematical Physics

2D Shapes.
Rectangle:
Area=length× breadth=l× b
Perimeter=2 ( length+breadth ) =2(l+w)

Square:
2 2
Area=Side =a
Perimeter=4 × side=4 a
Circle:
2
Area=π r
Circumference=2 πr
Parallelogram:
Area=base × height=b ×h
Perimeter=2 ( length+breadth ) =2(l+b)

Rhombus:
1 1
Area= (diagonal ¿ ¿ 1× diagonal2 )= (d 1)×(d 2) ¿
2 2
Perimeter=4 × side=4 a
Trapezium:
1
Area= ×(a+b)×h
2

Shoaib Akhtar | MPhil Mathematical Physics

3D Shapes.
Cube:
3 3
Volume=side =a
2 2
Surface Area=6 × sid e =6 a
Cuboid:
Volume=length ×breadth × height=l ×b × h
Surface Area=2(lb+bh+ hl)
Sphere:
4 3
Volume= π r
3
2
Surface Area=4 π r
Hemisphere:
2 3
Volume= π r
3
2
Surface Area=3 π r
Cone:
1 2
Volume= π r h
3
Surface Area=πr (r +l)
Cylinder:
2
Volume=π r h
Surface Area=2 πr (r +h)
Pyramid:
1 1
Volume= ×base area ×height = × A b × h
3 3
1
Surface Area=base area+ × perimeter of base × slant height
2

Shoaib Akhtar | MPhil Mathematical Physics

Rules of Indices
Product of Powers Rule:
m n m +n
a × a =a
Quotient of Powers Rule:
m
a m−n
n
=a
a
Power of a Power Rule:
n m× n
(a ¿¿ m) =a ¿

Power of a Product Rule:

m m m
(ab) =a × b

Power of a Quotient Rule:

()
m m
a a
= m
b b
Zero Exponent Rule:
0
a =1
Negative Exponent Rule:
−m 1
a = m
a
Fractional Exponent Rule:
m
a n =√ am
n

Shoaib Akhtar | MPhil Mathematical Physics

Statistics
Mean (The average of a set of numbers.)

Mean=
∑x
n
Mode (The value that occurs most frequently in a data set.)
Median (The middle value when the data is arranged in ascending (or descending) order. If
there is an even number of observations, the median is the average of the two middle
numbers.)
Range:
Range=MaximumValue−Minimum Value
Variance:

s=
∑2( x i−x )
2

n−1
Where x i are the data points, x is the mean, and n is the number of data points.

Mid-Point of the Class Interval:

Lower class boundary −Upper class boundary
MidPoint =
2
Estimated Mean:

Estimated Mean=
∑ f i . xi
∑ fi
 f i=frequency of each class interval
 x i=midpoint of each class interval
 ∑ f i=∑ of the frequencies
Frequency Density of Histogram:
Frequency of the class interval
Frequency Density=
Class width
 Frequency is the number of data points within a particular class interval.
 Class width is the difference between the upper and lower boundaries of the class
interval.

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Relative Frequency in Histogram:
Total frequency of class interval
Relative Frequency= × 100
Total Frequency

Coordinate Geometry
Distance:

√ 2
d= |x 2−x 1| +| y 2 − y 1|
2

Mid-Point:

M= ( x1 + x2 y1 + y2
2
,
2 )
Equation of Straight Line:
y=mx+c
Slope (Gradient):
y 2− y 1
m=
x 2−x 1

Equation of a Line through a Point with Slope:

y− y 1=m(x−x 1)

Parallel Lines:
m1=m2

Perpendicular Lines:
m1 × m2=−1

Collinearity of Three Points:

If ⃗
A, ⃗
B and ⃗
C are collinear.

m (⃗
AB ) =m(⃗
BC )

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Vectors
Magnitude:

| A|=√ x 2+ y 2
Equal Vectors:
Must satisfy the following condition,
 Equal Length.
 Parallel to each other.
Position Vector:
⃗
AB=⃗
OB −⃗
OA
Collinear Vectors:
Two or more vectors are collinear if they lie on the same straight line or are parallel to each
other. This means one vector is a scalar multiple of the other.
⃗
A=k ⃗
B
Parallel Vectors:
Parallel vectors are two or more vectors that have the same or exactly opposite direction. They
may have different magnitudes, but their directions must be aligned or anti-aligned.
⃗
A=k ⃗
B
Standard Coordinate System Rules:
 Horizontal (x-axis) Direction:
o → Right = + (Positive)
o ← Left = − (Negative)
 Vertical (y-axis) Direction:
o ↑ Up = + (Positive)

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 o ↓ Down = − (Negative)

Kinematics
Distance d
Average Speed= =
Time t
Dis placement d
Average Velocity= =
Time t
Velocity ∆ v v−u
Average Acceleration= = =
Time ∆t t
Uniformly Accelerated Motion (SUVAT Equations):
v=u+at
1 2
s=ut+ a t
2
2 2
2 as=u −v

Profit and Loss

Profit ( P )=Selling Price ( SP )−Cost Price (CP)
Profit ( L ) =Cost Price ( CP )−Selling Price ( SP )

( Profit
Profit ( % ) =
CP )
× 100 %

Loss ( % )=(
CP )
Loss
× 100 %

Discount Amount=Marked Price ( MP )−Selling Price(SP)

( Discount
Discount ( % )=
MP )
×100 %

Selling Price after Discount=MP × (1− )

Discount %
100

When Profit % is given:

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 (
SP=CP × 1+
Profit %
100 )
When Loss % is given:

(
SP=CP × 1−
Loss %
100 )

Rules for Identifying Significant Figures

Shoaib Akhtar | MPhil Mathematical Physics