Download Maths Rules

Math Rules
Factorisation:
For factorising a Quadratic Equation by Formula:
−(𝒃) ± √(𝒃)𝟐 − 𝟒(𝒂)(𝒄)
𝑿=
𝟐(𝒂)
Trignometry:
Pythagoras Theorem:
𝑨𝟐 = 𝑩𝟐 + 𝑪𝟐
𝑩𝟐 = 𝑨𝟐 − 𝑪𝟐
𝑪𝟐 = 𝑨𝟐 − 𝑩𝟐
B
A
Page
1
C
Kareem Mokhtar ©
Trignometry of a right angled Triangle:
𝑶𝒑𝒑𝒐𝒔𝒊𝒕𝒆
𝑺𝑶𝑯: 𝑺𝒊𝒏𝜽 =
𝑯𝒚𝒑𝒐𝒕𝒆𝒏𝒖𝒔𝒆
𝑨𝒅𝒋𝒂𝒄𝒆𝒏𝒕
𝑪𝑨𝑯: 𝑪𝒐𝒔𝜽 =
𝑯𝒚𝒑𝒐𝒕𝒆𝒏𝒖𝒔𝒆
𝑶𝒑𝒑𝒐𝒔𝒊𝒕𝒆
𝑻𝑶𝑨: 𝑻𝒂𝒏𝜽 =
𝑨𝒅𝒋𝒂𝒄𝒆𝒏𝒕
The Sine rule:
A,B & C are angles
A,b & c are sides
Page
𝒂
𝒃
𝒄
=
=
𝑺𝒊𝒏𝑨 𝑺𝒊𝒏𝒃 𝑺𝒊𝒏𝑪
2
The Sine Rule is used when:
 You are given ONE SIDE and TWO ANGLES, to find the
missing side
 You are given TWO SIDES and ONE ANGLE which is not
between the two sides, to find the missing angle.
Kareem Mokhtar ©
Examples on Sine rule:
1) Find X:
𝟓
𝑿
=
𝑺𝒊𝒏(𝟐𝟖) 𝑺𝒊𝒏(𝟏𝟎𝟎)
𝟓 × 𝑺𝒊𝒏(𝟏𝟎𝟎)
𝑿=
𝑺𝒊𝒏(𝟐𝟖)
𝑿 = 𝟏𝟎. 𝟓𝒄𝒎
2) Find 𝜽:
𝟏𝟐. 𝟓
𝟏𝟎. 𝟓
=
𝑺𝒊𝒏𝜽 𝑺𝒊𝒏(𝟑𝟐)
𝑺𝒊𝒏(𝟑𝟐) × 𝟏𝟐. 𝟓
𝑺𝒊𝒏𝜽 =
𝟏𝟎. 𝟓
−𝟏
𝜽 = 𝑺𝒊𝒏
𝑺𝒊𝒏(𝟑𝟐) × 𝟏𝟐. 𝟓
(
)
𝟏𝟎. 𝟓
Page
3
𝜽 = 𝟑𝟗. 𝟏°
Kareem Mokhtar ©
The Cosine Rule:
A,B & C are angles
A,b & c are sides
The Cosine Rule is used when:
 You are given TWO SIDES and ONE ANGLE which is
between the two sides, to get the side opposite to the
angle.
 You are given THREE SIDES, to find any angle.
𝒂𝟐 = 𝒃𝟐 + 𝒄𝟐 − 𝟐(𝒃)(𝒄)(𝑪𝒐𝒔𝜽)
𝒃 = 𝒂𝟐 + 𝒄𝟐 − 𝟐(𝒂)(𝒄)(𝑪𝒐𝒔𝜽)
Page
4
𝒄𝟐 = 𝒃𝟐 + 𝒂𝟐 − 𝟐(𝒃)(𝒂)(𝑪𝒐𝒔𝜽)
Kareem Mokhtar ©
Examples on the Cosine Rule:
1) Find X:
𝑿𝟐 = 𝟖𝟐 + 𝟏𝟐𝟐 − 𝟐(𝟖)(𝟏𝟐)(𝑪𝒐𝒔𝟏𝟎𝟎)
𝑿 = 𝟐𝟒𝟎. 𝟐𝟓
𝑿 = 𝟏𝟓. 𝟓𝒄𝒎
2) Find 𝜽:
Page
5
𝟏𝟕𝟐 = 𝟏𝟎𝟐 + 𝟏𝟐𝟐 − 𝟐(𝟏𝟎)(𝟏𝟐)(𝑪𝒐𝒔𝜽)
𝟐𝟖𝟗 = 𝟐𝟒𝟒 − 𝟐𝟒𝟎(𝑪𝒐𝒔𝜽)
𝟐𝟖𝟗 + 𝟐𝟒𝟎(𝑪𝒐𝒔𝜽) = 𝟐𝟒𝟒
𝟐𝟒𝟎(𝑪𝒐𝒔𝜽) = 𝟐𝟒𝟒 − 𝟐𝟖𝟗
𝟐𝟒𝟎 (𝑪𝒐𝒔𝜽) = −𝟒𝟓
−𝟒𝟓
𝑪𝒐𝒔𝜽 =
𝟐𝟒𝟎
−𝟒𝟓
𝜽 = 𝑪𝒐𝒔−𝟏 (
)
𝟐𝟒𝟎
𝜽 = 𝟏𝟎𝟎. 𝟖°
Kareem Mokhtar ©
Sine Rule of the Area of a triangle:
A,B & C are angles
A,b & c are sides
This Rule is used when:
 You are given TWO SIDES and ONE ANGLE which is
between the two sides.
𝟏
𝑨𝒓𝒆𝒂 = × 𝒃 × 𝒄 × 𝑺𝒊𝒏𝑨
𝟐
𝟏
𝑨𝒓𝒆𝒂 = × 𝒃 × 𝒂 × 𝑺𝒊𝒏𝑪
𝟐
𝟏
𝑨𝒓𝒆𝒂 = × 𝒂 × 𝒄 × 𝑺𝒊𝒏𝑩
𝟐
Examples on the Sine Rule of Area:
Find the area of this shape:
Kareem Mokhtar ©
Page
𝑨𝒓𝒆𝒂 = 𝟐𝟖𝟔𝟔. 𝟑𝟏𝒎𝟐
6
𝟏
𝟏
𝑨𝒓𝒆𝒂 = ( × 𝟏𝟐𝟎 × 𝟒𝟎 × 𝑺𝒊𝒏(𝟏𝟐𝟎)) + ( × 𝟐𝟎 × 𝟖𝟎 × 𝑺𝒊𝒏(𝟏𝟎𝟎))
𝟐
𝟐
Sine Curve Rule:
𝑺𝒊𝒏𝜽 = 𝑺𝒊𝒏(𝟏𝟖𝟎 − 𝜽)
Examples:
𝑺𝒊𝒏𝟏𝟎𝟎 = 𝑺𝒊𝒏𝟖𝟎
𝑺𝒊𝒏𝟕𝟒 = 𝑺𝒊𝒏𝟏𝟎𝟔
And so on…
Example on Sine Curve rule:
Calculate angle 𝜽 given that it is obtuse.
𝟏𝟐
𝟏𝟎
=
𝑺𝒊𝒏𝜽 𝑺𝒊𝒏𝟒𝟎
𝑺𝒊𝒏𝜽 =
𝟏𝟐 × 𝒔𝒊𝒏𝟒𝟎
𝟏𝟎
𝟏𝟐 × 𝒔𝒊𝒏𝟒𝟎
𝜽 = 𝑺𝒊𝒏−𝟏 (
)
𝟏𝟎
Kareem Mokhtar ©
Page
Back Bearing:
If the bearing of B from A is 𝜽, then the bearing of A from
B {Back Bearing} is:
𝜽 + 𝟏𝟖𝟎°(𝒊𝒇 𝜽 𝒊𝒔 𝒍𝒆𝒔𝒔 𝒕𝒉𝒂𝒏 𝟏𝟖𝟎)
𝜽 − 𝟏𝟖𝟎°(𝒊𝒇 𝜽 𝒊𝒔 𝒎𝒐𝒓𝒆 𝒕𝒉𝒂𝒏 𝟏𝟖𝟎)
7
𝜽 = 𝟓𝟎. 𝟒𝟕 … ° which is obtuse. We have to find the other angle
which has the same sine.
𝟏𝟖𝟎 − 𝟓𝟎. 𝟒𝟕 … = 𝟏𝟐𝟗. 𝟓
Co-ordinate Geometry and straight lines:
To calculate the distance between two given points:
𝑫𝒊𝒔𝒕𝒂𝒏𝒄𝒆 = √(𝑿𝟐 − 𝑿𝟏 )𝟐 + (𝒀𝟐 − 𝒀𝟏 )𝟐
To calculate the Co-ordinates of the mid-point between
two given points:
𝑿𝟏 + 𝑿𝟐 𝒀𝟏 + 𝒀𝟐
(
,
)
𝟐
𝟐
To calculate the gradient of a straight line:
we must have two points on the line (X1,Y1) and (X2,Y2)
the gradient (m) is:
𝒀𝟐 − 𝒀𝟏
𝒎=
𝑿 𝟐 − 𝑿𝟏
Matrices:
Multiplication of two Matrices |𝑴|:
𝑨𝟏 + 𝑩𝟑 𝑨𝟐 + 𝑩𝟒
𝑨 𝑩 𝟏 𝟐
(
)(
)=(
)
𝑪𝟏 + 𝑫𝟑 𝑪𝟐 + 𝑫𝟒
𝑪 𝑫 𝟑 𝟒
Determinant of a Matrix:
𝑨 𝑩
) → |𝑴| = 𝑨𝑫 − 𝑩𝑪
𝑪 𝑫
Page
8
𝑴=(
Kareem Mokhtar ©
Multiplicative Inverse of a Matrix (M-1):
𝑨
𝑴=(
𝑪
𝟏
𝑩
𝑫 −𝑩
−𝟏
) → |𝑴| = 𝑨𝑫 − 𝑩𝑪 → 𝑴 =
×(
)
|𝑴|
−𝑪 𝑨
𝑫
NOTE: M x M-1 = Identity Martix
Any matrix multiplied by its multiplicative inverse will give you
𝟏 𝟎
the identity matrix which is:(
)
𝟎 𝟏
Variations:
Direct proportion equation:
𝒀 = 𝑲(𝑿)
Y and X are the two variables and K is the constant of
variation which you will be given information to find.
Indirect proportion equation:
𝑲
𝒀=
𝑿
Page
9
Polygons:
To calculate the sum of interior angles of a regular
polygon:
(𝒏 − 𝟐) × 𝟏𝟖𝟎
Where n is the number of sides in the polygon.
Kareem Mokhtar ©
To calculate one interior angle of a regular polygon:
(𝒏 − 𝟐)𝟏𝟖𝟎
𝒏
To calculate the one exterior angle of a regular polygon:
𝟑𝟔𝟎
𝒏
Page
10
Note: the sum of exterior angles of any polygon is always
360.
Kareem Mokhtar ©