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Math Rules Factorisation: For factorising a Quadratic Equation by Formula: β(π) ± β(π)π β π(π)(π) πΏ= π(π) Trignometry: Pythagoras Theorem: π¨π = π©π + πͺπ π©π = π¨π β πͺπ πͺπ = π¨π β π©π B A Page 1 C Kareem Mokhtar © Trignometry of a right angled Triangle: πΆπππππππ πΊπΆπ―: πΊπππ½ = π―πππππππππ π¨π ππππππ πͺπ¨π―: πͺπππ½ = π―πππππππππ πΆπππππππ π»πΆπ¨: π»πππ½ = π¨π ππππππ The Sine rule: A,B & C are angles A,b & c are sides Page π π π = = πΊπππ¨ πΊπππ πΊπππͺ 2 The Sine Rule is used when: ο You are given ONE SIDE and TWO ANGLES, to find the missing side ο 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: π πΏ = πΊππ(ππ) πΊππ(πππ) π × πΊππ(πππ) πΏ= πΊππ(ππ) πΏ = ππ. πππ 2) Find π½: ππ. π ππ. π = πΊπππ½ πΊππ(ππ) πΊππ(ππ) × ππ. π πΊπππ½ = ππ. π βπ π½ = πΊππ πΊππ(ππ) × ππ. π ( ) ππ. π Page 3 π½ = ππ. π° Kareem Mokhtar © The Cosine Rule: A,B & C are angles A,b & c are sides The Cosine Rule is used when: ο You are given TWO SIDES and ONE ANGLE which is between the two sides, to get the side opposite to the angle. ο You are given THREE SIDES, to find any angle. ππ = ππ + ππ β π(π)(π)(πͺπππ½) π = ππ + ππ β π(π)(π)(πͺπππ½) Page 4 ππ = ππ + ππ β π(π)(π)(πͺπππ½) Kareem Mokhtar © Examples on the Cosine Rule: 1) Find X: πΏπ = ππ + πππ β π(π)(ππ)(πͺπππππ) πΏ = πππ. ππ πΏ = ππ. πππ 2) Find π½: Page 5 πππ = πππ + πππ β π(ππ)(ππ)(πͺπππ½) πππ = πππ β πππ(πͺπππ½) πππ + πππ(πͺπππ½) = πππ πππ(πͺπππ½) = πππ β πππ πππ (πͺπππ½) = βππ βππ πͺπππ½ = πππ βππ π½ = πͺππβπ ( ) πππ π½ = πππ. π° 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: ο You are given TWO SIDES and ONE ANGLE which is between the two sides. π π¨πππ = × π × π × πΊπππ¨ π π π¨πππ = × π × π × πΊπππͺ π π π¨πππ = × π × π × πΊπππ© π Examples on the Sine Rule of Area: Find the area of this shape: Kareem Mokhtar © Page π¨πππ = ππππ. ππππ 6 π π π¨πππ = ( × πππ × ππ × πΊππ(πππ)) + ( × ππ × ππ × πΊππ(πππ)) π π Sine Curve Rule: πΊπππ½ = πΊππ(πππ β π½) Examples: πΊπππππ = πΊππππ πΊππππ = πΊπππππ And so onβ¦ Example on Sine Curve rule: Calculate angle π½ given that it is obtuse. ππ ππ = πΊπππ½ πΊππππ πΊπππ½ = ππ × πππππ ππ ππ × πππππ π½ = πΊππβπ ( ) ππ Kareem Mokhtar © Page Back Bearing: If the bearing of B from A is π½, then the bearing of A from B {Back Bearing} is: π½ + πππ°(ππ π½ ππ ππππ ππππ πππ) π½ β πππ°(ππ π½ ππ ππππ ππππ πππ) 7 π½ = ππ. ππ β¦ ° which is obtuse. We have to find the other angle which has the same sine. πππ β ππ. ππ β¦ = πππ. π Co-ordinate Geometry and straight lines: To calculate the distance between two given points: π«πππππππ = β(πΏπ β πΏπ )π + (ππ β ππ )π To calculate the Co-ordinates of the mid-point between two given points: πΏπ + πΏπ ππ + ππ ( , ) π π 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: ππ β ππ π= πΏ π β πΏπ Matrices: Multiplication of two Matrices |π΄|: π¨π + π©π π¨π + π©π π¨ π© π π ( )( )=( ) πͺπ + π«π πͺπ + π«π πͺ π« π π Determinant of a Matrix: π¨ π© ) β |π΄| = π¨π« β π©πͺ πͺ π« Page 8 π΄=( Kareem Mokhtar © Multiplicative Inverse of a Matrix (M-1): π¨ π΄=( πͺ π π© π« βπ© βπ ) β |π΄| = π¨π« β π©πͺ β π΄ = ×( ) |π΄| βπͺ π¨ π« NOTE: M x M-1 = Identity Martix Any matrix multiplied by its multiplicative inverse will give you π π the identity matrix which is:( ) π π Variations: Direct proportion equation: π = π²(πΏ) 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: π² π= πΏ Page 9 Polygons: To calculate the sum of interior angles of a regular polygon: (π β π) × πππ Where n is the number of sides in the polygon. Kareem Mokhtar © To calculate one interior angle of a regular polygon: (π β π)πππ π To calculate the one exterior angle of a regular polygon: πππ π Page 10 Note: the sum of exterior angles of any polygon is always 360. Kareem Mokhtar ©