Emaths-Formula summary

Algebra:  𝑎(𝑏 + 𝑐) = 𝑎𝑏 + 𝑎𝑐  (𝑎 + 𝑏)2 = 𝑎2 + 2𝑎𝑏 + 𝑏2  (𝑎 − 𝑏)2 = 𝑎2 − 2𝑎𝑏 + 𝑏2  𝑎2 + 𝑏2 = (𝑎 + 𝑏)2 − 2𝑎𝑏  𝑎2 − 𝑏2 = (𝑎 + 𝑏)(𝑎 − 𝑏) Quadratic Equations: 𝑎𝑥2 + 𝑏𝑥 + 𝑐 = 0 Solve by formula: 𝑥 = −𝑏±√𝑏2−4𝑎𝑐 2𝑎 Complete the square: Make sure a = 1 𝑎𝑥2 + 𝑏𝑥 + ( 𝑏 2)2 − ( 𝑏 2)2 + 𝑐 = 0 Indices:  𝑎𝑚 × 𝑎𝑛 = 𝑎𝑚+𝑛  𝑎𝑚 ÷ 𝑎𝑛 = 𝑎𝑚−𝑛  (𝑎𝑚)𝑛 = 𝑎𝑚𝑛  𝑎0 = 1  𝑎−𝑛 = 1 𝑎𝑛  (𝑎 × 𝑏)𝑚 = 𝑎𝑚 × 𝑏𝑚  ( 𝑎 𝑏)𝑚 = 𝑎𝑚 𝑏𝑚  ( √𝑎 𝑛 )𝑚 = 𝑎 𝑚 𝑛  √𝑎 × √𝑏 = √𝑎 × 𝑏  √ 𝑎 𝑏 = √𝑎 √𝑏  (√𝑎)2 = 𝑎 Variation: 𝑦 is proportional to 𝑥: 𝑦 = 𝑘𝑥 𝑦 is inversely proportional to 𝑥: 𝑦 = 𝑘 𝑥 Simple Interest - To find interest: 𝑖 = 𝑃𝑅𝑇 100 Compound Interest: 𝐴 = 𝑃(1 + 𝑟 100)𝑛 Conversion of units: km/hr x 5 18 = m/sec ; m/sec x 18 5 = km/hr  Pythagoras Theorem: 𝑐𝟐 = 𝑎𝟐 + 𝑏𝟐 Trigonometry: TOA CAH SOH Cosine Rule: Length of a side: 𝑎2 = 𝑏2 + 𝑐2 − 2𝑏𝑐 cos 𝐴 Find an angle when all 3 sides are given: cos 𝐴 = 𝑏2 + 𝑐2 − 𝑎2 2𝑏𝑐 Co-ordinate Geometry: Eqn. of a straight line: 𝑦 = 𝑚𝑥 + 𝑐 Gradient of a straight line: 𝑚 = 𝑦2 − 𝑦1 𝑥2− 𝑥1 Midpoint: 𝑀 = ( 𝑥1+ 𝑥2 2 , 𝑦1 + 𝑦2 2 ) Distance between two points: 𝐴𝐵 = √(𝑥2 − 𝑥1)2 + (𝑦2 − 𝑦1)2 Matrices: Addition (𝑎 𝑏 𝑐 𝑑) + (𝑝 𝑞 𝑟 𝑠) = (𝑎 + 𝑝 𝑏 + 𝑞 𝑐 + 𝑟 𝑑 + 𝑠 ) Subtraction (𝑎 𝑏 𝑐 𝑑) − (𝑝 𝑞 𝑟 𝑠) = (𝑎 − 𝑝 𝑏 − 𝑞 𝑐 − 𝑟 𝑑 − 𝑠 ) Multiplication (𝑎 𝑏 𝑐 𝑑) × (𝑝 𝑞 𝑟 𝑠) = (𝑎𝑝 + 𝑏𝑟 𝑎𝑞 + 𝑏𝑠 𝑐𝑝 + 𝑑𝑟 𝑐𝑞 + 𝑑𝑠 ) 𝑘 × (𝑎 𝑏 𝑐 𝑑) = (𝑘𝑎 𝑘𝑏 𝑘𝑐 𝑘𝑑 ) Vectors: Parallelogram law of addition: 𝑂𝐵⃗⃗⃗⃗⃗ + 𝑂𝐴⃗⃗⃗⃗⃗ = 𝑂𝐶⃗⃗⃗⃗⃗ Polygons: Sum of Exterior angles = 360° One Exterior angle = 360° 𝑛 Sum of interior angles = (𝑛 − 2) × 180° Types of polygons: No. of sides 4 Quadrilateral 5 Pentagon 6 Hexagon 7 Heptagon 8 Octagon 9 Nonagon 10 decagon Angle Properties of Triangle: Sum of all angles = 180° Exterior angle (x) = Sum of opposite interior angles (𝑎 + 𝑏) Properties of Circles: -------------------------------------------- -------------------------------------------- -------------------------------------------- -------------------------------------------- -------------------------------------------- Chord of a Circle: -------------------------------------------- Tangent to a Circle: -------------------------------------------- -------------------------------------------- Alternate Segment Theorem: ∠QAB = ∠ACB (p = q) 𝑇𝑎𝑛 𝜃 = 𝑂𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝐴𝑑𝑗𝑎𝑐𝑒𝑛𝑡 = 𝑂 𝐴 𝐶𝑜𝑠 𝜃 = 𝐴𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝐻𝑦𝑝𝑜𝑡ℎ𝑒𝑛𝑢𝑠𝑒 = 𝐴 𝐻 𝑆𝑖𝑛 𝜃 = 𝑂𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝐻𝑦𝑝𝑜𝑡ℎ𝑒𝑛𝑢𝑠𝑒 = 𝑂 𝐻 Sine Rule: 𝑎 sin 𝐴 = 𝑏 sin 𝐵 = 𝑐 sin 𝐶 Triangular law of addition: 𝑂𝐴⃗⃗⃗⃗⃗ + 𝐴𝐶⃗⃗⃗⃗⃗ = 𝑂𝐶⃗⃗⃗⃗⃗ Angle at centre (2p) is twice angle at circumference (p) Angle in the same segment of a circle are equal Angle in a semicircle is a right angle. Opposite angles of a quadrilateral add up to 180o ∠A + ∠C = 180o ∠B + ∠D = 180o Exterior angle of a quadrilateral equals to interior opposite angle (∠b = ∠p) A line joining two points on a circle is called a chord (Line AB).