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Co-ordinate Geometry The angle sum of a quadrilateral is 360 Straight line Tests for special quadrilaterals: Gradient form: 𝑦 = 𝑚𝑥 + 𝑐 General form: 𝑎𝑥 + 𝑏𝑥 + 𝑐 = 0 Distance: 𝑑 = √(𝑥1 − 𝑥2 )2 + (𝑦1 − 𝑦2 )2 Midpoint: ( 1 2 2 , 1 2 2 ) Perpendicular distance of a point from a line. 𝑥 +𝑥 𝑑=| 𝑦 +𝑦 𝑎𝑥1 + 𝑏𝑦1 + 𝑐 √𝑎2 + 𝑏 2 | Relationship between gradient and angle 𝑚 = 𝑡𝑎𝑛 ∝ Angle between two lines 𝑚2 − 𝑚1 𝑡𝑎𝑛𝜑 = 1 + 𝑚1 𝑚2 𝑥= −𝑏±√𝑏2 −4𝑎𝑐 2𝑎 Geometrical Properties Complementary angles add to 90 Supplementary angle add to 180 Vertically opposite angles are equal Angles at a point add to 360 Angle sum of a triangle is 180 The exterior angle of a triangle is equal to the sum of the opposite interior angles An isosceles triangle has equal base angles Equilateral triangles have all angles 60 Alternate angles on parallel lines are equal Corresponding angles on parallel lines are equal Co-interior angles between parallel lines are supplementary The angle sum of a polygon is (n-2)x180 The sum of the exterior angles of any polygon is equal to 360 Parallelograms: Two opposite sides equal and parallel or Opposite sides are equal or Opposite angles are equal or Diagonals bisect each other Rhombus: All sides equal or Diagonals bisect each other at right angles Rectangle: All angles are right angles or Parallelogram with equal diagonals Square: All sides equal and one angle right or All angles right and two adjacent sides equal. Tests for congruent triangles SSS SAS AAS RHS Tests for similar shapes All angles are the same, therefore the overall shape is the same. All equivalent sides on each shape are in the same proportion to each other. Applications of Differentiation 𝑑𝑦 First derivative 𝑑𝑥 - Stationary point when equals 0 - Curve increasing>0 - Curve decreasing<0 -Max turning point if second derivative negative -Minimum turning point if second derivative positive Second derivative 2 𝑑𝑥 - Point of inflexion when equals 0 -Concave up when >0 -Concave down when <0 Horizontal point of inflexion if both first and second derivatives equal zero. 𝑑2 𝑦 1 Logarithmic Functions Integration 𝑛 ∫ 𝑥 𝑑𝑥 = 𝑥 𝑛+1 𝑑 1 (𝑙𝑛𝑥) = 𝑑𝑥 𝑥 𝑑 𝑓 ′ (𝑥) (𝑙𝑛𝑓(𝑥)) = 𝑑𝑥 𝑓(𝑥) +𝑐 Area between curve and axis 1 ∫ 𝑑𝑥 = ln 𝑥 + 𝑙𝑛𝐴 𝑥 𝑓 ′ (𝑥) ∫ 𝑑𝑥 = ln 𝑓(𝑥) + 𝑙𝑛𝐴 𝑓(𝑥) 𝑏 𝐴 = ∫ 𝑦𝑑𝑥 𝑎 Volume of revolution 𝑏 𝑉 = 𝜋 ∫ 𝑦 2 𝑑𝑥 Log laws 𝑎 Area between two curves A = top curve - bottom curve Volume between two curves A = (top curve)2 – (bottom curve)2 Approximating integrals 𝑙𝑛𝑒 2 = 2𝑙𝑛𝑒 = 2 𝑙𝑜𝑔𝑘𝑥 = 𝑙𝑜𝑔𝑘 + 𝑙𝑜𝑔𝑥 𝑘 𝑙𝑜𝑔 = 𝑙𝑜𝑔𝑘 − 𝑙𝑜𝑔𝑥 𝑥 𝑙𝑜𝑔𝑐 𝑏 𝑙𝑜𝑔𝑎 𝑏 = 𝑙𝑜𝑔𝑐 𝑎 2.5 Simpson’s Rule 𝒉 𝒂+𝒃 𝑨 = {𝒇(𝒂) + 𝟒 × 𝒇( + 𝒇(𝒃))} 𝟑 𝟐 2 1.5 1 Trapezium Rule 𝒉 𝑨 = (𝒚𝟎 + 𝒚𝒏 + 𝟐(𝒚𝟏 + 𝒚𝟐 … . +𝒚𝒏−𝟏 )) 𝟐 Logarithmic and Exponential Functions 0.5 0 0 2 4 6 8 10 Trigonometric Functions Exponential functions 𝑑 (𝑒 𝑥 ) 𝑑𝑥 𝐴𝑟𝑐 𝐿𝑒𝑛𝑔𝑡ℎ = 𝑟𝜃 = 𝑒𝑥 1 𝐴𝑟𝑒𝑎 𝑜𝑓𝑎 𝑠𝑒𝑐𝑡𝑜𝑟 = 𝑟 2 𝜃 2 y = Sin x Period = 2 Amplitude = 1 𝑑 𝑎𝑥 (𝑒 ) = 𝑎𝑒 𝑎𝑥 𝑑𝑥 𝑑 𝑓(𝑥) (𝑒 ) = 𝑓 ′ (𝑥)𝑒 𝑓(𝑥) 𝑑𝑥 8 1.5 7 6 1 5 0.5 4 0 3 -0.5 0 2 1 2 3 4 5 6 7 -1 1 -1.5 0 -4 -3 -2 -1 0 1 2 3 2 y = Cos x Period = 2 Amplitude = 1 Decay y = Ae-k 2.5 2 1.5 1.5 1 1 0.5 0.5 0 0 0.5 1 1.5 2 2.5 3 3.5 0 0 1 2 3 4 5 6 7 -0.5 Exponential Growth If the rate of change is proportional to P, ie dP/dt = kP Then P = Poekt Exponential Decay If dP/dt = -kP Then P = Poe-kt Where Po is the initial value of P k is the constant of proportionality P is the amount of quantity present at time t -1 -1.5 y = tan x Period = 𝑠𝑖𝑛𝑥 lim =1 𝑥→0 𝑥 Series and Applications Arithmetic Series 𝑛𝑡ℎ 𝑡𝑒𝑟𝑚 = 𝑎 + (𝑛 − 1)𝑑 𝑛 𝑆𝑛 = (𝑓𝑖𝑟𝑠𝑡 + 𝑙𝑎𝑠𝑡) 2 𝑛 𝑆𝑛 = (2𝑎 + (𝑛 − 1)𝑑) 2 Kinematics Displacement = x 𝑑𝑥 Velocity 𝑣 = 𝑥̇ = 𝑑𝑡 Acceleration 𝑎 = 𝑣 𝑑𝑣 𝑑𝑥 = 𝑥̈ = 𝑑2 𝑥 𝑑𝑡 2 Geometric Series 𝑛𝑡ℎ 𝑡𝑒𝑟𝑚 = 𝑎𝑟 𝑛−1 𝑎(𝑟 𝑛 − 1) 𝑆𝑛 = (𝑟 − 1) 𝑎 |𝑟| < 1 𝑠∞ = 1−𝑟 𝑥 = ∫ 𝑣𝑑𝑡 𝑣 = ∫ 𝑎𝑑𝑡 Exponential Growth and Decay If e = a, then = logea Growth y = aek Compound Interest A=P Superannuation If $P is invested at the beginning of each year in a superannuation fund earning interest at r% pa, the investment after n years will amount to T A1 = P A2 = P 3 Parabolas And so on, so that investment = A1 + A2… =P +P … (x-b)2 = 4a(y-c) where (b,c) is the vertex a is the focal length General Solutions of Trig Equations forms a geometric series with X = nπ+ (-1)nsin-1(k) a=P n = number of years X = 2nπ ± cos-1(k) and r = X = nπ + tan-1(k) Angle between two lines of slopes m1 and m2 Time payments A person borrows $P at r% per term, where the interest is compounded per term on the amount owing. If they pay off the loan in equal term instalments over n terms, their equal term instalment is M, where M= 𝑡𝑎𝑛∅ = | 𝑚1 − 𝑚2 | 1 + 𝑚1 𝑚2 Polynomials 𝛼 2 − 𝛽 2 = (𝛼 + 𝛽)2 − 4𝛼𝛽 𝛼 2 + 𝛽 2 = (𝛼 + 𝛽)2 − 2𝛼𝛽 Deriving the equation: An = P (rate)n – M (1 + rate + rate2…) After fully paid An = 0 Rearrange to find M, using (1 + rate + rate2…) as a geometric series. Probability Probability of an event occurring = The probability of two independent events A and B occurring is given by: 𝑃(𝐴 ∩ 𝐵) = 𝑃(𝐴) × 𝑃(𝐵) Sum and Difference of Two Cubes 𝑥 3 + 𝑦 3 = (𝑥 + 𝑦)(𝑥 2 − 𝑥𝑦 + 𝑦 2 ) 𝑥 3 − 𝑦 3 = (𝑥 − 𝑦)(𝑥 2 + 𝑥𝑦 + 𝑦 2 ) 4