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All rights reserved.  \nNo part of this publication may be reproduced by any means without the prior permission of the New Zealand Qualifications Authority.  \n2  \nMATHEMATICS – USEFUL FORMULAE  \nALGEBRA Quadratics  \nIf ax2 + bx + c = 0  \nthen x = −b ± b2 − 4ac  \n2a  \nLogarithms  \ny = logb x ⇔ x = by logb (xy) = logb x + logb y logb ⎜ ~~ ~~xy~~ ~~ ⎟ = logb x − logb y logb (xn ) = nlogb x logb x = log~~l~~ogaa~~ ~~xb  \nComplex numbers  \nz = x + iy  \n= r cisθ  \n= r(cosθ + isinθ )  \nz~~ ~~ = x − iy  \n= r cis (−θ )  \n= r(cosθ − isinθ )  \nr =  z = zz~~ ~~ = (x2 + y2 )  \nθ = arg z where cosθ = xrand sinθ = yr  \nDe Moivre’s Theorem  \nIf n is any integer, then (r cis θ )n = rn cis (nθ )  \nCOORDINATE GEOMETRY Straight Line  \nEquation y − y1 = m(x − x1 )  \nCALCULUS Differentiation  \n\n| y = f (x) | dydx = f ʹ(x) |\n| --- | --- |\n| ln x |  1\u003Cbr>x |\n| eax | ax ae |\n| sin x | cos x |\n| cos x | − sin x |\n| tan x | 2 sec x |\n| sec x | sec x tan x |\n| cosec x | −cosec x cotx |\n| cotx | 2 −cosec x |\n\nIntegration  \n\n| f (x) | ∫ f (x) dx |  |\n| --- | --- | --- |\n| xn\u003Cbr> 1\u003Cbr>x\u003Cbr>f ʹ(x)\u003Cbr>f (x) | n+1 x\u003Cbr>n + 1 | + c |\n|  | ( n ≠ −1) ln x  + c\u003Cbr>ln f (x) + c |  |\n\nParametric Function  \ndy  dy dt  \n= .  \ndx dt dx  \nd 2y d  ⎛ dy  ⎞ dtdx2 = dt ⎜⎝ dx ⎠⎟ . dx  \n3  \nProduct Rule  \n( f .g )′ = f .g′ + g.f ′ or if y = uv then  \nQuotient Rule  \n′  \n⎜ ~~ ~~fg~~ ~~ ⎟ = g.f~~ ~~′−g2f~~ ~~.g′ or if y = uv then  \nComposite Function or Chain Rule  \n′  \n( f ( g )) = f ′( g ) .g′  \nor if y = f (u) and u = g (x) then dy~~d~~x =  \ndy   dv   du dx = u dx + v dx  \ndu   dvdy = v dx − u dx dx v 2  \ndy dudu . dx  \nNUMERICAL METHODS  \nTrapezium Rule  \n∫ab f (x) dx ≈ 12 h y0 + yn + 2(y1 + y2 + ... + yn−1)  where h = b~~ ~~−n~~ ~~a andyr = f (xr )  \nSimpson’s Rule  \n∫ab f (x) dx ≈ 13h y0 + yn + 4(y1 + y3 + ... + yn−1) + 2(y2 + y4 + ... + yn−2)  where h = b~~ ~~−n~~ ~~a , yr = f (xr ) and n is even.  \n4  \nL 3–CAL CF  \nTRIGONOMETRY  \ncosec! = s~~i~~1n~~! ~~sec! = co1s~~! ~~cot! = ta1n~~!~~​  \ncot! = coss~~i~~n~~!~~​!  \nSine Rule  \na b c  \n~~ ~~ = ~~ ~~ = ~~  ~~sin A sinB sin C  \nCosine Rule  \nc2 = a2 + b2 ! 2ab cos C  \nIdentities  \ncos2 ! + sin2 ! = 1 tan2 ! + 1 = sec2 !  \ncot2 ! + 1 = cosec2 !  \nGeneral Solutions  \nIf sin ! = sin \" then ! = n! + (\"1)n \"If cos ! = cos \" then ! = 2n! ± \"If tan ! = tan \" then ! = n! + \"where n is any integer  \nCompound Angles  \nsin(A ± B) = sin AcosB ± cosAsin Bcos(A ± B) = cosAcos B! sin AsinB tan(A ± B) =  tan A ± tan B   \n1 ! tan AtanB  \nDouble Angles  \nsin 2 A = 2sin AcosA  \n2 tan Atan 2 A = ~~ ~~  \n1 ! tan2 Acos 2 A = cos2 A ! sin2 A  \n= 2cos2 A ! 1  \n= 1! 2sin2 A  \nProducts  \n2 sin AcosB = sin( A+ B) + sin (A − B)  \n2cosAsin B = sin( A+ B) − sin (A − B)  \n2cosAcos B = cos( A+ B) + cos( A − B)  \n2 sin Asin B = cos ( A − B) − cos( A+ B)  \nSums  \nsinC + sinD = 2sin ~~ ~~C~~ ~~+2~~ ~~D cos ~~ ~~C~~ ~~−2~~ ~~DsinC − sinD = 2cos ~~ ~~C~~ ~~+2~~ ~~D sinC~~ ~~−2~~ ~~DcosC + cosD = 2cos ~~ ~~C~~ ~~+2~~ ~~D cos ~~ ~~C~~ ~~−2~~ ~~DcosC − cosD = −2sin ~~ ~~C~~ ~~+2~~ ~~D sin ~~ ~~C~~ ~~−2~~ ~~D  \nMEASUREMENT  \nTriangle  \nArea = 12 absin C  \nTrapezium  \nArea = 12 (a + b)h  \nSector  \nArea = 12 r 2θ Arc length = rθ  \nCylinder  \nVolume = ! r 2 h  \nCurved surface area = 2! rh  \nCone  \nVolume = 13! r2 h  \nCurved surface area = ! rl where l = slant height  \nSphere  \nVolume = 43 ! r 3  \nSurface area = 4! r 2","cbCaitgzsMhlsZVs","https://ap.wps.com/l/cbCaitgzsMhlsZVs","pdf",894900,"English","# Mathematics – Useful Formulæ\n## Algebra: Quadratics, Logarithms, Complex numbers\n## Coordinate Geometry: Straight Line\n## Calculus: Differentiation and Integration\n## Numerical Methods: Trapezium Rule and Simpson’s Rule\n## Trigonometry: Sine Rule, Cosine Rule, Identities\n## Measurement: Triangle, Trapezium, Sector, Cylinder, Cone, Sphere","[{\"question\":\"What is this booklet used for during the exam?\",\"answer\":\"Use the booklet to answer questions in the related Question and Answer Booklets. Confirm the booklet has pages 2–4 in the correct order and none are blank.\"},{\"question\":\"Which calculus differentiation rules are included?\",\"answer\":\"It includes the product rule, quotient rule, and composite function (chain rule), along with a differentiation reference table for common functions.\"},{\"question\":\"What numerical integration methods are provided?\",\"answer\":\"It provides the trapezium rule and Simpson’s rule formulas, including the step size h and function values at points.\"}]","Level 3 Calculus 2022 - Formulas and Tables Booklet | PDF"]