MATHEMATICS: SPECIALIST UNITS 3C AND 3D FORMULA SHEET 05 Copyright School Curriculum and Standards Authority, 05 This document apart from any third party copyright material contained in it may be freely copied, or communicated on an intranet, for non-commercial purposes by educational institutions, provided that it is not changed in any way and that the School Curriculum and Standards Authority is acknowledged as the copyright owner. Copying or communication for any other purpose can be done only within the terms of the Copyright Act or by permission of the Authority. Copying or communication of any third party copyright material contained in this document can be done only within the terms of the Copyright Act or by permission of the copyright owners. This document is valid for teaching and examining until 3 December 05. TRIM 04/46494 Mathematics: Specialist 3C and 3D Formula Sheet updated January 05
MATHEMATICS: SPECIALIST Formula SHEET Index Vectors 3 Trigonometry 3 Functions 4 Matrices 5 Complex numbers 6 Exponentials and logarithms 7 Mathematical reasoning 7 Measurement 8
Formula SHEET 3 MATHEMATICS: SPECIALIST Vectors Magnitude: Dot product: (a, a, a ) = a + a 3 + a 3 a b = a b cosθ = a b + a b + a 3 b 3 Triangle inequality: Vector equation of a line in space: a + b a + b one point and the slope: two points A and B: r = a + λb r = a + λ(b a) x a Cartesian equations of a line in space: y a = z a = 3 b b Parametric form of vector equation of a line in space: x = a + λb...() y = a + λb...() z = a 3 + λb 3...(3) Vector equation of a plane in space: r n = c or r = a + λb + µc b 3 Trigonometry In any triangle ABC: a b c sin A = sin B = sin C a = b + c bc cosa cosa = b + c a bc Area = ab sin C In a circle of radius r, for an arc subtending angle θ (radians) at the centre: Length of arc = rθ Area of segment = r (θ sinθ) Area of sector = r θ Identities: cos θ + sin θ = cos (θ ± φ) = cosθ cosφ ± sinθ sinφ sin (θ ± φ) = sinθ cosφ ± cosθ sinφ tan (θ ± φ) = tanθ ± tanφ tanθ tanφ ± cos θ = cos θ sin θ = cos θ = sin θ sinθ = sinθ cosθ tan θ = tanθ tan θ lim x 0 sin x = x lim x 0 cos x = 0 x Simple Harmonic Motion: If = k d x dt x then x = Asin(kt +α) or x = Acos(kt +β) and v = k (A x ), where A is the amplitude of the motion, α and β are phase angles, v is the velocity and x is the displacement.
MATHEMATICS: SPECIALIST 4 Formula SHEET Functions Differentiation: If f(x) = y then f'(x) = dy If f(x) = e x then f'(x) = e x If f(x) = sin x then f'(x) = cos x If f(x) = tan x then f'(x) = sec x = cos x If f(x) = x n then f'(x) = nx n If f(x) = ln x then f'(x) = x If f(x) = cos x then f'(x) = sin x Product rule: If y = f (x) g(x) then y' = f'(x) g(x) + f(x) g'(x) or If y = uv dy du then = v + u dv Quotient rule: f(x) If y = g(x) then y' = f'(x) g(x) f(x) g'(x) (g(x)) or If y = u v dy then = du dv v u v Incremental formula: δy ~ dy δx or f (x + h) f (x) ~ f '(x)h Chain rule: If y = f(g(x)) Integration: then y' = f'(g(x)) g'(x) or If y = f (u) and u = g(x) then dy = dy du du Powers: x n = x n+ n + + c, n = Exponentials: e x = e x + c Logarithms: x = ln x + c Trigonometric: sin x = cos x + c cos x = sin x + c cos x = tan x + c Fundamental Theorem of Calculus: d x a f(t)dt = f (x) and b a f'(x) = f (b) f (a)
Formula SHEET 5 MATHEMATICS: SPECIALIST Functions Quadratic function: If y = ax + bx + c and y = 0, then x = b ± b 4ac a for x C Piecewise-defined functions: Absolute value function: x = x, for x 0 x, for x < 0 Sign function: sgn (x) =, for x > 0 0, for x = 0, for x < 0 Greatest integer function: int (x) = greatest integer x for all x Matrices If A = a b c d then A = deta = ad bc d b A = (ad bc) c a Dilation = a 0 a 0 a Shear = or 0 0 a Rotation = cos θ sin θ sin θ cos θ Reflection = cos θ sin θ sin θ cos θ
MATHEMATICS: SPECIALIST 6 Formula SHEET Complex numbers For z = a + ib, where i = b Argument: arg z = θ, where tan θ = a and π < θ π Modulus: mod z = z = a + ib = a + b = r Product: z = z arg (z ) = arg z + arg Quotient: z = z z arg z = arg z arg Polar form: For z = r cisθ, where r = z and θ = arg z: cis(θ + φ) = cis θ cis φ cis( θ) = cis θ z = r r cis (θ + φ) cis θ = cos θ + i sin θ cis(0) = z r z = cis (θ φ) r Exponential form: z = re iθ, where r = z and θ = arg z For complex conjugates: z = a + bi z = r cis θ z = reiθ z z = z z = a bi z = r cis ( θ) z = re iθ z z = = z z z + = z + z = z
Formula SHEET 7 MATHEMATICS: SPECIALIST Exponentials and logarithms For a, b > 0 and m, n real: a m a n = a m+n a 0 = a m a n = a m n a n = a n (a m ) n = a mn (ab) m = a m b m For m an integer and n a positive integer: a n = n a a m n = n a m = ( n a) m For a, b, y, m and n positive real and k real: = a 0 log a = 0 y = a x log a y = x log a mn = log a m + log a n a = a log a a = log a m = log b m log b a (change of base) log a (m k ) = k log a m If dp dt = kp, then P = P 0 e kt Mathematical reasoning De Moivre's theorem: (cis θ) n = (cos θ + isin θ) n (cis θ) n = cos nθ + isin nθ z n = z n cis (nθ) q q θ + πk θ + πk z = z cos + isin for k an integer. q q
MATHEMATICS: SPECIALIST 8 Formula SHEET Measurement Circle: C = πr = πd, where C is the circumference, r is the radius and D is the diameter A = πr, where A is the area Triangle: A = bh, where b is the base and h is the perpendicular height Parallelogram: A = bh Trapezium: A = (a + b)h, where a and b are the lengths of the parallel sides Prism: V = Ah, where V is the volume and A is the area of the base Pyramid: V = Ah 3 Cylinder: S = πrh + πr, where S is the total surface area V = πr h Cone: S = πrs + πr, where s is the slant height V = 3 πr h Sphere: S = 4πr V = πr 3 4 3 Note: Any additional formulas identified by the examination panel as necessary will be included in the body of the particular question.