Math135 Past Paper Summary | Macquarie University

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1. Learnyourunicourseinoneday.Checkspoonfeedme.com forfreevideosummaries,notesandcheatsheetsbytopstudents. CHEATSHEET MATH135 Mathematics 1A Macquarie University 1 Trigonometry Pythagorean Identity cos2 x + sin2 x = 1 Angle Sum Formulae sin(θ + φ) = sin θ cos φ + sin φ cos θ cos(θ + φ) = cos θ cos φ − sin θ sin φ tan(θ + φ) = tan θ + tan φ 1 − tan θ tan φ Double Angle Formula sin 2θ = 2 sin θ cos θ = 2 tan θ 1 + tan2 θ cos 2θ = cos2 θ − sin2 θ = 2 cos2 θ − 1 = 1 − 2 sin2 θ tan 2θ = 2 tan θ 1 − tan2 θ t-Formulae t = tan x 2 sin x = 2t 1 + t2 cos x = 1 − t2 1 + t2 tan x = 2t 1 − t2 Arc Length L = rθ (θ in radians) Cosine Rule a2 = b2 + c2 − 2bc cos A Sine Rule a sin A = b sin B = c sin C 2 Inverse Trigonometric Functions −1 −0.5 0.5 1 −π − π 2 π 2 π y = arcsin x −1 −0.5 0.5 1 π 2 π 3π 2 2π y = arccos x −3 −2 −1 1 2 3 − π 2 π 2 y = arctan x 3 Complex Numbers z = a + ib Where: i2 = −1 Modulus |z| = a2 + b2 Conjugate If z = a + ib then z = z − ib Properties of Modulus and Conjugate |zw| = |z||w| z w |z| |w| |z + w| ≤ |z| + |w| |z − w| ≥ |z| − |w| 1 2. Learnyourunicourseinoneday.Checkspoonfeedme.com forfreevideosummaries,notesandcheatsheetsbytopstudents. CHEATSHEET z ± w = z ± w zw = zw Triangle Inequality |z + w| ≤ |z| + |w| Polar Form z = rcisθ = r(cos θ + i sin θ) Where r is the modulus ||z|| and θ the argument arg z. Polar Form Operations and Properties rcis(θ) × pcis(φ) = rpcis(θ + φ) rcis(θ) pcis(φ) = r p cis(θ − φ) arg(zw) = arg(z) + arg(w) arg z w = arg(z) − arg(w) Euler’s Formula eix = cos x + i sin x Where e is Euler’s Number and x is real. De Moivre’s Theorem Given z = reiθ : zn = rn einθ 4 Vectors Length of a vector |v| = x2 + y2 + z2 Dot Product v · w = vxwx + vywy + vzwz = |v||w|cosθ Cross Product v × w = ˆi ˆj ˆk vx vy vz wx wy wz Angle Between Vectors cos θ = a · b a b Scalar projection v in direction of w vw = v · w |w| Vector projection v in direction of w vw = v · w |w|2 w Scalar Triple [u, v, w] = u · (v × w) v × w v u w V = [u, v, w] Volume of a parallelpiped 5 Lines in 3D Parametric Equation of a Line    x(t) = a + pt y(t) = b + qt z(t) = c + rt    Symmetric Form of Line Equation x − a p = y − b q = z − c r Vector Equation of a Line r(t) = d + tv 6 Planes in 3D Cartesian Equation of a Plane ax + by + cz = d 2 3. Learnyourunicourseinoneday.Checkspoonfeedme.com forfreevideosummaries,notesandcheatsheetsbytopstudents. CHEATSHEET Parametric Equation of a Plane    x(u, v) = a + pu + lv y(u, v) = b + qu + mv z(u, v) = c + ru + nv    Vector Equation of a Plane n · (r − d) = 0 7 Matrices Matrix Multiplication AB = a b c x y z   α ρ β σ γ τ   = aα + bβ + cγ aρ + bσ + cτ xα + yβ + zγ xρ + yσ + zτ Kronecker Delta δi,j = 1 if i = j 0 if i = j Determinants 2x2 Matrix det a b c d = a b c d = ad − cb 3x3 Matrix a b c d e f g h i = a e f h i − b d f g i + c d e g h 8 Partial Fractions: Important Rules Irreducible Quadratic Q(x) in Denominator: Factor on RHS is: Ax + B Q(x) Multiple Root (x + a)n in Denominator: Include on RHS: A1 x + a + A2 (x + a)2 + A3 (x + a)3 + ... + An (x + a)n 3 4. Learnyourunicourseinoneday.Checkspoonfeedme.com forfreevideosummaries,notesandcheatsheetsbytopstudents. CHEATSHEET