Advanced Higher Maths Formulas Flashcards
To memorize the formulas
Trigonometric Identities
Link between ratios (1)
Includes Cos and Sin
cos^2A + sin^2A = 1
Trigonometric Identiteis
Link between ratios (2)
Includes Sin, Cos and Tan
TanA = SinA / CosA
Trigonometric Identities
Squared (1)
Includes Cos
Cos^2x = 1/2(1+cos2x)
Trigonometric Identities
Squared (2)
Includes Sin and Cos
Sin^2x = 1/2(1-cos2x)
Trigonometric Identities
Compound Angle (1)
For Cos
Cos(A(+/-)B) = CosACosB (-/+) SinASinB
Trigonometric Identities
Compound Angle (2)
For Sin
Sin(A(+/-)B) = SinACosB (+/-) SinBCosA
Trigonometric Identities
Double Angle (1)
For sin
Sin(2A) = 2SinACosA
Trigonometric Identities
Double Angle (2)
For Cos
Cos(2A) = cos^2A - sin^2A
Trigonometric Identities
Link between ratio (3)
Includes tan and sec
Sec^2A = 1+Tan^2A
Exact Values
Sin 0, Cos 0, Tan 0
0, 1, 0
Exact Values
Sin π/6, Cos π/6, Tan π/6
1/2, √3/2, 1/√3
Exact Values
Sin π/4, Cos π/4, Tan π/4
1/√2, 1/√2, 1
Exact Values
Sin π/3, Cos π/3, Tan π/3
√3/2, 1/2, √3
Exact Values
Sin π/2, Cos π/2, Tan π/2
1, 0, undefined
Exact Values
Sin π, Cos π, Tan π
0, -1, 0
Exact Values
Sin 3π/2, Cos 3π/2, Tan 3π/2
-1, 0, undefined
Exact Values
Sin 2π, Cos 2π, Tan 2π
0, 1, 0
Complex Numebers
Complex Number Formula
z = a + bi
Complex Numbers
Modulus
l z l = √a^2+b^2
Complex Numbers
Argument
tan θ = b/a
Complex Numbers
Conjugate
z^- = a - bi
Differentiation
Speed
Parametric Equations
Speed = √(dy/dt)^2 + (dx/dt)^2
Differentiation
Gradient (direction of movement)
Parametric Equations
dy/dx = (dy/dt)/(dx/dt)
Intergration
Volume of solid of revolution about x axis
V = π ∫ y^2 dx
Intergration
Volume of solid of rotation about y axis
V = π ∫ x^2 dy
Properties of Functions
Odd function
f(-x) = -f(x)
Properties of functions
Even function
f(-x) = f(x)
Sequences and Series
Arithmetic Sequence
Un = a + (n-1)d
Sequences and Series
Geometric Sequence
Un = ar^n-1
Important Identities
Sigma Notation
Sigma notation of 1
Σ1 = n
Maclaurin Series
e^x
e^x = 1+x+(x^2/2!)+(x^3)/3!)+…+(x^n/n!)+…
Maclaurin Series
Sin x
sin x = x - (x^3/3!) + (x^5/5!) - (x^7/7!) + …
Maclaurins Series
Cos x
Cos x = 1 - (x^2/2!) + (x^4/4!) - (x^6/6!) + …
Maclaurin Series
tan ^-1
(NOT AS IMPORTANT)
Tan^-1 = x - (x^3/3) + (x^5/5) - (x^7/7) + …
Maclaurin Series
In(1+x)
(NOT AS IMPORTANT)
In(1+x) = x - (x^2/2) + (x^3/3) - (x^4/4) +…
Vectors, Lines and Planes
Parametric form of x
x = x1 + at
Vectors, Lines and Planes
Parametric form of y
y = y1 + bt
Vectors, Lines and Planes
Parametric form of z
z = z1 + ct
Vectors, Lines and Planes
Overall parametric equation
x = a + td
Vectors, Lines and Planes
Symmetric form
((x-x1)/a) = ((y-y1)/b) = ((z-z1)/c) = t
Vectors, Lines and Planes
Vector equation
x.a = a.n
Vector, Lines and Planes
Symmetric/Cartesian
lx + my + nz = k (where k = a.n)
Vectors, Lines and Places
Angle between 2 lines
Acute angle between their direction vectors
Vectors, Lines and Planes
Angle between 2 planes
Acute angle between their normals
Vectors, Lines and Planes
Angle between line and plane
90 - (Acute angle between n and d)
Matricies
l a b l
l c d l
2 x 2 Matricies
det A = ad - bc
and
A^-1 = 1/(ad - bc)
Differential Equations
Intergrating factor
μ(x) = e^∫P(x) dx
Differential Equations
Main Equation
dy/dx + P(x)y = Q(x)
Differential Equations
Solution
μ(x)y = ∫ μ(x)Q(x) dx
Nature of Roots
Two real and distinct roots (m and n)
Form of general solution
y = Ae^mx + Be^nx
Nature of Roots
Real and equal roots (m)
Form of general solution
y = Ae^mx + Bxe^mx
Nature of Roots
Complex conjugate
(m = p(+/-) iq)
Form of general solution
y = e^px (Acosq + Bsinqx)
Particular Intergral
Sin(ax) or Cos(ax)
If right hand sign contains … try …
y = Pcos(ax) + Qsin(ax)
Particular Intergral
e^ax
If right hand sign contains … try …
y = Pe^ax
Particular Intergral
Linear expression y = ax + b
If right hand sign contains … try …
y = Px + Q
Particular Intergral
Quadratic expression
y = ax^2 + bx + c
If right hand sign contains … try …
y = Px^2 + Qx + R
