core pure

| z - z₁ | = | z - z₂ |

| z - z₁ | = | z - z₂ |

perpendicular bisector between z₁ and z₂

| z - z₁ | = r

circle with radius r and centre z₁

arg( z - z₁ ) = θ

half line drawn from z₁, angled at θ to the horizontal [don’t fill in circle?]

exponential form of the complex number r(cosθ +isinθ)

re^iθ

de moivre’s theorem

zⁿ = rⁿe^niθ

steps to express cosKθ in terms of powers of cosθ

• use de moivres theorem (cosθ + isinθ)^k = coskθ + isinkθ

• use binomial expansion on LHS

• compare real values to cosKθ

• use cos²θ + isin²θ = 1 as necessary to remove any sinθ terms

zⁿ - z^-n

2isin(nθ)

zⁿ + z^-n

2cos(nθ)

steps to solve zⁿ = ω

• find the exponential form of ω

• raise to the power of 1/n

• add or subtract 2π/n to the power (ensuring -π < arg ≤ π)

roots of unity 1, ω, ω², … , ω^n-1

• form vertices of an n-sided polygon

• sum of roots is zero

• multiply by ω to move anticlockwise to the next vertex

steps to handle calculations for roots of unity that are not centred at the origin

• subtract the centre of the polygon from all known coordinates (so that it is centred at origin)

• perform calculations

• add centre back to coordinates

Σk (from 1 to n)

kn

Σr (from 1 to n)

½ n (n + 1)

Σα

-b/a

Σαβ

c/a

Σαβγ

-d/a

Σαβγδ

e/a

1/α + 1/β

Σα/αβ

1/α + 1/β + 1/γ

Σαβ/αβγ

1/α + 1/β + 1/γ + 1/δ

Σαβγ/αβγδ

α² + β² + …

(Σα)² - 2Σαβ

α³ + β³

(Σα)³ -3Σαβ(Σα)

α³ + β³ + γ³

(Σα)³ -3Σαβ(Σα) + 3αβγ

volume of revolution 2π about the x-axis

π ∫ y² dx

volume of revolution 2π about the y-axis

π ∫ x² dy

volume of a cylinder

πr²h

volume of a cone

1/3 πr²h

AA^-1 = ?

I = A^-1A

(AB)^-1 = ?

B^-1A^-1

determinant of 2 × 2 matrix

ad - bc

determinant of 3 × 3 matrix

(a x minor of a) - (b x minor of b) + (c x minor of c)

singular matrix

• determinant is 0

• has no inverse

steps to find inverse of 2 × 2 matrix

• find determinant

• switch a and d terms

• negate b and c terms

• multiply new matrix by reciprocal of the determinant

steps to find inverse of 3 × 3 matrix

• find determinant

• find matrix of minors

• negate the four items [in the internal cross]

• transpose the matrix (swap the rows and columns)

• multiply new matrix by reciprocal of the determinant

geometric interpretation:

• detM ≠ 0

planes meet at a single point

geometric interpretation:

• detM = 0

• equations are consistent

• equations are not multiples of each other

planes form a sheaf

geometric interpretation:

• detM = 0

• equations are consistent

• equations are multiples of each other

planes are all the same

geometric interpretation:

• detM = 0

• equations are inconsistent

• no planes are parallel

planes form a prism

geometric interpretation:

• detM = 0

• equations are inconsistent

• some planes are parallel (equations have the same LHS but different RHS)

some/all plane are parallel

invariant point multiplied by matrix

the same point

point on an invariant line multiplied by matrix

some other point on the line

area scale factor when a shape is multiplied by the matrix M

detM

a.b

|a||b| cosθ

a.b (if a and b are perpendicular)

0

r.n

a.n ( =c )

steps to find intersection between 2 lines

• equate r₁ = r₂

• find λ and μ

• lines will be intersecting, parallel or skew

steps to find intersection between a line and a plane

• find scalar product form of plane

• subsitute r from the line into r.n = d and solve

steps to find the line of intersection between 2 planes

• find 2 points that are on both planes

• find the equation of the line between these points

steps to find the shortest distance between a point and a line

• find vector between the point and a general point on the line

• make this vector perpendicular to the line direction (using dot product)

steps to find the shortest distance between parallel lines

• find vector between 2 general points on the lines

• make this vector perpendicular to the line direction (using dot product)

steps to find the shortest distance between skew lines

• find vector between 2 general points on the lines

• make this vector perpendicular to each of the line directions (using dot product)

• solve equations simultaneously

steps to find the shortest distance between 2 parallel planes

• find any point on the plane

• use the formula for distance between a point and a plane

steps to find a point reflected in a plane

• find intersection of normal to the plane and the point

• double the vector between the point and the normal intersection to find the reflected point

steps to find a line reflected in a plane

• reflect any point on the line

• find the point of intersection between the plane and the line

• find the equation of the line between the point of intersection and the reflected point

how to prove two lines are on the same plane

prove they intersect or prove they are parallel

sinhx

½ (e^x - e^-x)

coshx

½ (e^x + e^-x)

tanhx

(e^2x - 1)/(e^2x + 1)

osbourne’s rule

hyperbolic identities are the same as trig identities but negate any sin² or implied sin²

polar form of x

rcosθ

polar form of y

rsinθ

polar form of x² + y²

r²

tangent parallel to the initial line

dy/dθ = 0

tangent perpendicular to the initial line

dx/dθ = 0

integrating factor

e ^{∫ p(x) dx}

amplitude and period of motion described by this expression: asin(ωt+α)

amplitude: a, period: 2π/ω

general solution of second-order non-homogeneous differential equation

y = C.F. + P.I.

C.F. and type of damping when A.E. has 2 real roots (α, β)

• Ae^αx + Be^βx

• heavy damping

C.F. and type of damping when A.E. has one repeated real root (α)

• (A+Bx)e^αx

• critical damping

C.F. and type of damping when A.E. has 2 complex roots (p ± qi)

• e^px(Acosqx + Bsinqx)

• simple harmonic motion if p = 0, else light damping