Parametric Equations APCH

circle

x=h+rcost
y=k+rsint

horizontal ellipse

x=h+acost
y=k+bsint

vertical ellipse

x=h+bcost
y=k+asint

horizontal hyperbola

x=h+asect
y=k+btant

vertical hyperbola

x=h+btant
y=k+asect

cardioid (a to b)

|a| = |b|

limacon w/ loop (a to b)

|a| < |b|

limacon w/o loop (a to b)

|a| > |b|

rose curve # petals (r= asinkx)

if k is odd, k petals. if k is even, 2k petals.

conversions to and from polar and rectangular

x^2 + y^2 = r^2
tant = y/x
x = rcost
y = rsint

distance from center to directrix

a^2/c

p

focus to directrix

How to find angle between two vectors

cosθ = u•v / ||u|| ||v||

Properties of dot product

u•v = v•u
0•v = 0
u•(v+w) = u•v + u•w
v•v = ||v||²
c(v•w) = cv•w = v•cw

Equation of a sphere

(x - x₀)² + (y - y₀)² + (z - z₀)² = r²

Angles of 3D vectors

cosα = v₁/||v||
cosβ = v₂/||v||
cosγ = v₃/||v||

Unit vector for 3D

Multiplying complex numbers

(rcisα)(xcisβ) = rxcis(α+β)

De Moivre's Theorem for power

(rcisθ)^n = r^n(cisnθ)

De Moivre's Theorem for roots

(rcisθ)^1/n = r^1/n(cisθ+2πk/n) where k = 0 .... n-1

Roots of unity

The n nth roots of unity are the numbers, z, such that z^n = 1, where z = a +bi = rcisθ