Precalc

Coordinate Conversions: Rectangular → Polar

r=sqrt(x²+y²), θ=tan^1(y/x) w/ adjustment for quadrant

Coordinate Conversions: Polar → Rectangular

x=rcosθ, y=rsinθ

Equation Conversions between polar and rectangular

r²=x²+y², rcosθ=x, rsinθ=y, tanθ=y/x

Graph of Circle

r=acosθ, r=asinθ

cos vs sin

cos starts at a max, sin starts at 0

Rose Curves Equations

r=asin(nθ), r=acos(nθ)

Rose Curves Petals

if n is even, 2n petals. If n is odd, n petals

Limacons Equations

r=a+bsinθ, r=a-bsinθ, r=a+bcosθ, r=a-bcosθ

Limacon a/b<1

inner loop

Limacon a/b = 1

heart (cardioid)

Limacon 1<a/b<2

dimpled

Limacon a/b>2

no dimple nor loop

Limacon orientation + vs - for sin

+ is up, - is down

Limacon orientation + vs - for cos

+ is right, - is left

Lemniscates

r²=a²sin2θ, r²=a²cos2θ

Parametric Equations, how to eliminate a parameter

solve for t, put into other equation

Parametric Equations, how to eliminate a parameter with sin/cos

treat like a system of equations, add them together, use Pythagorean identity

Parameterizing a Line given P=(x1,y1), Q=(x2,y2)

x=(x2-x1)T+x1

y=(y2-y1)T+y1

Parameterizing a Circle w/ center (h,k)

x=rcosT+h, y=rsinT+k

Standard circle equation

(x-h)²+(y-k)²=r²

Modeling Linear/horizontal motion

x=vcosθ*T, v=initial velocity, T=time

Modeling vertical motion

y=-16T²+(vsinθ)T+h, h=initial height, v=initial velocity, T=time