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Module 8: Polar & Complex Quick Ref
Polar ↔ Rectangular
x = r cosθ y = r sinθ
r² = x²+y² tanθ = y/x
Polar Curves
r=a: circle | θ=c: line
r=a±acosθ: cardioid
r=acos(nθ): rose
n odd: n petals | even: 2n
Complex Trig Form
z = r(cosθ+isinθ) r=|z|=√(a²+b²)
Multiply: r₁r₂cis(θ₁+θ₂) | Divide: (r₁/r₂)cis(θ₁−θ₂)
DeMoivre's
[rcisθ]
n
= r
n
cis(nθ)
nth Roots
w
k
=r
1/n
cis((θ+2kπ)/n)
k = 0, 1, ..., n−1
Multiple representations: (r,θ) = (r,θ+2nπ) = (−r,θ+π)