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Polar Coordinates
Point (r, θ): r = distance from origin, θ = angle from positive x-axis. Multiple representations: (r,θ), (r,θ+2nπ), (−r,θ+π).
Conversions
Polar → Rect: x = r cosθ, y = r sinθ
Rect → Polar: r² = x²+y², tanθ = y/x
Common Polar Curves
r = a: circle | r = a cosθ, r = a sinθ: circles through origin
r = a ± a cosθ: cardioid | r = a ± b cosθ: limacon (loop if b>a)
r = a cos(nθ): rose (n petals odd, 2n petals even)
r² = a² cos(2θ): lemniscate
Complex Numbers in Trig Form
z = r(cosθ + i sinθ) where r = |z| = √(a²+b²), θ = arg(z)
Multiply: r₁r₂[cos(θ₁+θ₂)+isin(θ₁+θ₂)]
Divide: (r₁/r₂)[cos(θ₁−θ₂)+isin(θ₁−θ₂)]
DeMoivre's Theorem
[r(cosθ+isinθ)]n = rn(cos nθ + i sin nθ)
nth Roots
wk = r1/n[cos((θ+2kπ)/n)+isin((θ+2kπ)/n)], k=0,1,...,n−1
Every nonzero complex number has exactly n distinct nth roots.