University⏱ 4 min

De Moivre's Theorem

(cosθ + i·sinθ)ⁿ = cos(nθ) + i·sin(nθ)

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zz^3
r = 1.2
θ = 30°
n = 3
(r(cos⁡ heta+isin⁡ heta))n=rn(cos⁡n heta+isin⁡n heta)(r(\cos\ heta + i\sin\ heta))^n = r^n(\cos n\ heta + i\sin n\ heta)
(1.2(cos⁡30°+isin⁡30°))3=1.728(cos⁡90°+isin⁡90°)(1.2(\cos 30° + i\sin 30°))^{3} = 1.728(\cos 90° + i\sin 90°)

De Moivre's theorem: the nth power = raise modulus to rⁿ and rotate n times. Each step is a blue dot. Green is the result.