1.13: de Moivre's formula
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For positive integers n we have de Moivre’s formula:
(cos(θ)+isin(θ))n=cos(nθ)+isin(nθ)
- Proof
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This is a simple consequence of Euler’s formula:
(cos(θ)+isin(θ))n=(eiθ)n=einθ=cos(nθ)+isin(nθ)
The reason this simple fact has a name is that historically de Moivre stated it before Euler’s formula was known. Without Euler’s formula there is not such a simple proof.