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Mathematics LibreTexts

1.13: de Moivre's formula

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Moivre’s Formula

For positive integers n we have de Moivre’s formula:

(cos(θ)+isin(θ))n=cos(nθ)+isin(nθ)

Proof

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.


This page titled 1.13: de Moivre's formula is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Jeremy Orloff (MIT OpenCourseWare) via source content that was edited to the style and standards of the LibreTexts platform.

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