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Mathematics LibreTexts

1.13: de Moivre's formula

  • Page ID
    47199
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    de Moivre’s Formula

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

    \[(\cos (\theta) + i \sin (\theta))^n = \cos (n \theta) + i \sin (n \theta)\]

    Proof

    This is a simple consequence of Euler’s formula:

    \((\cos (\theta) + i \sin (\theta))^n = (e^{i \theta})^n = e^{i n \theta} = \cos (n \theta) + i \sin (n \theta)\)

    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.