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de Moivre identity
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(Theorem)
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From the Euler relation
\begin{equation*} e^{i\theta} = \cos \theta + i \sin \theta \end{equation*} it follows that
where $n\in\mathbb{Z}$ . This is called de Moivre's formula, and besides being generally useful, it's a convenient way to remember double- (and higher-multiple-) angle formulas. For example,
\begin{equation*} \cos 2 \theta + i \sin 2 \theta = (\cos \theta + i \sin \theta)^2 = \cos^2 \theta + 2 i \sin \theta \cos \theta - \sin^2 \theta. \end{equation*} Since the imaginary parts and real parts on each side must be equal, we must have \begin{equation*} \cos 2 \theta = \cos^2 \theta - \sin^2 \theta \end{equation*}and \begin{equation*} \sin 2 \theta = 2 \sin \theta \cos \theta. \end{equation*}
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"de Moivre identity" is owned by Daume. [ full author list (3) | owner history (2) ]
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Cross-references: side, real parts, imaginary parts, formulas, angle, Euler relation
There are 10 references to this entry.
This is version 8 of de Moivre identity, born on 2002-02-16, modified 2005-06-15.
Object id is 1994, canonical name is DeMoivreIdentity.
Accessed 17228 times total.
Classification:
| AMS MSC: | 12E10 (Field theory and polynomials :: General field theory :: Special polynomials) |
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Pending Errata and Addenda
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