PlanetMath: de Moivre identity

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de Moivre identity

(Theorem)

From the Euler relation

$\displaystyle e^{i\theta} = \cos \theta + i \sin \theta$

it follows that

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

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,

$\displaystyle \cos 2 \theta + i \sin 2 \theta = (\cos \theta + i \sin \theta)^2 = \cos^2 \theta + 2 i \sin \theta \cos \theta - \sin^2 \theta.$

Since the imaginary parts and real parts on each side must be equal, we must have

$\displaystyle \cos 2 \theta = \cos^2 \theta - \sin^2 \theta$

and

$\displaystyle \sin 2 \theta = 2 \sin \theta \cos \theta.$

"de Moivre identity" is owned by Daume. [ full author list (3) | owner history (2) ]

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Other names:

de Moivre's theorem, de Moivre's formula

Attachments:: proof of de Moivre identity (Proof) by rspuzio

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Cross-references: side, real parts, imaginary parts, angle, Euler relation
There are 7 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 14345 times total.

Classification:

AMS MSC:

12E10 (Field theory and polynomials :: General field theory :: Special polynomials)

Pending Errata and Addenda

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