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Direct Proof of De Moivre's Theorem

In §2.10, De Moivre's theorem was introduced as a consequence of Euler's identity:

$\displaystyle \zbox {\left[\cos(\theta) + j \sin(\theta)\right] ^n =
\cos(n\theta) + j \sin(n\theta), \qquad\hbox{for all $n\in{\bf R}$}}

To provide some further insight into the ``mechanics'' of Euler's identity, we'll provide here a direct proof of De Moivre's theorem for integer $ n$ using mathematical induction and elementary trigonometric identities.

Proof: To establish the ``basis'' of our mathematical induction proof, we may simply observe that De Moivre's theorem is trivially true for $ n=1$. Now assume that De Moivre's theorem is true for some positive integer $ n$. Then we must show that this implies it is also true for $ n+1$, i.e.,

$\displaystyle \left[\cos(\theta) + j \sin(\theta)\right] ^{n+1} = \cos[(n+1)\theta] + j \sin[(n+1)\theta]. \protect$ (3.2)

Since it is true by hypothesis that

$\displaystyle \left[\cos(\theta) + j \sin(\theta)\right] ^n =
\cos(n\theta) + j \sin(n\theta),

multiplying both sides by $ [\cos(\theta) + j \sin(\theta)]$ yields
$\displaystyle \left[\cos(\theta) + j \sin(\theta)\right] ^{n+1}$ $\displaystyle =$ $\displaystyle \left[\cos(n\theta) + j \sin(n\theta)\right]
\left[\cos(\theta) + j \sin(\theta)\right]$  
  $\displaystyle =$ $\displaystyle \qquad\!
\left[\cos(n\theta)\cos(\theta) -\sin(n\theta)\sin(\theta)\right]$  
    $\displaystyle \,+\, j \left[\sin(n\theta)\cos(\theta)+\cos(n\theta)\sin(\theta)\right].
\protect$ (3.3)

From trigonometry, we have the following sum-of-angle identities:

\sin(\alpha+\beta) &=& \sin(\alpha)\cos(\beta) + \cos(\alpha)\...
...pha+\beta) &=& \cos(\alpha)\cos(\beta) - \sin(\alpha)\sin(\beta)

These identities can be proved using only arguments from classical geometry.3.8Applying these to the right-hand side of Eq.$ \,$(3.3), with $ \alpha=n\theta$ and $ \beta=\theta$, gives Eq.$ \,$(3.2), and so the induction step is proved. $ \Box$

De Moivre's theorem establishes that integer powers of $ [\cos(\theta) + j \sin(\theta)]$ lie on a circle of radius 1 (since $ \cos^2(\phi)+\sin^2(\phi)=1$, for all $ \phi\in[-\pi,\pi]$). It therefore can be used to determine all $ N$ of the $ N$th roots of unity (see §3.12 above). However, no definition of $ e$ emerges readily from De Moivre's theorem, nor does it establish a definition for imaginary exponents (which we defined using Taylor series expansion in §3.7 above).

Previous: Roots of Unity
Next: Euler_Identity Problems

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About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See for details.


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