View source for De Moivre's Theorem

'''De Moivre's Theorem''' is a very useful theorem in the mathematical fields of [[complex numbers]]. It allows complex numbers in [[polar form]] to be easily raised to certain powers. It states that for <math>x\in\mathbb{R}</math> and <math>n\in\mathbb{Z}</math>, <math>\left(\cos x+i\sin x\right)^n=\cos(nx)+i\sin(nx)</math>.

== Proof ==
This is one proof of de Moivre's theorem by [[induction]].

*If <math>n\ge0</math>:

:If <math>n=0</math>, the formula holds true because <math>\cos(0x)+i\sin(0x)=1+0i=1=z^0.</math>

:Assume the formula is true for <math>n=k</math>. Now, consider <math>n=k+1</math>:

<cmath>\begin{align*}
(\cos x+i \sin x)^{k+1} & =(\cos x+i \sin x)^{k}(\cos x+i \sin x) & \text { by Exponential laws } \\
& =[\cos (k x)+i \sin (k x)](\cos x+i \sin x) & \text { by our assumption } \\
& =\cos (k x) \cos x-\sin (k x) \sin x+i[\cos (k x) \sin x+\sin (k x) \cos x] & \\
& =\operatorname{cis}((k+1)(x)) & \text { by various trigonometric identities }
\end{align*}</cmath>

:Therefore, the result is true for all nonnegative integers <math>n</math>.

*If <math>n<0</math>, one must consider <math>n=-m</math> when <math>m</math> is a positive integer.

<cmath>\begin{align*}
(\operatorname{cis} x)^{n} &=(\operatorname{cis} x)^{-m}  \\
&=\frac{1}{(\operatorname{cis} x)^{m}}  \\
&=\frac{1}{\operatorname{cis}(m x)}  \\
&=\cos (m x)-i \sin (m x) & \text { rationalization of the denominator } \\
&=\operatorname{cis}(-m x)  \\
&=\operatorname{cis}(n x) 
\end{align*}</cmath>

And thus, the formula proves true for all integral values of <math>n</math>. <math>\blacksquare</math>

==Generalization==

Note that from the functional equation <math>f(x)^n = f(nx)</math> where <math>f(x) = \cos x + i\sin x</math>, we see that <math>f(x)</math> behaves like an exponential function. Indeed, [[Euler's identity]] states that <math>e^{ix} = \cos x+i\sin x</math>. This extends de Moivre's theorem to all <math>n\in \mathbb{R}</math>.

==See Also==
[[Category:Theorems]]
[[Category:Complex numbers]]