De Moivre's Theorem states that

(cos(x) + isin(x))^{n} = cos(nx) + isin(nx).

Here n and x are any complex number and

i is the square root of -1.

It is a simple consequence of

Euler's formula (exp(ix) = cos(x) + isin(x)) and the fact that exp(ab) = (exp(a))

^{b}. It may also be verified for the special case when n is a

natural number using

induction.

It is particularly useful for quick generation for

trigonometric identities involving cos(nx) and sin(nx). For example, when n=3,

cos(3x) + isin(3x) = (cos(x)+isin(x))^{3}
cos(3x) + isin(3x) = cos^{3}(x) + 3icos^{2}(x)sin(x)
- 3cos(x)sin^{2}(x) - isin^{3}(x).

Equating real and imaginary parts gives

cos(3x) = cos^{3}(x) - 3cos(x)sin^{2}(x)
sin(3x) = 3cos^{2}(x)sin(x) - sin^{3}(x).