de Moivre's Theorem

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) = cos3(x) + 3icos2(x)sin(x)
                     - 3cos(x)sin2(x) - isin3(x).

Equating real and imaginary parts gives

cos(3x) = cos3(x) - 3cos(x)sin2(x)
sin(3x) = 3cos2(x)sin(x) - sin3(x).