De Moivres formula

De Moivre's formula states that for any real number x and any integer n,

(cos x + i sin x)ⁿ = cos (nx) + i sin (nx).

The formula is important because it connects complex numbers (i stands for the imaginary unit) and trigonometry.

By expanding the left hand side and then comparing real and imaginary parts, it is possible to derive useful expressions for cos(nx) and sin(nx) in terms of sin(x) and cos(x).

De Moivre's formula was first proved by Roger Cotes; it can be derived from (but historically preceded) Euler's Theorem e^ix = cos x + i sin x and the exponential law (e^ix)ⁿ = e^inx.

The complex number e^ix = cos x + i sin x is oftentimes refered to as cis x for short.

De Moivre's formula is actually true for all complex numbers x and all real numbers n, but this requires careful extension of several functions to the complex plane. <-- Loisel I'm not so sure this makes any sense.