eix = cosx + isinx
e^(i*x) = cos(x) + i*sin(x)
Where e is the base of the natural logarithm, i is defined as the square root of -1, and x is any real number.
de Moivre's formula gives the definition of e raised to any imaginary power ix, thus the restriction of x to be among the reals. It follows from the Taylor series for ex, cosx, and sinx. It gives rise to Euler's Relation, which when written eπi - 1 = 0 relates the five fundamental constants of mathematics: 0, 1, π, e, and i. Quite possibly the most beautiful single formula in mathematics.
de Moivre's formula can also be abbreviated using the cis function: eix = cisx
The following nodes contain more information about the derivation of de Moivre's formula: Euler's Relation, e ^ pi * i, and de Moivre's Theorem. Note that since it is a mathematical identity relation, it is not generally considered to be a theorem. The web page http://www.math.toronto.edu/mathnet/questionCorner/epii.html gives an excellent explanation for the less calculus inclined.