*e*^{ix} = cos*x* + *i*sin*x*

`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 *e*^{x}, cos*x*, and sin*x*. It gives rise to Euler's Relation, which when written *e*^{πi} - 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: *e*^{ix} = cis*x*

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.