**De Moivre's Theorem is a formula that is used to calculate the power of complex numbers.** The formula of the theorem states that any real number x and any integer n that (cosx + isinx) to the nth power would equal cos (nx) + isin(nx).

De Moivre's Theorem is important due to its ability to connect complex numbers and trigonometry. Through expanding the left hand side and then comparing both the real and imaginary parts while assuming that x is real, it is possible to get an expression for cos(nx) and sin(nx).

Although it is a theorem, De Moivre's formula can be proven mathematically through induction for natural numbers and extended to all integers from that induction. There is a single limitation in the theorem - it does not work for non-integer powers.

When an equation has a complex number that is taken to a non-integer power, De Moivre's theorem cannot be used due to the fact that the solution can have many different values. Despite this shortcoming, it is generally accepted that the right-hand side of an expression that meets these standard can still be a possible value of the power.

De Moivre's theorem and the formula that accompanies it is an integral part of trigonometry.