De Moivre's Formula

De Moivre's Formula

In mathematics, de Moivre's formula (a.k.a. De Moivre's theorem and De Moivre's identity), named after Abraham de Moivre, states that for any complex number (and, in particular, for any real number) x and integer n it holds that

While the formula was named after De Moivre, he never explicitly stated it in his works.

The formula is important because it connects complex numbers (i stands for the imaginary unit (i2 = −1.)) and trigonometry. The expression cos x + i sin x is sometimes abbreviated to cis x.

By expanding the left hand side and then comparing the real and imaginary parts under the assumption that x is real, it is possible to derive useful expressions for cos(nx) and sin(nx) in terms of cos x and sin x. Furthermore, one can use a generalization of this formula to find explicit expressions for the nth roots of unity, that is, complex numbers z such that zn = 1.