In Mathematics, **de Moivre’s formula**, 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

(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. The expression cos *x* + *i* sin *x* is sometimes abbreviated **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 *n*th roots of unity, that is, complex numbers *z* such that *z ^{n}* = 1.

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