# De Moivre's formula

## Theorem: (cos x + i sin x)^n = cos nx + i sin nx / From Wikipedia, the free encyclopedia

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In mathematics, **de Moivre's formula ** (also known as **de Moivre's theorem** and **de Moivre's identity**) states that for any real number x and integer n it holds that

- ${\big (}\cos x+i\sin x{\big )}^{n}=\cos nx+i\sin nx,$

where i is the imaginary unit (*i*^{2} = −1). The formula is named after Abraham de Moivre, although he never stated it in his works.[1] The expression cos *x* + *i* sin *x* is sometimes abbreviated to cis *x*.

The formula is important because it connects complex numbers and trigonometry. 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*.

As written, the formula is not valid for non-integer powers n. However, there are generalizations of this formula valid for other exponents. These can be used to give explicit expressions for the nth roots of unity, that is, complex numbers z such that *z ^{n}* = 1.

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