In mathematics, de Moivre’s formula (also known as de Moivre’s theorem and de Moivre’s identity) states that for any real numberx and integern it holds that

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 cisx.

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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