de Moivre’s formula

Portrait of Abraham de Moivre, French mathemat...
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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 nth roots of unity, that is, complex numbers z such that zn = 1.

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