de Moivre's theorem

A theorem, named after Abraham de Moivre, that links complex numbers and trigonometry. It states that for any real number x and any integer n,

(cos x + isin x)n = cos(nx) + isin(nx)
By expanding the left hand side and then comparing real and imaginary parts, it is possible to derive useful expressions for cos(nx) and sin(nx) in terms of sin(x) and cos(x). Furthermore, the formula can be used to find explicit expressions for the nth roots of unity: complex numbers z such that zⁿ = 1. It can be derived from (but historically preceded) Euler's formula e^ix = cos x + i sin x and the exponential law (e^ix)ⁿ = e^inx.

Related category

• COMPLEX NUMBERS

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