De Moivre's formula

In mathematics, de Moivre's formula, named after Abraham de Moivre, states that for any complex number z and integer n,

[math]\displaystyle{ z^n = (|z|e^{i.\arg z})^n = (|z|(\cos \arg z + i \sin \arg z))^n = |z|^n (\cos (n \arg z) + i \sin (n \arg z)), }[/math]

where [math]\displaystyle{ |z| }[/math] is the modulus of z, and [math]\displaystyle{ \arg z }[/math] is the argument of z. Here, [math]\displaystyle{ e }[/math] is Euler's number, with [math]\displaystyle{ |z|e^{i.\arg z} }[/math] often being called the polar form of a complex number.

The formula is very important because it connects complex numbers and trigonometry. It can be easily proved using the trigonometry form of complex numbers, induction, and some trigonometrical identities. Furthermore, using this formula any equation of [math]\displaystyle{ z^n = w }[/math], where w is complex, can be solved.