Trigonometry/Trigonometric Form of the Complex Number

$z=a+bi= r \left ( \cos \phi \ + i\sin \phi \right )$

where

• i is the imaginary number $\left (i\ = \sqrt{-1}\right )$
• the modulus $r=\operatorname{mod}(z)=|z|=\sqrt{a^2+b^2}$
• the argument ϕ = arg(z) is the angle formed by the complex number on a polar graph with one real axis and one imaginary axis. This can be found using the right angle trigonometry for the trigonometric functions.

This is sometimes abbreviated as $r\left ( \cos \phi \ + i\sin \phi \right )=r\operatorname{cis}\phi$ and it is also the case that $r\operatorname{cis}\phi=re^{i\phi}$ (provided that ϕ is in radians). The latter identity is called Euler's formula.

Euler's formula can be used to prove DeMoivre's formula: $( \cos \phi \ + i\sin \phi )^n=\cos(n\phi)+i\sin(n\phi).$ This formula is valid for all values of n, real or complex.