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 $\phi =\arg(z)$ 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 $\phi$ 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.