Trigonometry/Trigonometric Form of the Complex Number

$z=a+bi=r(\cos \varphi +i\sin \varphi )$

where

• i is the imaginary number $(i=\sqrt{-1})$
• the modulus $r=\mathrm{mod}(z)=|z|=\sqrt{a^2+b^2}$
• the argument $\varphi =\arg (z)$ 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 $r(\cos \varphi +i\sin \varphi )=r\mathrm{cis}\varphi$ and it is also the case that $r\mathrm{cis}\varphi =r{e}^{i\varphi }$ (provided that $\varphi$ is in radians). The latter identity is called Euler's formula.

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

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