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.