Trigonometry/Trigonometric Form of the Complex Number

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


  • 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.