de Moivre's Theorem

Description:

• Let be a complex number, $z=r(\cos \theta +i\sin \theta )$
• Then $z^2=r^2(\cos \theta -\sin \theta +2i\cos \theta \sin \theta )=z^2=r^2(\cos (2\theta )+i\sin (2\theta ))$
  • by trigonometry law
• Keep powering, we have $z^n=\cos (n\theta )+i.\sin (n\theta )$
  • when $r=1$
• $(x+yi{)}^n=R^n.(\cos \theta +i\sin \theta )$