Polar Form

.# What is the polar form for the complex plane Complex Numbers

• what is the polar form of a complex number

  • $z=a+bi,⟹z=r(\cos \theta +i\sin \theta )$
    • where $r=|z|$ and $\theta$ is an argument of z satisfying $a=r\cos \theta ,b=r\sin \theta$

Multiplication in polar form

• let $z_1=r_1(\cos {\theta }_1+i\sin {\theta }_1)$ and $z_2=r_2(\cos {\theta }_2+i\sin {\theta }_2)$

  • $z_1{z}_2=r_1{r}_2(\cos ({\theta }_1+{\theta }_2)+i\sin ({\theta }_1+{\theta }_2))$

De Moivre’s formula

• let $z=r(\cos \theta +i\sin \theta )=/=0$ and n any integer

  • $z^n=r^n(\cos (n\theta )+i\sin (n\theta ))$