Question

Represent the complex numbers in polar form
$1-i$

Answer 1

Given,

$z=1-i$

Polar form $r.\arg \left(z\right)=r(\cos \theta +i\sin \theta )$

where $r=\sqrt{x^2+y^2}$ where $z=x+iy$

$\therefore r=\sqrt{1^2+(-1{)}^2}=\sqrt{1+1}=\sqrt{2}$

$\therefore 1=\sqrt{2}\cos \theta -1=\sqrt{2}\sin \theta$

$\cos \theta =\frac{1}{\sqrt{2}}\sin \theta =\frac{-1}{\sqrt{2}}$

$\therefore \theta$ lies in $4^{th}$ quad where $\cos$ is $+ve$

and $\sin$ is $-ve$

$\therefore \arg =-\theta$

$=-{45}^o=\frac{-\pi }{4}$

Hence $r=2;\theta .\frac{-\pi }{4}$

Polar form of $1-i=\sqrt{2}(\left(\frac{-\pi }{4}\right)+i\sin \left(\frac{-\pi }{4}\right))$