Question

Find the modulus and argument of mentioned below complex number and hence express in polar form:

$1-i$

Answer 1

Let $z=1-i$

$\therefore$ Modulus of $z$,

$|z|=|1-i|$

$\Rightarrow |z|=\sqrt{(1{)}^2+(-1{)}^2}$

$\Rightarrow |z|=\sqrt{2}\dots \left(1\right)$

Let $\alpha$ be acuteangle such that

$\tan \alpha =\frac{|\text{Im}\left(z\right)|}{|\text{Re}\left(z\right)|}$

$\Rightarrow \tan \alpha =\frac{|-1|}{|1|}$

$\Rightarrow \tan \alpha =1$

$\Rightarrow \alpha =\frac{\pi }{4}$

Since, $z$ lies in the $4^{th}$ quadrant,

$\therefore arg\left(z\right)=-\alpha$

$arg\left(z\right)=\frac{\pi }{4}$


Hence, Polar form of $z$

$z=|z|\left[\cos arg\right(z)+i\sin arg(z\left)\right]$

$\Rightarrow z=\sqrt{2}[\cos (-\frac{\pi }{4})+i\sin (-\frac{\pi }{4})]$

$\Rightarrow z=\sqrt{2}(\cos \frac{\pi }{4}-i\sin \frac{\pi }{4})$

Therefore, Polar form of $(1-i)$ is $\sqrt{2}(\cos \frac{\pi }{4}-i\sin \frac{\pi }{4})$