Question

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

$1+i$

Answer 1

Let $z=1+i$


$\therefore \text{Modulus of}z$,

$|z|=|1+i|$

$\Rightarrow \left|z\right|=\sqrt{1^2+1^2}$

$\Rightarrow |z|=\sqrt{2}\dots 1$
Let $\alpha$ be acute angle 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 first quadrant,

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

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

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

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

Hence, Polar form of $z$

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

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

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