# Solve the following

Question:

If $|z|=2$ and $\arg (z)=\frac{\pi}{4}$, find $z$

Solution:

We know that,

$z=|z|\{\cos [\arg (z)]+i \sin [\arg (z)]\}$

$=2\left(\cos \frac{\pi}{4}+i \sin \frac{\pi}{4}\right)$

$=2\left(\frac{1}{\sqrt{2}}+i \frac{1}{\sqrt{2}}\right)$

$=\sqrt{2}(1+i)$

Hence, $z=\sqrt{2}(1+i)$