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