Question:
If $|z|=2$ and $\arg (z)=\frac{\pi}{4}$, then $z=$
Solution:
for $|z|=2=r \arg z=\frac{\pi}{4}$
$z=r(\cos (\arg z)+i \sin (\arg z))$
i. e $z=2\left(\cos \frac{\pi}{4}+i \sin \frac{\pi}{4}\right)$
$=2\left(\frac{1}{\sqrt{2}}+i \frac{1}{\sqrt{2}}\right)$
$z=\sqrt{2}+i \sqrt{2}$
hence, $z=\sqrt{2}(1+i)$