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

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