Question:
Find $z$, if $|z|=4$ and $\arg (z)=\frac{5 \pi}{6}$.
Solution:
We know that,
$z=|z|\{\cos [\arg (z)]+i \sin [\arg (z)]\}$
$\Rightarrow z=4\left(\cos \frac{5 \pi}{6}+i \sin \frac{5 \pi}{6}\right)$
$=4\left(-\cos \frac{\pi}{6}+i \sin \frac{\pi}{6}\right)$
$=4\left(-\frac{\sqrt{3}}{2}+\frac{1}{2} i\right)$
$=-2 \sqrt{3}+2 i$
Thus, $z=-2 \sqrt{3}+2 i$.