Solve the following

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, $z=-2 \sqrt{3}+2 i$.

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