Question

If $|z|=4$, $argz=\frac{5\pi }{6}$, then $z=$

• A. $\sqrt{3}+i$
• B. $-\sqrt{3}+i$
• C. $2(\sqrt{3}+i)$
• D. $2(-\sqrt{3}+i)$

Answer 1

The correct option is D $2(-\sqrt{3}+i)$

For any complex number
$z=|z|e^{iarg\left(z\right)}$
Here $|z|=4$ and $arg\left(z\right)=\frac{5\pi }{6}$.
Hence, $z=|z|e^{iarg\left(z\right)}$
$=4e^{i\frac{5\pi }{6}}$
$=4(\cos \left(\frac{5\pi }{6}\right)+i\sin \left(\frac{5\pi }{6}\right))$
$=4\left(\frac{-\sqrt{3}+i}{2}\right)$
$=2(-\sqrt{3}+i)$