Question

The $|z|=4$ and $arg\left(z\right)=\frac{5\pi }{6}$, then $z=$ _______

Answer 1

Given that-

$|z|=4$ and $arg\left(z\right)=\frac{5\pi }{6}$

$\because z=|z|\left[\cos arg\right(z)+i\sin arg(z\left)\right]$

$\Rightarrow z=4[\cos \frac{5\pi }{6}+i\sin \frac{5\pi }{6}]$

$\Rightarrow z=4[\cos (\frac{\pi }{2}+\frac{\pi }{3})+i\sin (\frac{\pi }{2}+\frac{\pi }{3})]$

$\Rightarrow z=4[-\sin \frac{\pi }{3}+i\cos \frac{\pi }{3}]$

$\Rightarrow z=4[-\frac{\sqrt{3}}{2}+\frac{1}{2}i]$

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

The $|z|=4$ and $arg\left(z\right)=\frac{5\pi }{6}$, the $z=-2\sqrt{3}+2i$