Question

If z be a complex number such that $\left|z\right|=4$ and arg $\left(z\right)=\frac{5\pi }{6}$,then $z$ is equal to

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

Answer 1

The correct option is A $-2\sqrt{3}+2i,2\sqrt{3}-2i$

$|z|=4$

$arg\left(Z\right)=\frac{5\pi }{6},z=|z|(cos\frac{5\pi }{6}+isin\frac{5\pi }{6})$

$z=|z|\left(cos\right(\pi -\frac{\pi }{6})+isin(\pi -\frac{\pi }{6}\left)\right)$

$=|z|=(-cos\frac{\pi }{6}+isin\frac{\pi }{6})$

$=|z|(-\frac{\sqrt{3}}{2}+\frac{i}{2})$

$=4(-\frac{\sqrt{3}}{2}+\frac{i}{2})=-2\sqrt{3}+2i$