Question

$\text{Find z,~if}|z|=4andarg\left(z\right)=\frac{5\pi }{6}$

Answer 1

$Letz=r(cos\theta +isin\theta )\text{Also,}|z|=r=4and\theta =\frac{5\pi }{6}[\because arg(z)=\theta ]\therefore z=4[cos\frac{5\pi }{6}+isin\frac{5\pi }{6}][\because z=r(cos\theta +isin\theta \left)\right]=4[cos(\pi -\frac{\pi }{6})+isin(\pi -\frac{\pi }{6})]=4[-cos\frac{\pi }{6}+isin\frac{\pi }{6}]=4[-\frac{\sqrt{3}}{2}+i\frac{1}{2}]=2\sqrt{3}+2i$