Question

The argument of $\frac{1-i\sqrt{3}}{1+i\sqrt{3}}$ is

• A. $210^{\circ}$
• B. $180^{\circ}$
• C. $240^{\circ}$
• D. $60^{\circ}$

Answer 1

Answer: (C)

Let $z=\frac{1-i\sqrt{3}}{1+i\sqrt{3}}=\frac{1-i\sqrt{3}}{1+i\sqrt{3}}\times\frac{1-i\sqrt{3}}{1-i\sqrt{3}}$
$=\frac{1-3-2\sqrt{3}\,i}{1+3}$
$=\frac{-2-2\sqrt{3}\,i}{4}$
$=\frac{-1}{2}-\frac{\sqrt{3}}{2} i$
$\because\, x< 0 ,y< 0$
$\therefore \,$ $z$ lies in $3^{rd}$ Quadrant
$\therefore \, tan\,\theta=\frac{-\sqrt{3}/2}{-1/ 2}=\sqrt{3}$
$=tan\, 60^{\circ}$
$=tan \left(180^{\circ}+60^{\circ}\right)$
=tan $240^{\circ}$
$\Rightarrow \, \theta=240^{\circ}$