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
$\therefore tan\theta =\frac{-\sqrt{3}/2}{-1/2}=\sqrt{3}$
$=tan60^{\circ }$
$=tan\left(180^{\circ }+60^{\circ }\right)$
=tan $240^{\circ }$
$\Rightarrow \theta =240^{\circ }$