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Argument of $\frac{1-i \sqrt{3}}{1+i \sqrt{3}}$ is
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$240^{\circ}$
$z=\frac{1-i \sqrt{3}}{1+i \sqrt{3}} \times \frac{1-i \sqrt{3}}{1-i \sqrt{3}}=\frac{-2-2 \sqrt{3} i}{4}=-\frac{1}{2}-\frac{1}{2} \sqrt{3} i$
$\begin{aligned} & \text { Now Arg }(Z)=\pi+\tan ^{-1}\left|\frac{\operatorname{Im}(Z)}{\operatorname{Re}(Z)}\right| \\ & =180^{\circ}+\tan ^{-1}(\sqrt{3})=180^{\circ}+60^{\circ}=240^{\circ}\end{aligned}$
$\begin{aligned} & \text { Now Arg }(Z)=\pi+\tan ^{-1}\left|\frac{\operatorname{Im}(Z)}{\operatorname{Re}(Z)}\right| \\ & =180^{\circ}+\tan ^{-1}(\sqrt{3})=180^{\circ}+60^{\circ}=240^{\circ}\end{aligned}$
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