Question

The principal argument of the complex number $\frac{{(1+i)}^5{(1+\sqrt{3i})}^2}{-2i(-\sqrt{3}+i)}$ is

• A. $\frac{19\pi }{12}$
• B. $\frac{-7\pi }{12}$
• C. $\frac{-5\pi }{12}$
• D. $\frac{5\pi }{12}$

Answer 1

Answer: (C)

$\frac{-5\pi }{12}$

Let $z=\frac{(1+i{)}^5(1+\sqrt{3}{)}^2}{-2i(-\sqrt{3}+i)}$

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

$\Rightarrow z=\frac{8+8\sqrt{3}+i(8-8\sqrt{3})}{2+2\sqrt{3}i}$

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

$\Rightarrow arg\left(z\right)={\tan }^{-1}\left(\frac{-2-2\sqrt{3}}{-2+2\sqrt{3}}\right)$

$\Rightarrow arg\left(z\right)={\tan }^{-1}\left(\frac{1+\sqrt{3}}{1-\sqrt{3}}\right)$

$\Rightarrow arg\left(z\right)=\frac{-5\pi }{12}$