Question

The principal argument of the complex number

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

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

Answer 2

Sol.

$\frac{(1 + i)^{5}(1 + \sqrt{3}i)^{2}}{- 2i( - \sqrt{3} + i)} = \frac{(\sqrt{2})^{5}\left( \frac{1}{\sqrt{2}} + \frac{i}{\sqrt{2}} \right)^{5}.2^{2}\left( \frac{1}{2} + \frac{\sqrt{3}}{2}i \right)^{2}}{(2i)2\left( \frac{\sqrt{3}}{2} - \frac{i}{2} \right)}$

\ argument = $\frac{5\pi}{4} + \frac{2\pi}{3} - \frac{\pi}{2} + \frac{\pi}{6} = \frac{19\pi}{12}$

\ principal argument is $- \frac{5\pi}{12}$