Question

Argument of the complex number $\frac{1}{(\sqrt{3}-i{)}^{25}}$ is

• A. $\frac{\pi }{6}$
• B. $\frac{\pi }{4}$
• C. $\frac{\pi }{3}$
• D. $\frac{\pi }{2}$

Answer 1

The correct option is A $\frac{\pi }{6}$

Let $z=\frac{1}{(\sqrt{3}-i{)}^{25}}$
Then
$\arg \left(z\right)=\arg \left(1\right)-\arg \left(\right(\sqrt{3}-i{)}^{25})+2k\pi ,k\in Z\Rightarrow \arg (z)=0-25\arg (\sqrt{3}-i)+2k\pi ,k\in Z\Rightarrow \arg (z)=0-25(-\frac{\pi }{6})+2k\pi ,k\in Z\Rightarrow \arg (z)=25\times \frac{\pi }{6}+2k\pi ,k\in Z$
If $k=-2$
$\arg \left(z\right)=\frac{\pi }{6}\in (-\pi ,\pi ]$