Question

If $z_1$ and $z_2$ are two $nth$ roots of unity, then $arg\left(\frac{z_1}{z_2}\right)$ is a multiple of

• A. $n\pi$
• B. $\frac{3\pi }{n}$
• C. $\frac{2\pi }{n}$
• D. None of these.

Answer 1

The correct option is C $\frac{2\pi }{n}$

We have $z_1=e^{i\frac{2r\pi }{n}}$ and $z_2=e^{i\frac{2s\pi }{n}}$,
where $r$ and $s$ are integers lying between $0$ and $n$, including $0$ and $n$
$\therefore$ $arg\left(\frac{z_1}{z_2}\right)=\frac{2r\pi }{n}-\frac{2s\pi }{n}=2(r-s)\frac{\pi }{n}$,
Clearly, $arg\left(\frac{z_1}{z_2}\right)$ is a multiple of $\frac{2\pi }{n}$