Question

If $z_1$ and $z_2$ are two $n^{\text{th}}$ 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. $\frac{\pi }{n}$

Answer 1

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

We have, $z_1=e^{i\frac{2k\pi }{n}}$ and $z_2=e^{i\frac{2m\pi }{n}}$ where $k$ and $m$ are integers lying between $0$ and $(n-1)$, including $0$ and $(n-1)$ and $k\ne m$.
$\therefore \arg \left(\frac{z_1}{z_2}\right)=\frac{2k\pi }{n}-\frac{2m\pi }{n}$
$=\frac{2\pi }{n}(k-m)$
Clearly, $\arg \left(\frac{z_1}{z_2}\right)$ is a multiple of $\frac{2\pi }{n}$