Question

If $z_1$ and $z_2$ are two $n^{th}$ 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. None of the above

Answer 1

Answer: (C)

$\frac{2\pi }{n}$

$z={\left(1\right)}^{1/n}=(\cos \frac{2r\pi }{n}+i\sin \frac{2r\pi }{n})=e^{\frac{2r\pi }{n}i}$

$z_1=e^{\frac{2p\pi }{n}i},z_2=e^{\frac{2q\pi }{n}i}$

$\therefore \arg z_1=\frac{2p\pi }{n}$

$\therefore \arg \left(\frac{z_1}{z_2}\right)=\arg z_1-\arg z_2$

$=(p-q)\frac{2\pi }{n}=$ multiple of $\frac{2\pi }{n}$