Question

if $z_{\gamma }=\cos \frac{2\gamma \pi }{5}+i\sin \frac{2\gamma \pi }{5}=0$, $\gamma =0,1,2,3,\mathrm{4.....}$ then $z_1{z}_2{z}_3{z}_4{z}_5$ is equal to

• A. $-1$
• B. $0$
• C. $1$
• D. $noneofthese$

Answer 1

The correct option is C $1$

Given, $z_{\gamma }=\cos \frac{2\pi \gamma }{5}+i\sin \frac{2\pi \gamma }{5}$ where $\gamma =1,2,3,4,5.$
Thus, $z_{\gamma }=e^{(}i\frac{2\pi \gamma }{5})$
Thus, $z_1{z}_2{z}_3{z}_4{z}_5=e^{(}\frac{2\pi }{5}+\frac{4\pi }{5}+\frac{6\pi }{5}+\frac{8\pi }{5}+\frac{10\pi }{5})$
$=e^{6\pi }$
$=\cos \left(6\pi \right)+i\sin \left(6\pi \right)$
$=1$