Question

If $z_r=\cos \left(r\frac{\pi }{10}\right)+i\sin \left(\frac{r\pi }{10}\right)$ then
$\prod _{r=1}^4{z}_r$ equals

• A. 0
• B. 1
• C. -1
• D. None of these

Answer 1

The correct option is C -1

$z_1{z}_2{z}_3{z}_4=\prod _{r=1}^4{z}_r$ $\therefore z_1{z}_2{z}_3{z}_4=(\cos \frac{\pi }{10}+i\sin \frac{\pi }{10})(\cos \frac{2\pi }{10}+i\sin \frac{2\pi }{10})$ $\times (\cos \frac{3\pi }{10}+i\sin \frac{3\pi }{10})(\cos \frac{4\pi }{10}+i\sin \frac{4\pi }{10})$
$\cos (\frac{\pi }{10}+\frac{2\pi }{10}+\frac{3\pi }{10}+\frac{4\pi }{10})+i\sin (\frac{\pi }{10}+\frac{2\pi }{10}+\frac{3\pi }{10}+\frac{4\pi }{10})=\cos \pi +i\sin \pi =-1$