Question

If $z_k=\cos \left(\frac{k \pi}{10}\right)+i \sin \left(\frac{k \pi}{10}\right)$, then $z_1 z_2 z_3 z_4$ is equal to

• A. -1
• B. 2
• C. -2
• D. 1

Answer 1

Answer: (A)

We have $z_k=\omega^k$ where
$\omega=\cos \left(\frac{\pi}{10}\right)+i \sin \left(\frac{\pi}{10}\right)$
Thus, $z_1 z_2 z_3 z_4=\omega \cdot \omega^2 \cdot \omega^3 \cdot \omega^4=\omega^{10}$
$=\cos \left(\frac{10 \pi}{10}\right)+i \sin \left(\frac{10 \pi}{10}\right)$
[by the De Movire's Theorem]
$=\cos \pi+i \sin \pi=-1$