Q. If $\omega$ is a complex cube root of unity, then $\sin\left\{ (\omega^{10} + \omega^{23}\pi - \frac{\pi}{4}) \right\}$ is equal to
A) $\frac{1}{\sqrt{2}}$
B) $\frac{1}{2}$
C) 1
D) $\frac{\sqrt{3}}{2}$
Explanation
(a)
$\sin\left\{ {(\omega}^{10} + \omega^{23})\pi - \frac{\pi}{4} \right\} = \sin\left\{ \left( \omega + \omega^{2} \right)\pi - \frac{\pi}{4}\ \right\}$
$= \sin\left( - \pi - \frac{\pi}{4} \right) = \sin{\frac{\pi}{4} = \frac{1}{\sqrt{2}}}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.