Question

If $\omega$ is a complex cube root of unity, then the value of $\sin \left\{({\omega }^{10}+{\omega }^{23})\pi -\frac{\pi }{6}\right\}$ is

• A. $\frac{1}{\sqrt{2}}$
• B. $\frac{\sqrt{3}}{2}$
• C. $-\frac{1}{\sqrt{2}}$
• D. $-\frac{\sqrt{3}}{2}$
• E. $\frac{1}{2}$

Answer 1

Answer: (E)

Since, $\omega$ is a complex cube root of unity.
$\therefore$ ${\omega }^{10}+{\omega }^{23}={({\omega }^3)}^3\omega +{({\omega }^3)}^7{\omega }^2$
$=\omega +{\omega }^2=-1$
$\therefore$ $\sin \left\{({\omega }^{10}+{\omega }^{23})\pi -\frac{\pi }{6}\right\}$
$=\sin \left(-\pi -\frac{\pi }{6}\right)=-\sin \left(\pi +\frac{\pi }{6}\right)$
$=\sin \frac{\pi }{6}=\frac{1}{2}$