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} $