A) \[\frac{1}{\sqrt{2}}\]
B) \[\frac{\sqrt{3}}{2}\]
C) \[-\frac{1}{\sqrt{2}}\]
D) \[-\frac{\sqrt{3}}{2}\]
E) \[\frac{1}{2}\]
Correct Answer: E
Solution :
Since, \[\omega \] is a complex cube root of unity. \[\therefore \]\[{{\omega }^{10}}+{{\omega }^{23}}={{({{\omega }^{3}})}^{3}}\omega +{{({{\omega }^{3}})}^{7}}{{\omega }^{2}}\] \[=\omega +{{\omega }^{2}}=-1\] \[\therefore \] \[\sin \left\{ ({{\omega }^{10}}+{{\omega }^{23}})\pi -\frac{\pi }{6} \right\}\] \[=\sin \left( -\pi -\frac{\pi }{6} \right)=-\sin \left( \pi +\frac{\pi }{6} \right)\] \[=\sin \frac{\pi }{6}=\frac{1}{2}\]
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