Question

If $\omega$ is an imaginary cube root of unity, then the value of
$\sin \left\{\left({\omega }^{10}+{\omega }^{23}\right)\pi -\frac{\pi }{4}\right\}$ is

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

Answer 1

Answer: (C)

Using ${\omega }^{10}=\omega ,{\omega }^{23}={\omega }^2$, we get
$\sin \left\{\left({\omega }^{10}+{\omega }^{23}\right)\pi -\frac{\pi }{4}\right\}=\sin \left(-\pi -\frac{\pi }{4}\right)=\sin \frac{\pi }{4}=\frac{1}{\sqrt{2}}$