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