If $\omega$ is a complex cube root of unity, then $\sin \left\{\left(\omega^{10}+\omega^{23}\right)…
If $\omega$ is a complex cube root of unity, then $\sin \left\{\left(\omega^{10}+\omega^{23}\right) \pi-\frac{\pi}{4}\right\}=$
$\frac{1}{\sqrt{2}}$
$\frac{1}{2}$
1
$\frac{\sqrt{3}}{2}$
Solution
It is given that $w$ is a complex cube root of unity, so $w^3=1$ and $w^2+w+1=0$.
So, $\sin \left\{\left(w^{10}+w^{23}\right) \pi-\frac{\pi}{4}\right\}$
$
\begin{aligned}
& =\sin \left\{\left(w+w^2\right) \pi-\frac{\pi}{4}\right\} \\
& =-\sin \left(\pi+\frac{\pi}{4}\right)=\sin \frac{\pi}{4}=\frac{1}{\sqrt{2}}
\end{aligned}
$
Hence, option (1) is correct