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(C) Given that $\omega$ is an imaginary cube root of unity,we know that $\omega^3 = 1$ and $1 + \omega + \omega^2 = 0$.First,simplify the powers of $\

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$\frac{1}{\sqrt{2}}$

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(C) Given that $\omega$ is an imaginary cube root of unity,we know that $\omega^3 = 1$ and $1 + \omega + \omega^2 = 0$.First,simplify the powers of $\omega$:$\omega^{10} = (\omega^3)^3 \cdot \omega = 1^3 \cdot \omega = \omega$$\omega^{23} = (\omega^3)^7 \cdot \omega^2 = 1^7 \cdot \omega^2 = \omega^2$Substitute these into the expression:$\sin \left[ (\omega + \omega^2)\pi - \frac{\pi}{4} ight]$Since $1 + \omega + \omega^2 = 0$,we have $\omega + \omega^2 = -1$.So,the expression becomes $\sin \left[ -1 \cdot \pi - \frac{\pi}{4} ight] = \sin \left( -\pi - \frac{\pi}{4} ight)$.Using the property $\sin(-	heta) = -\sin(	heta)$:$-\sin \left( \pi + \frac{\pi}{4} ight) = -(-\sin \frac{\pi}{4}) = \sin \frac{\pi}{4} = \frac{1}{\sqrt{2}}$.

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

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