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(C) Given that $\omega$ is a complex cube root of unity, we have $\omega^3 = 1$ and $1 + \omega + \omega^2 = 0$.We simplify the powers of $\omega$:$\o

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

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(C) Given that $\omega$ is a complex cube root of unity, we have $\omega^3 = 1$ and $1 + \omega + \omega^2 = 0$.We 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:$\sin [\pi(\omega + \omega^2) - \frac{\pi}{4}]$.Since $1 + \omega + \omega^2 = 0$, we have $\omega + \omega^2 = -1$.Now, substitute this value:$\sin [\pi(-1) - \frac{\pi}{4}] = \sin [-\pi - \frac{\pi}{4}] = \sin [-\frac{5\pi}{4}]$.Using the property $\sin(-	heta) = -\sin(	heta)$:$-\sin(\frac{5\pi}{4}) = -\sin(\pi + \frac{\pi}{4}) = -(-\sin(\frac{\pi}{4})) = \sin(\frac{\pi}{4}) = \frac{1}{\sqrt{2}}$.

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

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