Step 1: Powers of omega repeat:
The powers cycle with period 3: $\omega^1, \omega^2, 1$. Since $10 = 3\times 3 + 1$ and $23 = 3\times 7 + 2$, we get $\omega^{10} = \omega$ and $\omega^{23} = \omega^2$.
Step 2: Angle simplification:
The sum is $-1$, so the argument is $-\pi - \frac{\pi}{4} = -\frac{5\pi}{4}$.
\[ \sin\left(-\frac{5\pi}{4}\right) = -\sin\frac{5\pi}{4} = -\left(-\frac{1}{\sqrt{2}}\right) = \frac{1}{\sqrt{2}} \]
Final Answer:
$\frac{1}{\sqrt{2}}$.
\[ \boxed{\frac{1}{\sqrt{2}}} \]