To solve $\sin \left(-\frac{11\pi}{12} \right)$, we will use trigonometric identities and simplifications. Here's the step-by-step process:

Step 1: Simplify the angle

We know: $\sin(-x) = -\sin(x)$ So, $\sin \left(-\frac{11\pi}{12} \right) = -\sin \left(\frac{11\pi}{12} \right)$

Step 2: Rewrite $\frac{11\pi}{12}$

Express $\frac{11\pi}{12}$ as: $\frac{11\pi}{12} = \pi - \frac{\pi}{12}$ This means: $\sin \left( \frac{11\pi}{12} \right) = \sin \left( \pi - \frac{\pi}{12} \right)$ Using the identity $\sin(\pi - x) = \sin(x)$, we have: $\sin \left( \frac{11\pi}{12} \right) = \sin \left( \frac{\pi}{12} \right)$ Thus: $\sin \left(-\frac{11\pi}{12} \right) = -\sin \left( \frac{\pi}{12} \right)$

Step 3: Exact value of $\sin \left( \frac{\pi}{12} \right)$

The angle $\frac{\pi}{12}$ can be expressed as: $\frac{\pi}{12} = \frac{\pi}{3} - \frac{\pi}{4}$ Now, use the sine subtraction formula: $\sin(a - b) = \sin(a)\cos(b) - \cos(a)\sin(b)$ Substituting $a = \frac{\pi}{3}$ and $b = \frac{\pi}{4}$, we get: $\sin \left( \frac{\pi}{12} \right) = \sin \left( \frac{\pi}{3} \right) \cos \left( \frac{\pi}{4} \right) - \cos \left( \frac{\pi}{3} \right) \sin \left( \frac{\pi}{4} \right)$

Step 4: Substitute exact values

From known trigonometric values: