Evaluate. \[ \sin \left(\frac{5 \pi}{3}\right) \cos \left(-\frac{\pi}{6}\right)=\square \]

To evaluate the expression \(\sin \left(\frac{5 \pi}{3}\right) \cos \left(-\frac{\pi}{6}\right)\), we will find the values of the sine and cosine functions involved. First, we evaluate \(\sin \left(\frac{5 \pi}{3}\right)\). The angle \(\frac{5 \pi}{3}\) is equivalent to \(360^\circ - 60^\circ = 300^\circ\). The sine of an angle in the fourth quadrant (like \(300^\circ\)) is negative, and it is equal to the sine of its reference angle, which is \(60^\circ\). \[ \sin \left(300^\circ\right) = -\sin \left(60^\circ\right) = -\frac{\sqrt{3}}{2} \] Next, we evaluate \(\cos \left(-\frac{\pi}{6}\right)\). The cosine function is even, so \(\cos(-\theta) = \cos(\theta)\). Thus, \[ \cos \left(-\frac{\pi}{6}\right) = \cos \left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2} \] Now, we can substitute these values back into our original expression: \[ \sin \left(\frac{5 \pi}{3}\right) \cos \left(-\frac{\pi}{6}\right) = \left(-\frac{\sqrt{3}}{2}\right) \left(\frac{\sqrt{3}}{2}\right) \] Multiplying these two fractions together yields: \[ -\frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3}}{2} = -\frac{3}{4} \] Thus, the final answer is: \[ \boxed{-\frac{3}{4}} \]