1. **State the problem:** Evaluate the expression $$\frac{2 \cos \left(\frac{8\pi}{6}\right) - 5 \sin \left(-\frac{5\pi}{2}\right)}{3 \tan \left(\frac{3\pi}{4}\right)}$$. 2. **Simplify the angles:** - $$\frac{8\pi}{6} = \frac{4\pi}{3}$$. - $$-\frac{5\pi}{2}$$ can be simplified by adding multiples of $$2\pi$$: $$-\frac{5\pi}{2} + 2\pi = -\frac{5\pi}{2} + \frac{4\pi}{2} = -\frac{\pi}{2}$$. - $$\frac{3\pi}{4}$$ remains as is. 3. **Evaluate each trigonometric function:** - $$\cos \left(\frac{4\pi}{3}\right) = \cos 240^\circ = -\frac{1}{2}$$. - $$\sin \left(-\frac{\pi}{2}\right) = -\sin \left(\frac{\pi}{2}\right) = -1$$. - $$\tan \left(\frac{3\pi}{4}\right) = \tan 135^\circ = -1$$. 4. **Substitute values back into the expression:** $$\frac{2 \times \left(-\frac{1}{2}\right) - 5 \times (-1)}{3 \times (-1)} = \frac{-1 + 5}{-3} = \frac{4}{-3} = -\frac{4}{3}$$. 5. **Final answer:** $$-\frac{4}{3}$$.