1. **Problem:** Find the exact value of $\cos\left(\frac{11\pi}{6}\right)$.\n\n2. **Formula and rules:** The cosine function on the unit circle corresponds to the $x$-coordinate of the point at the given angle. Familiar angles include $\frac{\pi}{6}$, $\frac{\pi}{4}$, $\frac{\pi}{3}$, etc. The angle $\frac{11\pi}{6}$ is in the fourth quadrant where cosine is positive.\n\n3. **Step-by-step solution:**\n- Note that $\frac{11\pi}{6} = 2\pi - \frac{\pi}{6}$, so it is $\frac{\pi}{6}$ less than a full circle.\n- Using the identity $\cos(2\pi - \theta) = \cos(\theta)$, we get:\n$$\cos\left(\frac{11\pi}{6}\right) = \cos\left(\frac{\pi}{6}\right)$$\n- The exact value of $\cos\left(\frac{\pi}{6}\right)$ is $\frac{\sqrt{3}}{2}$.\n\n4. **Final answer:**\n$$\cos\left(\frac{11\pi}{6}\right) = \frac{\sqrt{3}}{2}$$