1. The problem is to find the value of $\cos\left(\frac{\sqrt{11}}{6}\right)$.\n\n2. Here, $\frac{\sqrt{11}}{6}$ is the angle in radians. We need to evaluate the cosine of this angle.\n\n3. Since $\sqrt{11} \approx 3.3166$, the angle is approximately $\frac{3.3166}{6} \approx 0.5528$ radians.\n\n4. Using a calculator or cosine function approximation, $\cos(0.5528) \approx 0.8507$.\n\n5. Therefore, the value of $\cos\left(\frac{\sqrt{11}}{6}\right)$ is approximately $0.8507$.