To find the exact value of $\cos\left(\frac{11\pi}{6}\right)$, follow these steps:

1. Convert the angle $\frac{11\pi}{6}$ into degrees: $\frac{11\pi}{6} \times \frac{180^\circ}{\pi} = 330^\circ$

2. Determine the reference angle: The reference angle for $330^\circ$ is $360^\circ - 330^\circ = 30^\circ$.

3. Evaluate the cosine of the reference angle: $\cos(30^\circ) = \frac{\sqrt{3}}{2}$.

4. Determine the sign of the cosine in the fourth quadrant: Since $330^\circ$ is in the fourth quadrant, where the cosine function is positive, $\cos(330^\circ) = \frac{\sqrt{3}}{2}$.

So, the exact value of $\cos\left(\frac{11\pi}{6}\right)$ is $\frac{\sqrt{3}}{2}$.

Would you like any more details, or do you have any questions?

Here are some related questions:

1. What is the exact value of $\sin\left(\frac{11\pi}{6}\right)$?
2. How do you find the sine and cosine of an angle greater than $2\pi$?
3. What is the reference angle for $\frac{13\pi}{6}$?
4. How does the unit circle help in finding trigonometric values?
5. What are the signs of trigonometric functions in each quadrant?

Tip: Always consider the quadrant in which an angle lies when determining the sign of its trigonometric functions.