We are tasked with finding the exact value of $\cos(210^\circ)$.

Step 1: Find the reference angle

The reference angle is the smallest positive acute angle formed with the x-axis. For $210^\circ$, we subtract 180° from it because it is in the third quadrant:

$210^\circ - 180^\circ = 30^\circ$

So, the reference angle is $30^\circ$.

Step 2: Determine the cosine value for the reference angle

The cosine of $30^\circ$ is $\frac{\sqrt{3}}{2}$. Since $210^\circ$ is in the third quadrant, the cosine value is negative. Therefore:

$\cos(210^\circ) = -\frac{\sqrt{3}}{2}$

Step 3: Rationalize the denominator

The denominator is already rationalized because there is no square root in the denominator. The expression for $\cos(210^\circ)$ is already in its simplest form.

Thus, the exact value of $\cos(210^\circ)$ is:

$\boxed{-\frac{\sqrt{3}}{2}}$

Let me know if you'd like more details or if you have any questions! Here are a few questions for further exploration:

1. What other trigonometric values can be found for angles in the third quadrant?
2. How does the reference angle method work for angles in different quadrants?
3. Can you apply the unit circle to find the cosine of other angles like $150^\circ$?
4. What is the significance of rationalizing denominators in trigonometric expressions?
5. How would you determine the cosine of $330^\circ$?

Tip: For angles in the third quadrant, both sine and cosine values are negative.