The question asks us to find the exact value of $\cos\left(\frac{\pi}{6}\right)$, ensuring that the result has a rational denominator.

Step-by-Step Solution:

1. Recall the Trigonometric Identity: The cosine of $\frac{\pi}{6}$ (which is equivalent to 30 degrees) is a well-known value from the unit circle. Specifically: $\cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2}.$

2. Expressing with a Rational Denominator: To ensure a rational denominator, we rationalize it by multiplying numerator and denominator by $\sqrt{3}$: $\frac{\sqrt{3}}{2} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{3}{2\sqrt{3}} = \frac{\sqrt{3}}{2}.$ The result remains $\frac{\sqrt{3}}{2}$, already rational.

Final Answer:

$\cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2}.$

Would you like further details or have additional questions?

Here are five related questions to expand your understanding:

1. What is the sine value of $\frac{\pi}{6}$, and how does it relate to cosine?
2. How do you compute cosine values for other common angles like $\frac{\pi}{4}$ or $\frac{\pi}{3}$?
3. How can the unit circle help find trigonometric values of non-standard angles?
4. Why is rationalizing denominators important in trigonometry and mathematics in general?
5. Can you explain the difference between exact values and approximations in trigonometry?

Tip: Memorize the values of sine and cosine for standard angles ($0$, $30^\circ$, $45^\circ$, $60^\circ$, $90^\circ$) using the unit circle—it saves time during calculations!