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

1. Express the angle in degrees: $\frac{11\pi}{6} \times \frac{180^\circ}{\pi} = 330^\circ$ So, $\frac{11\pi}{6}$ radians is equivalent to $330^\circ$.

2. Locate the angle on the unit circle: The angle $330^\circ$ lies in the fourth quadrant. In this quadrant, tangent is negative because tangent is the ratio of sine to cosine, and sine is negative while cosine is positive in the fourth quadrant.

3. Use a reference angle: The reference angle for $330^\circ$ is $360^\circ - 330^\circ = 30^\circ$. The tangent of the reference angle $30^\circ$ is: $\tan(30^\circ) = \frac{1}{\sqrt{3}}$

4. Apply the sign based on the quadrant: Since $330^\circ$ is in the fourth quadrant where tangent is negative, we get: $\tan\left(\frac{11\pi}{6}\right) = -\frac{1}{\sqrt{3}}$

Thus, the exact value of $\tan\left(\frac{11\pi}{6}\right)$ is $-\frac{1}{\sqrt{3}}$.

Would you like more details or have any questions?

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Here are 5 related questions you might find helpful:

1. How do you determine the reference angle for any given angle?
2. What are the signs of trigonometric functions in different quadrants?
3. How can you convert radians to degrees and vice versa?
4. What is the exact value of $\tan\left(\frac{5\pi}{6}\right)$?
5. How can you use the unit circle to find trigonometric values?

Tip: Always remember that in the fourth quadrant, sine is negative, cosine is positive, and tangent is negative.