To determine [tex]\(\tan \left(\frac{11 \pi}{6}\right)\)[/tex], let's analyze the angle [tex]\(\frac{11\pi}{6}\)[/tex] in the context of the unit circle.
1. Convert the Angle to Degrees:
[tex]\[
\frac{11\pi}{6} \text{ radians} \times \frac{180^\circ}{\pi} = \frac{11 \times 180^\circ}{6 \times 1} = 330^\circ
\][/tex]
So, [tex]\(\frac{11\pi}{6}\)[/tex] radians is equivalent to [tex]\(330^\circ\)[/tex].
2. Determine the Reference Angle:
Since [tex]\(330^\circ\)[/tex] is in the fourth quadrant, we find the reference angle by subtracting [tex]\(330^\circ\)[/tex] from [tex]\(360^\circ\)[/tex]:
[tex]\[
360^\circ - 330^\circ = 30^\circ
\][/tex]
Therefore, the reference angle is [tex]\(30^\circ\)[/tex].
3. Find Tangent of the Reference Angle:
We know from trigonometric ratios that:
[tex]\[
\tan(30^\circ) = \frac{\sqrt{3}}{3}
\][/tex]
4. Determine the Sign:
The angle [tex]\(330^\circ\)[/tex] is in the fourth quadrant, where the tangent function is negative. Therefore, we have:
[tex]\[
\tan(330^\circ) = -\tan(30^\circ) = -\frac{\sqrt{3}}{3}
\][/tex]
So, the value of [tex]\(\tan \left(\frac{11 \pi}{6}\right)\)[/tex] is:
[tex]\[
-\frac{\sqrt{3}}{3}
\][/tex]
Therefore, the correct answer is:
C. [tex]\(-\frac{\sqrt{3}}{3}\)[/tex]
