Question

what is tan \\( \frac { 11 \pi } { 6 } \\) ?

Explanation:

Step1: Find the reference angle

The angle \(\frac{11\pi}{6}\) is in the fourth - quadrant. The reference angle \(\theta_{r}\) for an angle \(\theta=\frac{11\pi}{6}\) is \(\theta_{r}=2\pi-\frac{11\pi}{6}=\frac{12\pi - 11\pi}{6}=\frac{\pi}{6}\).

Step2: Determine the sign of the tangent function

In the fourth - quadrant, \(\tan\theta=\frac{\sin\theta}{\cos\theta}\). Since \(\sin\theta\lt0\) and \(\cos\theta\gt0\) in the fourth - quadrant, \(\tan\theta\lt0\).

Step3: Calculate the value of \(\tan\theta\)

We know that \(\tan\frac{\pi}{6}=\frac{\sqrt{3}}{3}\). Using the identity \(\tan\theta =-\tan\theta_{r}\) (because \(\theta\) is in the fourth - quadrant), so \(\tan\frac{11\pi}{6}=-\tan\frac{\pi}{6}\).

Answer:

\(-\frac{\sqrt{3}}{3}\), so the number in the green box is \(3\)