find the exact value of \\( \\tan \\frac { 7 \\pi } { 6 } \\) in simplest form with a rational denominator.

Answer:

$\frac{\sqrt{3}}{3}$

Explanation:

Step1: Find the reference angle

The angle $\frac{7\pi}{6}$ is in the third quadrant. The reference angle $\theta'$ for an angle $\theta=\frac{7\pi}{6}$ is $\theta'=\frac{7\pi}{6}-\pi=\frac{\pi}{6}$.

Step2: Determine the sign of the tangent function

In the third quadrant, $\tan\theta=\frac{\sin\theta}{\cos\theta}$. Both $\sin\theta$ and $\cos\theta$ are negative in the third quadrant, so $\tan\theta=\frac{\sin\theta}{\cos\theta}> 0$.

Step3: Use the tangent of the reference angle

We know that $\tan\frac{\pi}{6}=\frac{\sin\frac{\pi}{6}}{\cos\frac{\pi}{6}}=\frac{\frac{1}{2}}{\frac{\sqrt{3}}{2}}=\frac{1}{\sqrt{3}}$.

Step4: Rationalize the denominator

Multiply the numerator and denominator of $\frac{1}{\sqrt{3}}$ by $\sqrt{3}$: $\frac{1\times\sqrt{3}}{\sqrt{3}\times\sqrt{3}}=\frac{\sqrt{3}}{3}$.

So, $\tan\frac{7\pi}{6}=\frac{\sqrt{3}}{3}$.