Evaluate. Write your answer in simplified, rationalized form. Do not round.tan–7​𝜋6=

To evaluate $\tan\left(-\frac{7\pi}{6}\right)$, we follow these steps:

1. Break Down the Problem

Identify the properties of the tangent function and how it behaves with negative angles and angle periodicity.

2. Relevant Concepts

The tangent function is periodic with a period of $\pi$: $\tan(x) = \tan(x + n\pi)$ For negative angles, we can use the property: $\tan(-x) = -\tan(x)$

3. Analysis and Detail

Evaluating $\tan\left(-\frac{7\pi}{6}\right)$:

1. Convert to a Positive Angle: $-\frac{7\pi}{6} + \pi = -\frac{7\pi}{6} + \frac{6\pi}{6} = -\frac{\pi}{6}$

2. Evaluate Tangent: $\tan\left(-\frac{7\pi}{6}\right) = \tan\left(-\frac{\pi}{6}\right) = -\tan\left(\frac{\pi}{6}\right)$

3. Find $\tan\left(\frac{\pi}{6}\right)$: $\tan\left(\frac{\pi}{6}\right) = \frac{1}{\sqrt{3}}$

4. Substituting Back: $\tan\left(-\frac{7\pi}{6}\right) = -\frac{1}{\sqrt{3}} = -\frac{\sqrt{3}}{3}$

4. Verify and Summarize

The final evaluation follows from the periodic properties and the simplicity of the trigonometric function. The evaluated angle leads directly back to a well-known value and reveals the behavior of tangent with respect to negative angles.

Final Answer

$\tan\left(-\frac{7\pi}{6}\right) = -\frac{\sqrt{3}}{3}$