Evaluate \( \tan^{-1}(\tan \frac{7\pi}{6}) \)
Step-by-Step Solution
We need to evaluate:
\[ \tan^{-1}\left(\tan \frac{7\pi}{6}\right) \]
Step 1: Principal value range
The principal value range of \( \tan^{-1}x \) is:
\[ \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \]
Step 2: Adjust the angle
Since \( \frac{7\pi}{6} \in \left(\frac{\pi}{2}, \frac{3\pi}{2}\right) \), use:
\[ \tan(x – \pi) = \tan x \]
\[ \frac{7\pi}{6} – \pi = \frac{7\pi – 6\pi}{6} = \frac{\pi}{6} \]
Step 3: Apply inverse tangent
\[ \tan^{-1}\left(\tan \frac{7\pi}{6}\right) = \tan^{-1}\left(\tan \frac{\pi}{6}\right) \]
Since \( \frac{\pi}{6} \in \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \), we get:
\[ \tan^{-1}(\tan \frac{7\pi}{6}) = \frac{\pi}{6} \]
Final Answer
\[ \boxed{\frac{\pi}{6}} \]