Evaluate \( \tan^{-1}(\tan \frac{6\pi}{7}) \)
Step-by-Step Solution
We need to evaluate:
\[ \tan^{-1}\left(\tan \frac{6\pi}{7}\right) \]
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 \( \frac{6\pi}{7} > \frac{\pi}{2} \), bring it into principal range using:
\[ \tan(x – \pi) = \tan x \]
\[ \frac{6\pi}{7} – \pi = \frac{6\pi – 7\pi}{7} = -\frac{\pi}{7} \]
Step 3: Apply inverse tangent
\[ \tan^{-1}\left(\tan \frac{6\pi}{7}\right) = \tan^{-1}\left(\tan \left(-\frac{\pi}{7}\right)\right) \]
Since \( -\frac{\pi}{7} \in \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \), we get:
\[ \tan^{-1}(\tan \frac{6\pi}{7}) = -\frac{\pi}{7} \]
Final Answer
\[ \boxed{-\frac{\pi}{7}} \]