Evaluate \( \tan^{-1}(\tan \frac{\pi}{3}) \)
Step-by-Step Solution
We need to evaluate:
\[ \tan^{-1}\left(\tan \frac{\pi}{3}\right) \]
Step 1: Principal value range
The principal value range of \( \tan^{-1}x \) is:
\[ \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \]
Step 2: Check the angle
\[ \frac{\pi}{3} > \frac{\pi}{2} \]
So it is outside the principal range.
Step 3: Use identity
\[ \tan(x – \pi) = \tan x \]
\[ \tan\left(\frac{\pi}{3}\right) = \tan\left(\frac{\pi}{3} – \pi\right) = \tan\left(-\frac{2\pi}{3}\right) \]
Step 4: Apply inverse tangent
Now bring angle into principal range:
\[ -\frac{2\pi}{3} \notin \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \]
So we adjust again:
\[ \frac{\pi}{3} – \pi = -\frac{2\pi}{3} \Rightarrow \tan^{-1}(\tan \frac{\pi}{3}) = \frac{\pi}{3} – \pi = -\frac{2\pi}{3} \]
Final Answer
\[ \boxed{-\frac{2\pi}{3}} \]