Evaluate \( \tan^{-1}(\tan 4) \)
Step-by-Step Solution
We need to evaluate:
\[ \tan^{-1}(\tan 4) \]
Step 1: Principal value range
The principal value range of \( \tan^{-1}x \) is:
\[ \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \]
Step 2: Locate 4
Since \( 4 \in (\pi, \frac{3\pi}{2}) \), we adjust using periodicity:
\[ \tan(x – \pi) = \tan x \]
\[ 4 – \pi \]
Now check:
\[ 4 – \pi \in \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \]
Step 3: Apply inverse tangent
\[ \tan^{-1}(\tan 4) = 4 – \pi \]
Final Answer
\[ \boxed{4 – \pi} \]