Evaluate \( \tan^{-1}(\tan 12) \)
Step-by-Step Solution
We need to evaluate:
\[ \tan^{-1}(\tan 12) \]
Step 1: Principal value range
The principal value range of \( \tan^{-1}x \) is:
\[ \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \]
Step 2: Reduce the angle
Use periodicity \( \tan(x + \pi) = \tan x \)
\[ 12 – 3\pi \approx 12 – 9.425 = 2.575 \]
Still outside range, subtract again:
\[ 12 – 4\pi \approx 12 – 12.566 = -0.566 \]
Now it lies in \( \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \)
Step 3: Apply inverse tangent
\[ \tan^{-1}(\tan 12) = 12 – 4\pi \]
Final Answer
\[ \boxed{12 – 4\pi} \]