Find the Principal Value of sec-1(2tan 3π/4)
Solution:
Given:
\[ y = \sec^{-1}\left(2\tan \frac{3\pi}{4}\right) \]
Step 1: Evaluate tan(3π/4)
\[ \tan \frac{3\pi}{4} = -1 \]
So,
\[ 2\tan \frac{3\pi}{4} = -2 \]
Step 2: Convert to cosine
\[ \sec y = -2 \Rightarrow \cos y = -\frac{1}{2} \]
Step 3: Find angle
\[ \cos\left(\frac{2\pi}{3}\right) = -\frac{1}{2} \]
Principal value range of sec-1(x):
\[ [0,\pi], \quad y \ne \frac{\pi}{2} \]
Therefore,
\[ y = \frac{2\pi}{3} \]
Final Answer:
Principal Value = \[ \frac{2\pi}{3} \]