Find the Principal Value of sec-1(2)
Solution:
Let
\[ y = \sec^{-1}(2) \]
Then,
\[ \sec y = 2 \]
Taking reciprocal:
\[ \cos y = \frac{1}{2} \]
We know:
\[ \cos\left(\frac{\pi}{3}\right) = \frac{1}{2} \]
Principal value range of sec-1(x):
\[ [0,\pi], \quad y \ne \frac{\pi}{2} \]
Therefore,
\[ y = \frac{\pi}{3} \]
Final Answer:
Principal Value = \[ \frac{\pi}{3} \]