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