Find the Principal Value of cosec-1(2cos 2π/3)
Solution:
Given:
\[ y = \csc^{-1}\left(2\cos \frac{2\pi}{3}\right) \]
Step 1: Evaluate cos(2π/3)
\[ \cos \frac{2\pi}{3} = -\frac{1}{2} \]
So,
\[ 2\cos \frac{2\pi}{3} = -1 \]
Step 2: Convert to sine
\[ \csc y = -1 \Rightarrow \sin y = -1 \]
Step 3: Find principal value
\[ \sin y = -1 \Rightarrow y = -\frac{\pi}{2} \]
But principal value range of cosec-1(x) is:
\[ \left[-\frac{\pi}{2}, 0\right) \cup \left(0, \frac{\pi}{2}\right] \]
Since \( -\frac{\pi}{2} \) is included,
\[ y = -\frac{\pi}{2} \]
Final Answer:
Principal Value = \[ -\frac{\pi}{2} \]