Find the Principal Value of cosec-1(2/√3)
Solution:
Let
\[ y = \csc^{-1}\left(\frac{2}{\sqrt{3}}\right) \]
Then,
\[ \csc y = \frac{2}{\sqrt{3}} \Rightarrow \sin y = \frac{\sqrt{3}}{2} \]
We know:
\[ \sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2} \]
Principal value range of cosec⁻¹(x):
\[ \left[-\frac{\pi}{2}, 0\right) \cup \left(0, \frac{\pi}{2}\right] \]
Since \( \frac{\pi}{3} \) lies in \( (0, \frac{\pi}{2}) \),
\[ y = \frac{\pi}{3} \]
Final Answer:
Principal Value = \[ \frac{\pi}{3} \]