Find the Principal Value of cosec-1(−√2)
Solution:
Let
\[ y = \csc^{-1}(-\sqrt{2}) \]
Then,
\[ \csc y = -\sqrt{2} \Rightarrow \sin y = -\frac{1}{\sqrt{2}} \]
We know:
\[ \sin\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}} \]
So,
\[ \sin y = -\frac{1}{\sqrt{2}} = \sin\left(-\frac{\pi}{4}\right) \]
Principal value range of cosec⁻¹(x):
\[ \left[-\frac{\pi}{2}, 0\right) \cup \left(0, \frac{\pi}{2}\right] \]
Since \(-\frac{\pi}{4}\) lies in this range,
\[ y = -\frac{\pi}{4} \]
Final Answer:
Principal Value = \[ -\frac{\pi}{4} \]