Find the Principal Value of cos-1(−1/√2)
Solution:
Let
\[ y = \cos^{-1}\left(-\frac{1}{\sqrt{2}}\right) \]
Then,
\[ \cos y = -\frac{1}{\sqrt{2}} \]
We know that:
\[ \cos\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}} \]
So,
\[ \cos y = \cos\left(\pi – \frac{\pi}{4}\right) = \cos\left(\frac{3\pi}{4}\right) \]
Since the principal value range of cos-1(x) is:
\[ [0, \pi] \]
Therefore,
\[ y = \frac{3\pi}{4} \]
Final Answer:
Principal Value = \[ \frac{3\pi}{4} \]