Find the principal value of sec^{-1} (-\sqrt{2)}

Given sec^-1 \( (-\sqrt{2})\)

Now let y = sec^-1 \( (-\sqrt{2})\)

Sec y = \( -\sqrt{2}\)

We know that  \( sec \dfrac{\pi}{4} =\sqrt{2}\)

Therefore,  \( -sec (\dfrac{\pi}{4}) =- \sqrt{2}\)

= \( sec (π – \dfrac{\pi}{4})\)

= \( sec (\dfrac{3\pi}{4})\)

Thus the range of principal value of sec^-1 is [0, π] – {\( \dfrac{\pi}{2}\)}

And \( sec (\dfrac{3\pi}{4})=- \sqrt{2}\)

Hence the principal value of \( sec^{-1 }(-\sqrt{2})\) is \( \dfrac{3\pi}{4}\)