The principal value of\(cos^{-1}\left(\frac{-1}{\sqrt{2}}\right)\)isA.

1.	The principal value of\(cos^{-1}\left(\frac{-1}{\sqrt{2}}\right)\)isA. \(-\frac{\pi}{4}\)B. \(\frac{\pi}{4}\)C. \(\frac{3\pi}{4}\)D. \(\frac{5\pi}{4}\)
Answer» To Find: The Principle value of \(cos^{-1}\left(\frac{-1}{\sqrt{2}}\right)\) Let the principle value be given by x Now, let x = \(cos^{-1}\left(\frac{-1}{\sqrt{2}}\right)\) ⇒cos x = \(\frac{-1}{\sqrt{2}}\) ⇒cos x= - cos( \(\frac{\pi}{4}\)) ( \(\because cos\left(\frac{\pi}{4}\right)\)= \(\frac{1}{\sqrt{2}}\)) ⇒cos x=cos( \(\pi-\frac{\pi}{4}\)) ( \(\because -cos(\theta)=cos(\pi-\theta\))) ⇒x =\(\frac{3\pi}{4}\)

The principal value of\(cos^{-1}\left(\frac{-1}{\sqrt{2}}\right)\)isA. \(-\frac{\pi}{4}\)B. \(\frac{\pi}{4}\)C. \(\frac{3\pi}{4}\)D. \(\frac{5\pi}{4}\)

Answer»

To Find: The Principle value of \(cos^{-1}\left(\frac{-1}{\sqrt{2}}\right)\)

Let the principle value be given by x

Now, let x = \(cos^{-1}\left(\frac{-1}{\sqrt{2}}\right)\)

⇒cos x = \(\frac{-1}{\sqrt{2}}\)

⇒cos x= - cos( \(\frac{\pi}{4}\)) ( \(\because cos\left(\frac{\pi}{4}\right)\)= \(\frac{1}{\sqrt{2}}\))

⇒cos x=cos( \(\pi-\frac{\pi}{4}\)) ( \(\because -cos(\theta)=cos(\pi-\theta\)))

⇒x =\(\frac{3\pi}{4}\)