Find the principal value of each of the following: \(sin^{-1}(cos\fra

1.	Find the principal value of each of the following: \(sin^{-1}(cos\frac{3\pi}4)\)
Answer» Let \(sin^{-1}(cos\frac{3\pi}4)\) = y Then sin y = cos \(\frac{3\pi}4=-sin(\pi-\frac{3\pi}4)\)\(=-sin(\frac{\pi}4)\) We know that the principal value of sin^-1 is \([-\frac{\pi}2,\frac{\pi}2]\) \(-sin(\frac{\pi}4)=cos\frac{3\pi}4\) Therefore the principal value of sin^-1(cos\(\frac{3\pi}4\)) is \(-\frac{\pi}4\).

Find the principal value of each of the following: \(sin^{-1}(cos\frac{3\pi}4)\)

Answer»

Let \(sin^{-1}(cos\frac{3\pi}4)\) = y

Then sin y = cos \(\frac{3\pi}4=-sin(\pi-\frac{3\pi}4)\)\(=-sin(\frac{\pi}4)\)

We know that the principal value of sin^-1 is \([-\frac{\pi}2,\frac{\pi}2]\)

\(-sin(\frac{\pi}4)=cos\frac{3\pi}4\)

Therefore the principal value of sin^-1(cos\(\frac{3\pi}4\)) is \(-\frac{\pi}4\).