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

1.	Find the principal value of each of the following:\(sin^{-1}(tan\frac{5\pi}4)\)
Answer» Let y = \(sin^{-1}(tan\frac{5\pi}4)\) Therefore, sin y = \((tan\frac{5\pi}4)=tan(\pi+\frac{\pi}4)\) \(=tan\frac{\pi}4=1=sin(\frac{\pi}2)\) We know that the principal value of sin^-1 is \([-\frac{\pi}2,\frac{\pi}2]\) And sin\((\frac{\pi}2)=tan(\frac{5\pi}4)\) Therefore the principal value of sin^-1\((tan\frac{5\pi}4)\) is \(\frac{\pi}2\).

Find the principal value of each of the following:\(sin^{-1}(tan\frac{5\pi}4)\)

Answer»

Let y = \(sin^{-1}(tan\frac{5\pi}4)\)

Therefore, sin y = \((tan\frac{5\pi}4)=tan(\pi+\frac{\pi}4)\)

\(=tan\frac{\pi}4=1=sin(\frac{\pi}2)\)

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

And sin\((\frac{\pi}2)=tan(\frac{5\pi}4)\)

Therefore the principal value of sin^-1\((tan\frac{5\pi}4)\) is \(\frac{\pi}2\).