Find the principal value of each of the following: sin^-1(tan 5π/4)

Question

Find the principal value of each of the following:

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

Answer 1

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 \(\frac{\pi}2\).