Question

On what interval is the identity sin^-1(sin (x)) = x valid?

Answer 1

It is valid in the interval
`[(-\pi )/(2),(+\pi )/(2)]`

Explanation:

Let
`f(x)=sin^(-1)(sin(x))`

The domain of the given function is

`sin(x)\in [-1,1]\\\\(\because sin^(-1)x`is defined only for x in the set[-1,+1])

Thus we have

`-1\leq sin(x)\leq +1\\\\\Rightarrow sin^(-1)(-1)\leq sin^(-1)(sin(x))\leq sin^(-1)1\\\\\therefore (-\pi )/(2)\leq sin^(-1)(sin(x))\leq (+\pi )/(2)\\`

`\therefore x\in [-\pi /2,+\pi /2]`