Question

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

Answer 1

Answer:

It is valid in the interval $[\frac{-\pi }{2},\frac{+\pi }{2}]$

Step-by-step 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

$-1\leq sin(x)\leq +1\\\\\Rightarrow sin^{-1}(-1)\leq sin^{-1}(sin(x))\leq sin^{-1}1\\\\\therefore \frac{-\pi }{2}\leq sin^{-1}(sin(x))\leq \frac{+\pi }{2}$

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