Question

Range of $x$ satisfying the inequality $\left[{\sin }^{-1}x\right]>\left[{\cos }^{-1}x\right]$ is
([.] represents the Greatest Integer function)

• A. $[\sin 1,1]$
• B. $(\cos 1,1]$
• C. $[0,\cos 1)$
• D. $(\sin 1,1)$

Answer 1

The correct option is A $[\sin 1,1]$

We know, $\left[{\sin }^{-1}x\right]=-2,-1,0,1;\left[{\cos }^{-1}x\right]=0,1,2,3$
So, $\left[{\sin }^{-1}x\right]>\left[{\cos }^{-1}x\right]$ is possible only when
$\left[{\sin }^{-1}x\right]=1$ and $\left[{\cos }^{-1}x\right]=0$
$\Rightarrow 1\le {\sin }^{-1}x<2$ and $0\le {\cos }^{-1}x<1$
$\Rightarrow \sin 1\le x\le 1$ and $\cos 1<x\le 1$
$\therefore$ $x\in [\sin 1,1]$