Question

If $\left[si{n}^{-1}x\right]>\left[co{s}^{-1}x\right]$ where [.] denotes the greatest integer function, then the set of values of x is:

• A. [cos 1, 1]
• B. [cos 1, sin 1]
• C. [sin 1, 1]
• D. None of these

Answer 1

The correct option is C [sin 1, 1]

$\left[si{n}^{-1}x\right]=-2,-1,0,1[\because -\frac{\pi }{2}\le si{n}^{-1}x\le \frac{\pi }{2}]0\le co{s}^{-1}x\le \pi \Rightarrow \left[co{s}^{-1}x\right]=0,1,2,3$.
$\therefore$ Possible values for $\left[si{n}^{-1}x\right]$ and $\left[co{s}^{-1}x\right]$ satisfying $\left[si{n}^{-1}x\right]>\left[co{s}^{-1}x\right]$ are $\left[si{n}^{-1}x\right]=1\text{and}\left[co{s}^{-1}x\right]=0$
i.e., $1\le si{n}^{-1}x\le \frac{\pi }{2},0\le co{s}^{-1}x<1\Rightarrow sin1\le x\le 1,cos1\le x\le 1$.
Since $1>\frac{\pi }{4},sin1>cos1$
Hence, required solution is $x\in [sin1,1]$