Question

Let $x\in \left(0,1\right).$ The set of all x such that $si{n}^{-1}x>co{s}^{-1}x,$ is the interval:

• A. $\left(\frac{1}{2},\frac{1}{\sqrt{2}}\right)$
• B. $\left(\frac{1}{\sqrt{2}},1\right)$
• C. (0,1)
• D. $\left(0,\frac{\sqrt{3}}{2}\right)$

Answer 1

Answer: (B)

Given $si{n}^{-1}x>co{s}^{-1}x$ where $x\in \left(0,1\right)$
$\Rightarrow si{n}^{-1}x>\frac{\pi }{2}-si{n}^{-1}x$
$\Rightarrow 2si{n}^{-1}x>\frac{\pi }{2}\Rightarrow si{n}^{-1}x>\frac{\pi }{4}$
$\Rightarrow x>sin\frac{\pi }{4}\Rightarrow x>\frac{1}{\sqrt{2}}$
Maximum value of $si{n}^{-1}xis\frac{\pi }{2}$
So, maximum value of x is 1. So,
$x\in \left(\frac{1}{\sqrt{2}},1\right).$