Question

Inverse trigonometric functions have restricted domain and range, [.] and |.| denotes greatest integer function and modulus function respectively, then
$\begin{array}{ccc}InverseTrigonometric& Domain& Range\\ Function& & \\ y=si{n}^{-1}x& xϵ[-1,1]& yϵ[\frac{-\pi }{2},\frac{\pi }{2}]\\ y=co{s}^{-1}x& xϵ[-1,1]& yϵ[0,\pi ]\\ y=ta{n}^{-1}x& xϵR& yϵ(\frac{-\pi }{2},\frac{\pi }{2})\\ y=co{t}^{-1}x& xϵR& yϵ(0,\pi )\\ y=se{c}^{-1}x& xϵR-(-1,1)& yϵ[0,\pi ]-\left\{\frac{\pi }{2}\right\}\\ y=cose{c}^{-1}x& xϵR-(-1,1)& yϵ[\frac{-\pi }{2},\frac{\pi }{2}]\end{array}$
The equation $si{n}^{-1}x=|x-a|$ will have at least one solution, if

• A. $aϵ[-1,1]$
• B. $aϵ[\frac{\pi }{2},\frac{\pi }{2}]$
• C. $aϵ[-1,\frac{1+\pi }{2}]$
• D. $aϵ[\frac{1-\pi }{2},\frac{1+\pi }{2}]$

Answer 1

The correct option is A $aϵ[-1,1]$

$si{n}^{-1}x=|x-a|$

For at least one solution |x - a| y must the graph of $si{n}^{-1}$ at least once and