Question

Let $g: R \rightarrow R$ be defined as $g(x)=\operatorname{sgn}\left(x^2-5 x+6\right)$, then find the number of solutions of equation $\sin x=\cos ^{-1}\left(g\left(\sin ^{-1} x\right)\right)$ lying in interval$ [0,314].$
[Note: $\operatorname{sgn}( k )$ denotes the signum function of $k$.]

Answer 1

As, $g\left(\sin ^{-1} x\right)=\operatorname{sgn}\left(\left(\sin ^{-1} x-2\right) \cdot\left(\sin ^{-1} x-3\right)\right)=1$
$\therefore$ We get, $\sin x=\cos ^{-1}(1) \Rightarrow \sin x=0$
$\Rightarrow x = n \pi, n \in I$
But domain of equation is $[-1,1]$.
$\therefore$ The possible solution is $x =0$.
Hence, the number of solutions of given equation are one.