Answer:
[tex]f^{-1}(x)=\sin ^{-1}(x-\frac{\pi}{2})[/tex]
Step-by-step explanation:
The given function is
[tex]y=\frac{\pi}{2}+\sin x[/tex]
To find the inverse of this function, we interchange x and y.
[tex]x=\frac{\pi}{2}+\sin y[/tex]
we now solve for y.
[tex]x-\frac{\pi}{2}=\sin y[/tex]
Take the sine inverse of both sides to obtain;
[tex]\sin ^{-1}(x-\frac{\pi}{2})=y[/tex]
Hence the inverse of the given function is;
[tex]f^{-1}(x)=\sin ^{-1}(x-\frac{\pi}{2})[/tex]
where [tex]\frac{\pi}{2}-1\le x\le \frac{\pi}{2}+1[/tex]
