9. (2nd "Hope Cup" Grade 1 High School Competition Question) Extend the graph of $y=\sin x$ on the interval $\left[0, \frac{\pi}{2}\right)$ to the entire real axis, making it a function with a period of $\frac{\pi}{2}$, as shown in Figure 1-1-10. The analytical expression of this new function is $\qquad$ .
9. (Induction)

On the interval $\left[0, \frac{\pi}{2}\right)$, $f(x)=\sin x$;
On the interval $\left[\frac{\pi}{2}, \pi\right)$, $f(x)=\sin \left(x-\frac{\pi}{2}\right)$;
On the interval $\left[\pi, \frac{3}{2} \pi\right)$, $f(x)=\sin \left(x-\frac{2}{2} \pi\right)$;
.
On the interval $\left[-\frac{\pi}{2}, 0\right)$, $f(x)=\sin \left[x-\left(-\frac{\pi}{2}\right)\right]$;
On the interval $\left[-\pi,-\frac{\pi}{2}\right)$, $f(x)=\sin \left[x-\left(-\frac{2}{2} \pi\right)\right]$;
$\cdots$
Therefore, on the interval $\left[\frac{k \pi}{2}, \frac{k+1}{2} \pi\right)$, $f(x)=\sin \left(x-\frac{k \pi}{2}\right),(k \in \mathbf{Z})$.