Example 9 Numbers $a, b, c$ are within the interval $\left(0, \frac{\pi}{2}\right)$, and satisfy $a=\cos a$, $b=\sin (\cos b)$, $c=\cos (\sin c)$, try to compare their sizes.
Solution: First, draw the graphs of the functions $y=\cos x, y=\sin x, y=\sin (\cos x)$, and $y=\cos (\sin x)$. When $x \in\left(0, \frac{\pi}{2}\right)$, the graphs are as shown in Figure 5-4.

Since $\cos a=a, \sin (\cos b)=b$, and $\cos (\sin c)=c$, $a, b, c$ are the x-coordinates of the points where the curves intersect with $y=x$. From the graph, it is clear that $b<\sin x$ leads to $b=$
Figure 5-4
$\sin (\cos b)\cos c \geqslant \cos a=a$, which is a contradiction. Therefore, $c>a$. Hence, $b<a<c$.