Example 1. The numbers $a, b, c$ belong to the open interval $\left(0, \frac{\pi}{2}\right)$ and satisfy the following equations
$$
\begin{array}{l}
\cos a=a, \\
\sin \cos b=b, \\
\cos \sin c=c .
\end{array}
$$

Arrange these three numbers in ascending order.
Solution 1. First, separately plot the functions
$$
\begin{array}{l}
y=\cos x, \\
y=\sin \cos x, \\
y=\cos \sin x
\end{array}
$$

(see Figure 1), then the x-coordinates of their intersections with the line $y=x$ are $a$, $b$, and $c$, respectively. From the graph, it can be seen that the order should be
$$
b<a<c.
$$

Then, it is not difficult to prove this by contradiction, as referenced in [1].
Reviewing the above solution, there is an obvious drawback: plotting the composite functions $y=\sin \cos x$ and $y=\cos \sin x$ is not only cumbersome but also, due to a lack of in-depth understanding of the graph, it does not help us make the necessary judgment, thus forcing us to resort to proof by contradiction to confirm the conclusion derived from the graph.

If the graphical method is used, and (1)-(3) are first transformed equivalently, then the above defects will be completely overcome.
Solution 2. Within $\left(0, \frac{\pi}{2}\right)$, (1)-(3) can be respectively transformed into
$$
\begin{array}{l}
a=\operatorname{arc} \cos a, \\
\cos b=\arcsin b, \\
\sin c=\arccos c,
\end{array}
$$

At this point, the functions on both sides
$$
\begin{array}{l}
y=x, \quad y=\sin x, \quad y=\cos x, \\
y=\arcsin x, \quad y=\arccos x,
\end{array}
$$

have graphs whose characteristics we are very familiar with, making them convenient to use.
First, the x-coordinates of the intersections of the lines $y=\arcsin x$ and $y=x$ with the cosine curve $y=\cos x$ can respectively determine $b$ and $a$.
Since $y=\arcsin x$ is above $y=x$, and the cosine function is decreasing, it is known that $b<a$.
Second, the x-coordinates of the intersections of the lines $y=x$ and $y=\sin x$ with the inverse cosine curve $y=\arccos x$ can determine $a$ and $c$.
Since $y=x$ is above $y=\sin x$, and the inverse cosine function is decreasing, it is known that $a<c$.
Thus, the order should be $b<a<c$.
Comparing the two solutions, it is not difficult to see: when using the geometric proof method, appropriate transformations should be made first to ensure that the geometric images we are familiar with are assigned. This helps us utilize their characteristics to discover certain necessary connections, thereby making the required judgments.