Example 10. The cosine value of the angle between the median on one leg of an isosceles right triangle and the hypotenuse is $\qquad$
(1986, Jilin Province Eight
Cities Junior High School Competition)
Given $\triangle A B C$ is an isosceles right triangle with $A B$ as the base, $A D$ is the median of side $B C$, and $\angle D A B = \alpha$.
Then, for the angle bisector, we have
$$
\begin{array}{l}
D B = \frac{A D \cdot \sin \alpha}{A C \cdot \cos \angle C A B} \\
= A D \cdot \sin \alpha \\
A C \cdot \sin 45^{\circ}
\end{array}
$$

But $B C = A C$, $C D = D B$, and by the Pythagorean theorem, $A D = \sqrt{5} C D = \sqrt{5} D B$. Therefore, we immediately get
$$
\begin{aligned}
\sin \alpha & = \frac{D B}{A D} \cdot \sin 45^{\circ} = \frac{1}{\sqrt{5}} \cdot \frac{\sqrt{2}}{2} \\
& = \frac{\sqrt{10}}{10} .
\end{aligned}
$$