$7 \cdot 32$ Given that the base $BC$ and the height $AD$ of isosceles $\triangle ABC$ are both integers, then among $\sin A$ and $\cos A$,
(A) one is rational and the other is irrational.
(B) both are rational.
(C) both are irrational.
(D) whether they are rational or irrational depends on the values of $BC$ and $AD$.
(China High School Mathematics League, 1983)
[Solution] As shown in the figure, let $\alpha=\frac{1}{2} A$, then
$$
\tan \alpha=\frac{B D}{A D}=\frac{B C}{2 \cdot A D}
$$

is a rational number.
And $\sin A=\frac{2 \tan \alpha}{1+\tan^{2} \alpha}$, also $\cos A=\frac{1-\tan^{2} \alpha}{1+\tan^{2} \alpha}$, so $\sin A, \cos A$ are both rational numbers.
Therefore, the answer is $(B)$.