Analysis: By the uniqueness of the answer, the size relationship must satisfy the special value $x_{0}=-\frac{1}{4} \in\left(-\frac{1}{2}, 0\right)$. At this time,
$$\begin{aligned} a_{1} & =\cos \left[\sin \left(-\frac{\pi}{4}\right)\right]=\cos \left(-\frac{\sqrt{2}}{2}\right) \\ & =\cos \frac{\sqrt{2}}{2}, \\ a_{2} & =\sin \left[\cos \left(-\frac{\pi}{4}\right)\right]=\sin \frac{\sqrt{2}}{2}, \\ a_{3} & =\cos \left(\frac{3}{4} \pi\right)=-\frac{\sqrt{2}}{2}\sin \frac{\sqrt{2}}{2}. \end{aligned}$$
Then $a_{1}>a_{2}>a_{3}$. Therefore, the answer is (A).