4. As shown in Figure 2, the base edge length of the regular triangular prism $A B C-A_{1} B_{1} C_{1}$ is $a$, and the side edge length is $\sqrt{2} a$. Then the angle between $A C_{1}$ and the side face $A B B_{1} A_{1}$ is ( ).
(A) $30^{\circ}$
(B) $45^{\circ}$
(C) $60^{\circ}$
(D) $75^{\circ}$
4. A.

Take the midpoint $D$ of $A_{1} B_{1}$, and connect $A D$ and $C_{1} D$.
Since $C_{1} D \perp A_{1} B_{1}$, we have $C_{1} D \perp$ plane $A B B_{1} A_{1}$. Therefore, the angle formed by $A C_{1}$ and $A D$ is the required angle.
Given $C_{1} D=\frac{\sqrt{3}}{2} a$ and $A D=\frac{3}{2} a$, we have
$$
\tan \angle D A C_{1}=\frac{C_{1} D}{A D}=\frac{\sqrt{3}}{3}, \angle D A C_{1}=30^{\circ},
$$

Thus, the angle between $A C_{1}$ and the side face $A B B_{1} A_{1}$ is $30^{\circ}$.