Find the smaller angle between the hour and minute hands of a clock at $3:38$.

The standard formula to calculate the absolute angle $\theta$ between the hour hand and the minute hand of a clock is:
$$\theta =\left|30H-\frac{11}{2}M\right|$$

Given time is $3:38$, so $H=3$ and $M=38$. Substituting these values into the equation:

$$\theta =\left|30(3)-\frac{11}{2}(38)\right|$$

$$\theta =\left|90-11\times 19\right|$$

$$\theta =\left|90-209\right|$$

$$\theta =\left|-119\right|=119^{\circ }$$

Since $119^{\circ }\le 180^{\circ }$, it is the smaller angle between the two hands.