3. Let $M$ be the midpoint of side $BC$ of $\triangle ABC$, $AB=4$, $AM=1$. Then the minimum value of $\angle BAC$ is $\qquad$ .
3. $150^{\circ}$.

Let the symmetric point of $A$ with respect to $M$ be $N$. Then quadrilateral $A B N C$ is a parallelogram. Therefore,
$$
\angle B A C=180^{\circ}-\angle A B N \text {. }
$$

Draw a circle with center at $A$ and radius 2. Then when $B N$ is tangent to this circle, $\angle A B N$ is maximized and equals $30^{\circ}$.
Thus, the minimum value of $\angle B A C$ is $150^{\circ}$.