3. In $\triangle A B C$, $\angle B=60^{\circ}, \angle C=75^{\circ}, S_{\triangle A B C}=\frac{1}{2}(3+\sqrt{3}), B C=$
3. 2 .
$$
\angle A=180^{\circ}-60^{\circ}-75^{\circ}=45^{\circ}, S_{\triangle A B C}=\frac{1}{2} a c \sin B=\frac{1}{2} a c \sin B=\frac{1}{2} a c \sin 60^{\circ}=\frac{1}{2}(3+
$$
$\sqrt{3}) \Rightarrow a c=2(1+\sqrt{3}), \frac{a}{c}=\frac{\sin A}{\sin C}=\frac{\sin 45^{\circ}}{\sin 75^{\circ}}=\frac{1}{\sin 30^{\circ}+\cos 30^{\circ}} \Rightarrow c=\frac{\sqrt{3}+1}{2} a$, so
$$
\frac{\sqrt{3}+1}{2} a^{2}=2(\sqrt{3}+1) \Rightarrow a=2
$$