Example 13. Let $\alpha, \beta$ be acute angles, and $3 \sin \alpha=2 \sin \beta$, $3 \cos \alpha+2 \cos \beta=3$. Find the degree measure of $\beta + \frac{\alpha}{2}$.
Solve for $3 \sin \alpha = 2 \sin \beta$, we get $\frac{\sin \alpha}{2} = \frac{\sin \beta}{3}$. From this, by inversely applying the Law of Sines, construct $\triangle ABC$ such that $\angle A = \alpha, \angle B = \beta, \angle C = 180^{\circ} - (\alpha + \beta), BC = 2, AC = 3$.

In $\triangle ABC$, using the projection theorem, we get
$$
AB = 3 \cos \alpha + 2 \cos \beta = 3 = AC.
$$

Thus, $\angle B = \angle C$, i.e., $\beta = 180^{\circ} - (\alpha + \beta)$, hence
$$
\beta + \frac{\alpha}{2} = 90^{\circ}.
$$

From the above, it is evident that using the construction method to solve related value problems is novel and concise, and is beneficial for training students' thinking abilities.