  Models with two-Higgs-doublets and natural flavour conservation contain $\tan
\beta = v_2 / v_1$ as a physical parameter. We offer here a generalization of a
recently proposed idea where only the Cabibbo angle, $\theta_\text{c} \simeq
0.22$, was related to $\tan \beta$ by virtue of the $\mathbb{D}_{4}$ dihedral
symmetry group. The original proposal consisted of a massless first generation
of quarks and no mixing with the third generation. In our case, through the
addition of a third Higgs doublet with a small vacuum-expectation-value but
very large masses, thus later decoupling, all quarks become massive and quark
mixing is fully reproduced. In fact, all quark mixing angles are expressed in
terms of $\tan \beta$ and one recovers trivial mixing in the limit $\beta
\rightarrow 0$. We also explore the consequences in lepton mixing by adopting a
type I seesaw mechanism with three heavy right-handed neutrinos.
