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When $\tan β$ meets all the mixing angles

Models with two-Higgs-doublets and natural flavour conservation contain $\tan β= v_2 / v_1$ as a physical parameter. We offer here a generalization of a recently proposed idea where only the Cabibbo angle, $θ_\text{c} \simeq 0.22$, was related to $\tan β$ 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 $\tan β$ and one recovers trivial mixing in the limit $β\rightarrow 0$. We also explore the consequences in lepton mixing by adopting a type I seesaw mechanism with three heavy right-handed neutrinos.