Cherenkov photon-cone and wavefront-cone angles

For a charge moving at $v=\beta c$, the Cherenkov radiation photon direction makes angle ${\theta }_C$ to the trajectory with $\cos {\theta }_C=1/(n\beta )$. The full cone of photon directions has opening $2{\theta }_C$. The corresponding conical wavefront is perpendicular to those directions and has half-angle $\mu =\pi /2-{\theta }_C$, satisfying $\sin \mu =1/(n\beta )$. Therefore a full opening angle $\alpha$ gives $n=1/[\beta \cos (\alpha /2)]$ for the photon-direction cone, but $n=1/[\beta \sin (\alpha /2)]$ for the wavefront cone. These are different geometrical definitions, not competing approximations to the same angle.