= 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.
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