 Solution

In time $\Delta t$ the particle advances $\beta c\Delta t$, while a light wavelet travels $c\Delta t/n$. Coherent addition along the direction of emitted photons gives the Cherenkov cone angleThe angle is measured between photon propagation and the particle velocity. It is complementary to the angle between the conical wavefront surface and the trajectory; confusing those two angles exchanges sine and cosine. At threshold the photon cone closes, $\alpha \to 0$.