Strong oblique-shock angle branches (${\beta }_{\mathrm{weak},\mathrm{strong}}=[\theta +\arcsin (\gamma \sin \theta ),\theta +\pi -\arcsin (\gamma \sin \theta )]/2$)

Measure shock angle $\beta$ from the incident flow to the shock front, and deflection $\theta$ towards it. The strong normal-compression ratio and continuity of tangential velocity give $\tan \beta =[(\gamma +1)/(\gamma -1)]\tan (\beta -\theta )$. Angle-addition identities reduce this to $\sin (2\beta -\theta )=\gamma \sin \theta$. The two title branches exist for $0<\theta <\arcsin (1/\gamma )$ and merge at $\beta =\pi /4+\theta /2$. As $\theta \to 0$, their formal limits are $(\gamma +1)\theta /2$ and $\pi /2-(\gamma -1)\theta /2$. The weak-branch expansion requires $M\sin \beta \gg 1$; at finite Mach number its true zero-deflection limit is the Mach angle.