= Strong oblique-shock angle branches
{title2=$\beta_{\rm weak,strong}=[\theta+\arcsin(\gamma\sin\theta),\ \theta+\pi-\arcsin(\gamma\sin\theta)]/2$}
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 $\theta\to0$, their formal limits are $(\gamma+1)\theta/2$ and $\pi/2-(\gamma-1)\theta/2$. The weak-branch expansion requires $M\sin\beta\gg1$; at finite <Mach number> its true zero-deflection limit is the <Mach angle>.
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