= Solution
Let $U_1,U_2$ denote the total flow speeds, keeping the normal components distinct from them. The original diagram measures $\beta$ from the initial wall direction to the <shock wave> itself. The outgoing <velocity> is turned by $\theta$, so it makes angle $\beta-\theta$ with the shock.
Resolve each <velocity> tangent and normal to the <oblique shock>. Tangential <momentum conservation>, with nonzero mass flux and no tangential surface force, makes the tangential <velocity> continuous:
$$
U_1\cos\beta=U_2\cos(\beta-\theta).
$$
The normal flow obeys the strong <normal shock wave> result from part (b):
$$
U_2\sin(\beta-\theta)=\frac{\gamma-1}{\gamma+1}U_1\sin\beta.
$$
Divide the second relation by the first to obtain
$$
\boxed{\tan\beta=\frac{\gamma+1}{\gamma-1}\tan(\beta-\theta).}
$$
The approximation is controlled by the upstream normal <Mach number> $M\sin\beta\gg1$, not just by $M\gg1$. This distinction matters on the small-angle weak branch in part (d). The angle convention is also important: formulas using the angle to the shock normal instead would interchange sine and cosine.
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