= Solution
For fixed $\beta$, differentiate the apparent <speed> with respect to angle:
$$
\frac{d\beta_{\rm app}}{d\theta}=\frac{\beta(\cos\theta-\beta)}{(1-\beta\cos\theta)^2}.
$$
The maximum lies at $\cos\theta=\beta$ and has value $\beta_{\rm app,max}=\beta/\sqrt{1-\beta^2}=\gamma\beta$. Hence strict <apparent superluminal motion> requires
$$
\boxed{\beta>\frac1{\sqrt2}\simeq0.7071.}
$$
The threshold is an infimum: equality yields exactly $\beta_{\rm app}=1$ at $\theta=\pi/4=45^\circ$, rather than a strictly superluminal value. For a chosen larger $\beta$, the optimal inclination is $\theta=\arccos\beta$, and the full superluminal interval satisfies $\sin\theta+\cos\theta>1/\beta$.
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