The conjunction of this answer and this one would seem to imply that for $\theta_1,\theta_2,\theta_3 \in(-\pi/2,+\pi/2),$ $$ |\theta_1+\theta_2+\theta_3| < \frac \pi 2 \text{ if and only if } \tan\theta_1\tan\theta_2 + \tan\theta_1\tan\theta_3 + \tan\theta_2\tan\theta_3<1. \tag 1 $$ And a bit of number crunching and graphing via software seems to suggest that maybe $$ \sup\big\{ \arctan( \tan\theta_1\tan\theta_2 + \tan\theta_1\tan\theta_3 + \tan\theta_2\tan\theta_3 ) : \theta_1+\theta_2+\theta_3=s \big\} \\[8pt] = \frac\pi4(1-\cos s) \text{ if $|s|<\pi/2$ ??} \tag 2 $$ or some function similar to that. And if $|s|>\pi/2,$ put $\inf$ there instead of $\sup.$
The equality in $(2)$ seems far-fetched, and concerning $(1)$ I wonder if there's an efficient way of showing it, as opposed to juxtaposing the two answers I linked. If the hour were not late, I would figure this all out myself, except for the answer to this question: Is this stuff known?
