TITLE: Is spherical trigonometry a dead research area?
QUESTION [36 upvotes]: When I was an undergrad, the field of spherical trigonometry was cited as a once-popular area of math that has since died. Is this true? Are the results from spherical trigonometry relevant for contemporary research?

REPLY [2 votes]: Other answers give examples of advanced research, but I thought I'd point out a modest result from my somewhat recreational investigations in the area:

If $W$ is area of the "hypotenuse-face" of a spherical tetrahedron with right "leg-faces" of area $X$, $Y$, $Z$, then
$$\cos\tfrac12W = \cos\tfrac12X\cos\tfrac12Y\cos\tfrac12Z+\sin\tfrac12X\sin\tfrac12Y\sin\tfrac12Z \tag{$\star$}$$

This is a counterpart of de Gua's theorem (aka, the "Pythagorean Theorem for Euclidean tetrahedra"):
$$W^2 = X^2 + Y^2+Z^2$$
Of course, a hyperbolic counterpart exists as well; just append "h"s to all the trig functions in $(\star)$, and change "$+$" to "$-$". There are even associated Laws of Cosines, but I won't go into that here.
I will give a shout-out to an ancient, still-open question of mine: "Pythagorean theorem for right-corner hyperbolic simplices?
", which specifically seeks the $4$-dimensional hyperbolic counterpart of $(\star)$. (Since spherical and hyperbolic relations are likely comparable, this also counts as an open question in spherical geometry.)