Given two sides $a,b$ of a spherical triangle and the angle $C$ between them, the spherical law of cosines gives an elegant formula for the missing edge length $c$:
$\cos c = \cos a \cos b+ \sin a \sin b \cos C.$
I have a spherical quadrilateral and know the lengths of three consecutive edges $a,b,c$, and the angles between them $\theta_{ab}$ and $\theta_{bc}$. The last edge length (and its two adjacent angles) are unknown.
Is there an elegant formula for $\cos d$ in terms of the known lengths and angles, and preferably needing only the cosine (and not the sine) of $\theta_{ab}$ and $\theta_{bc}$?
I know it is possible to solve for $d$, for instance by arbitrarily diagonalizing the quad and then "cutting the ear" by solving for the new angles. But when I do this I get a horrible mess of trig, and am hoping that a simplified formula is known.