Areas of tetrahedron faces in proportion to opposite solid angles?

Question

Is there a relationship analogous to the law of sines for triangles, but for tetrahedra? A natural generalization would be $$ a : b : c : d \;=\; \sin A : \sin B : \sin C : \sin D $$ where $a,b,c,d$ are face areas, and $A,B,C,D$ are opposite solid angles. If the above does not hold, does some other similar relationship hold connecting $\{a,b,c,d\}$ to $\{A,B,C,D\}$?