Hello. So I don't know that anyone cares about where this problem comes from, but I'd be happy to explain: it's based on a new seven sided solid called the

Chestahedron, which has all seven faces of equal area (four equilateral triangles and three 'kite' shapes).

The formula below comes from setting the area of the unit equilateral triangles equal to the area of the kite as expressed in terms of the dihedral angle of the base equilateral triangle to the 'petal' equilateral triangles (the other 3).
The long and the short of it is that I need help solving for the angle theta in the following equation:

Any help would be appreciated.

p.s. If this isn't solvable, then there's something wrong with the way the equation was derived, so help in that direction could be helpful as well. (Smirk)