There is a [lively site][1] which publishes research questions, some of which essentially involve spherical geometry and trigonometry. Recent examples include:

 - [Cutting a spherical surface into mutually non-congruent pieces of
   equal area][2]
 - [Can you perturb an inscribed polytope so all its edges grow?][3]


  [1]: https://mathoverflow.net/
  [2]: https://mathoverflow.net/questions/415631/cutting-a-spherical-surface-into-mutually-non-congruent-pieces-of-equal-area
  [3]: https://mathoverflow.net/questions/422290/can-you-perturb-an-inscribed-polytope-so-all-its-edges-grow