Could you build an inverse kinematics solver with this? (I recently watched a youtube video of someone iteratively working out the solutions for a robotic arm, by alternating modifying the inputs and the results)

That's an interesting example I hadn't thought of. Probably? I'll need to try it. Thank you for the suggestion!

I think one issue will be that trig functions are kinda weird because they are non-injective. So they work but they are awkward (try solving cos(A1) = 0.5). Inverse kinematics is so well studied, you're probably better off using a dedicated algorithm.

Closed form solutions might be more efficient (in time, energy) and perhaps more numerically stable.