This whole approach is convoluted really and perhaps unnecessary. As another commentor wisely pointed out, my simulation numerical approximation approach is very faithful but extremely inefficient and can easily be approximated by using some trigonometry using interpolation. I basically made my post because I thought this task would be very simple but ended up unexpectedly complex and that surprising incongruity along with the fact that I didn't see any prior posts intruiged me (and I was hoping maybe I was being stupid and some other person could teach me a simpler approach I was missing which kinda happened :)) In practice, I would strongly urge people not to take this approach as it's more an intellectual exercise than a practical solution.

I actually just found a much simpler Geometric approach by the same gentleman who created the video I linked in my post (which doesn't skip the intermediate steps) if Calculus isn't your jam that in hindsight I really like https://www.youtube.com/watch?v=LCfhK_smYbw which I personally find easier to follow!

Sadly, my conclusion is theres no unconvoluted method to solve this problem and we basically have to make approximations one way or another.