After watching one of Peyam's recent videos (this one: https://www.youtube.com/watch?v=NVC1w4_ulzI) i was curious if you could do something similar for sine and cosine. Turns out you can!
Now this only really works for theta in quadrant 1, but you could probably split it into 4 cases and use similar proofs to get other cases.
Wow, this looks really really cool. I will have to read over it tomorrow during the day. Thanks for sharing. And I also sent this to Dr. P too.