I also invented this! There is cool stuff like angle adding and angle doubling formulas, but the main downside is that you can only directly encode 180 degrees

I also invented this! There is cool stuff like angle adding and angle doubling formulas, but the main downside is that you can only directly encode 180 degrees of rotation. I use it for FOV in my games internally! (With degrees as user input of course.) In order to actually use it to replace angles, I assume you'd want to use some sort of half angle system like quaternions. Even then you still have singularities, so it does have its warts.