I have a circle like so
Given a rotation θ and a radius r, how do I find the coordinate (x,y)? Keep in mind, this rotation could be anywhere between 0 and 360 degrees.
For example, I have a radius r of 12 and a rotation θ of 115 degrees. How would you find the point (x,y)?
From the picture, it seems that your circle has centre the origin, and radius $r$. The rotation appears to be clockwise. And the question appears to be about where the point $(0,r)$ at the top of the circle ends up.
The point $(0,r)$ ends up at $x=r\cos\theta$, $y=r\sin\theta$.
In general, suppose that you are rotating about the origin clockwise through an angle $\theta$. Then the point $(s,t)$ ends up at $(u,v)$ where $$u=s\cos\theta+t\sin\theta\qquad\text{and} \qquad v=-s\sin\theta+t\cos\theta.$$