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I have a circle like so

circle with a given radius $r$, with angle $\theta$ to the $y$-axis

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 of 12 and a rotation θ of 115 degrees. What would the point (x,y) be?

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Do you recall polar coordinates? $(r,\theta)$ [polar]=$(r\cos(\theta),r\sin(\theta))$ [cartesian]. – Clayton Dec 16 '12 at 17:30
Nope but that does look promising! Thanks – CoderTheTyler Dec 16 '12 at 17:33
up vote 8 down vote accepted

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 $x=r\sin\theta$, $y=r\cos\theta$.

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 $$u=s\cos\theta+t\sin\theta\qquad\text{and} \qquad v=-s\sin\theta+t\cos\theta.$$

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Thanks for the clear answer. It works like a charm! – CoderTheTyler Dec 16 '12 at 18:49

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