Move a point with known angle on a circle

Question

Having a circle of radius $R$ with the center in $O(0, 0)$, a starting point on the circle (e.g. $(0, R)$) and an angle $\alpha$, how can I move the point on the circle with $\alpha$ degrees? I need to get the second point where it was moved.

Example

The red point is on $(0, R)$ and $\alpha$ is $90$ degrees. The violet circle is where the first point is supposed to be moved, and its coordinates are $(R, 0)$. Then we consider the violet point as starting point and move it with $45$ degrees. The new position will be where the blue circle is.

Answer 1

As you can see from the image, The point you require is at $-45 $ Degree from $0$ . So your required point is $R \cos (-\pi /4), R \sin (-\pi/4)$

Which can be equivalently written as?

Answer 2

well you can consider the following linear transformation given by:

T : $R^2$ $\rightarrow$ $R^2$

given by T(x,y) = (cos$\theta$x - sin$\theta$y,sin$\theta$x + cos$\theta$y).

This does what you asked rotates your point counter-clockwise about the origin.