Finding the points of a circle by using one set of coordinates and an angle

Question

I know the image below isn't to scale and that the angle isn't quite at the centre point but can we just imagine it is picture perfect....

I know the coordiantes and point (x,y) lets say they are (1,1) for simplicity, I also know the angle at centre point is 15 degrees. What is the method I use to find the coordinates at x1,y1.

Please note this is not doing my 'homework' for me, despite it being broad. It is part of a project I am on at work and I have simply came to a mind block with this common calculation - cant seem to find it on the internet without it going into arc lengths and pythagoras (which I will later need)

This is impossible without some more information. Do you know the radius, or the center point, or anything else? — bubba, Mar 05 '14 at 11:17
centre point is 0,0 and yes the radius is known, for simplicity call it 10 arbitrary units — Amie-lea Sambrook, Mar 05 '14 at 11:21

score 2 · Answer 1 · edited Sep 26 '15 at 01:53

Let's call the radius $r$. Since the original point $(x,y)$ is on the circle, you can find some angle $\alpha$ such that $x = r\cos\alpha$ and $y = r\sin\alpha$. You could get $\alpha$ by calculating atan2(x,y), for example.

Then the new point $(x_1,y_1)$ is at angle $\beta = \alpha - 15^\circ$, so its coordinates are $x_1 = r\cos\beta$ and $y_1 = r\sin\beta$.

Or, a bit more directly: $$ x_1 = \; x\cos 15^\circ + y\sin 15^\circ \\ y_1 = -x\sin 15^\circ + y\cos 15^\circ \\ $$ This assumes that positive angles (like your 15 degrees) correspond to clockwise movement.

Finding the points of a circle by using one set of coordinates and an angle

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