Everything seemed so simple and simple until I had to program it.
What I have
I uploaded an image to better explain it.
I want to be able when I rotate the image of the gray circle with 10 degrees to calculate the red buttons with the new coordinates (x1y1, x2y2).
This should not be difficult to achieve for those who know math, but I have not been able to find a suitable solution. I also searched here and did not find a working solution. Any help is appreciated. Thanks you

Working solution as below:
- first take care of the rotation angle, at each redraw just increase it
angle = angle+mainRotationAngle;
float x = (float) (center.X + Math.cos(angle*Math.PI / 180F) * radius
float y = (float) (center.Y + Math.sin(angle*Math.PI / 180F) * radius
button.setX(x);
button.setY(y);