Geometric Verification of Rotation of a Point
Manipulate the sliders to change the point and the angle of rotation.
The various figures show the product of x and y and the sine and cosine of the angle of rotation by constructing right triangles with the length of x and y as the hypotenuse.
Those triangles are then translated to create the sum that the derivation suggests. From that translation, you can see the location of the rotated point.