Free Rotate does not have any angular control, it is completely free movement. If you want to control the angle, you will need to add some sort of angular constraint, and specify the angle off of some reference, such as an origin or work plane. For example, an angular constraint between a plane on the round object, and a plane passing through the axis of rotation.
This is a very bad tool, almost like it was only developed halfway. Not all parts are cylindrical or simple cubes. We work with such weirdly shaped automotive components and its very difficult to predict your constraint beforehand, exactly why you need a incremented move to degree of rotation or move per grid x/y input. Perhaps the software developers should spend some time on Solid Edge and see how easy life can be. I really don't understand why simple tools have to be so difficult to use. You don't always know where you are heading with your design, and to plan accordingly is almost impossible.