Let's consider my avatar as the original image. I make sure it is cropped into a circle.
Then in photoshop, I do Filter->Distort->Polar coordinates->Polar to rectangular:
Then, I resize the contents (but not the container [e.g. select all, edit->transform->scale]) of the image horizontally. The percentage I reduce is the percentage of degrees I will remove:
Finally, I go back to polar coordinates, and retrim as a circle: (OP: I don't know what they mean by "retrim as a circle," though, so I stop after selecting "rectangular to polar")
Wrapping the image around the removed wedge is an exercise left to the reader.