Make sure that you have osnap enabled, and it is set to pick on the center point of a circle, also make sure that polar tracking is enabled.
Draw a circle using any method you like, then draw a line with the starting point at the center of the circle, extend horizontally using polar tracking to an intersect with the edge of the circle. Close the line by pressing enter.
Now using the "rotate" command, select the line you just drew, and it will ask you for a start point, and once again select the center of the circle/end point of the line. You can now rotate this line to what ever angle you like.
Now you can mirror this line using the "mirror" command, select the start point of the mirror line from the center of the circle, extend horizontally using polar tracking again along the path where your original line was. Hit enter and it will ask you if you want to delete the original object, select no by just hitting enter.
Now you can trim the section of circle between the two lines using "trim", select both lines and hit enter, then select the section of circle you want to remove between the two lines. Hit enter.
You're done.