Measure the slope or angle of the sloped ceiling that runs into the flat part, using an angle finder.
Set the miter saw at 1/2 the angle of the slope and cut the molding, using a miter saw.
Make a transition piece. This will be a small, triangular piece that will connect the horizontal piece of crown molding to the sloped piece to be attached to the vaulted or cathedral part of the ceiling. Remove some material in back of the transition piece with a utility knife so that it will sit flush on the ceiling.
Cut the horizontal piece of crown molding so that it will connect to the transition piece. Set the miter saw to 45 degrees to make the cut on the horizontal piece.
Apply construction adhesive or polyurethane glue to the wall and both flat sides of the crown molding and the angled cuts that will make the joints. Pre-drill nail holds with a drill/driver. Fasten the crown molding to wall studs and ceiling joists with finishing nails for extra support, using a hammer. Use 4d, 6d, or 8d finishing nails depending on the thickness of the crown molding.
Measure the slope of one part of your vault or cathedral ceiling with the angle finder. Set the miter saw to 1/2 the sloped angle and cut the crown molding at the place where it will meet the second slope.
Cut the piece of crown molding for the second slope that will meet the piece from the first slope. Always make a right-sided cut on one piece and a left-handed cut on the other piece.
Attach the crown molding with adhesive, finishing nails and a hammer. Always drill pilot holes for the nails so you won't damage the molding.
Fill nail holes with putty and use caulk or joint compound to fill in imperfections in the joints. Sand the joints.