You do have a missing return... All the returns in the block you pasted occur inside of conditionals. The compiler isn't able to guarentee that any of those conditionals will be executed.
Some soultions...
After all the if statements, have something like: return "error!!!Helpme!!";
Or, define String retval; Then, let each if statement say something like retval="east"; and just return retval; at the bottom.
Better still. Keep a Map called direction as a member of your class.
Code:
direction = new HashMap()
direction.put(new Integer(0), "north");
direction.put(new Integer(90), "east");
...
Then, your method gets reduced to return direction.get(new Integer(degrees));