How to surgically remove a fractured tooth root from sinus cavity?

My daughter has had 2 root canals on tooth #3 and recently, it cracked and got so infected that it could not be saved and we had it extracted. Unfortunately, at some point in time, a piece of the root tip has made its way up into the sinus cavity and may need to be surgically removed. We won't know yet until we have give the extracted tooth time to heal. Does anyone know what the symptoms might be when you have a root tip stuck there? My daughter says her jaw burns like someone is inside of it holding a match to it. Also, what does the surgery entail....and is there a name for it?