The predicate of the sentence is the part of it that modifies or gives information about the subject, so it must include a verb and it could also include some modifier. A simple predicate has only a verb or verb phrase; a compound predicate includes two or more verbs; a complete predicate includes not only a verb but also other modifiers that give more information.
Taking all this into consideration, tipped his hat politely is the complete predicate of the sentence.