The Law of Sines and The Law of Cosines are of paramount importance in the field of trigonometry because these two theorems establish relationships satisfied by the three sides and the three angles of any triangle. In this article, the authors use these two laws to discover a host of other trigonometric relationships that exist within any triangle. These other relationships are not found in most algebra and trigonometry textbooks, but can easily be derived using elementary methods, like substitution and algebraic manipulation, that are familiar to students in an algebra and trigonometry class. (Contains 1 figure.)