We are asked to address the underlying basis for the equivalence of the Law of Cosines, the Law of Sines, and PT...

As I see it (others may differ), the "deep" core of the connections between these theorems is that they are simply alternative manifestations of the "flatness" of the Euclidean plane. (I have alluded to this in my CTK pages on PT and the Parallel Postulate.)

It is well-known that, in the context of "absolute" geometry, the existence of similar, non-congruent triangles is equivalent to the Parallel Postulate. (Only a single pair of them is needed! -- I have written a CTK page with the proof.) It is also well-known that PT is a feature of Euclidean geometry -- indeed, (again, in the context of absolute geometry), PT is equivalent to the Parallel Postulate (I wrote a CTK page on this, too.) That is, the existence (and theory of) similar triangles and PT are both (equivalent) manifestations of the "flatness" of the Euclidean Plane.

But the very definitions of the trigonometric functions (in so far as they describe the triangles of our geometry, and are not simply analytic objects) presuppose the existence and properties of similar (right) triangles -- that is, the flatness of the plane ties together the whole discussion.

"Hope this helps".

Scott.