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CTK Exchange
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jmolokach
Member since Jan-11-11
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Feb-22-11, 11:12 PM (EST) |
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3. "RE: Equivalence of the Law of Sines, PT, and Law of Cosines"
In response to message #2
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>John, you may want to seek a bigger community. Not every one
>shares your interests. This must be clear and reasonably
>expected. Understood, I just thought I might ask about this continuity issue since you mentioned in the other thread that all I needed was continuity to prove the PT from this approach. It'seems like a very simple proof to say that sin(A+B) = sinC is the PT - but perhaps this is just a restatement of the same and not really a proof. I was hoping I had at least done an adequate proof. But I will move on to other things if this all is trivial. Thanks for the reply. molokach |
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sbrodie
Member since Dec-28-10
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Mar-03-11, 05:23 PM (EST) |
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4. "RE: Equivalence of the Law of Sines, PT, and Law of Cosines"
In response to message #0
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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.
Scott |
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jmolokach
Member since Jan-11-11
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Mar-07-11, 10:00 AM (EST) |
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5. "RE: Equivalence of the Law of Sines, PT, and Law of Cosines"
In response to message #4
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Yes, it does actually. I set out to do a completely algebraic proof that a->b->c->a (a the law of sines, b the PT, c the law of cosines). All of which assumes the Euclidean plane (ie triangles angles sum to 180). But I have since taken two extra rather important points from my paper after writing it. 1) The paper is of little value since all three are more easily derived "from scratch." The fact that they circularly imply each other is not a big surprise, as Alex said, as this quite often happens in mathematics. There is also still this unanswered question as to whether one has license to use sin(A+B) = sin(C) = 1 to prove the PT. I set out to defend this in extending the definition of sine to angles that are not acute (90 in particular). So I suppose one could say that instead of proving the PT, I just picked out a special case of the law of sines which happens to lead to the PT (again no surprise). 2) The PT is not really necessary in this at all. Drawing the three altitudes in a triangle both leads to the law of sines and the law of cosines, without recourse to the PT (see my other post where I have linked to my PWW of the cosine law independent of the PT). That same diagram (without the exterior rectangles and square) is also what is used to derive the law of sines. And so the law of sines and the law of cosines both stem from drawing altitudes in a triangle. The PT is just a special case where the orthocenter is also a vertex of an angle. If you follow the other thread, you will see that I am attempting to create (or convince Alex to create) a droodle that illustrates this fact from my aforementioned PWW. Thanks for your insight.
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