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0|0|0|||||Pythagorean Theorem Proof -- Trig or Not%3F|Grapes||18:21:58|02/08/2008|Over at %5Ba href%3D%22http%3A%2F%2Fwww.bautforum.com%2F%22%5DBAUT%5B%2Fa%5D%2C we%27ve been reading the Pythagorean Theorem page%0D%0Ahttp%3A%2F%2Fwww.cut-the-knot.org%2Fpythagoras%2Findex.shtml%0D%0A%0D%0AVery nice%2C and I appreciate all the work that has been done.  But a question has come up%2C about the following proof%2C which doesn%27t seem to appear on the webpage in this form%3A%0D%0A%0D%0ATake right triangle ABC with opposite sides a%2C b%2C and c%2C C the right angle.%0D%0A%0D%0ADrop a perpendicular from C to the side c. That divides c into two parts%2C which are easily shown to measure b cos A and a cos B%2C by the definition of cosine. In other words%2C %0D%0Ac %3D b cos A %2B a cos B%0D%0ABut%2C from the original triangle%2C it%27s easy to see that cos A %3D b%2Fc and cos B %3D a%2Fc so%0D%0Ac %3D b %28b%2Fc%29 %2B a %28a%2Fc%29%0D%0A%0D%0AThere is a comment at the webpage that says that no trigonometric proof is possible.   Other than the basic definition of cosine%2C this proof does not use any trig identities %28many of which are based upon the Pythagorean theorem of course%29.  It is equivalent%2C geometrically%2C to a couple of the proofs%2C I think%2C but a lot of the proofs differ by subtle steps.%0D%0A%0D%0AThe question then is%3A  Is that a trigonometric proof of the Pythagorean Theorem%3F%0D%0A%0D%0A
2|1|0|||||RE%3A Pythagorean Theorem Proof -- Trig or Not%3F|alexb||20:00:47|02/08/2008|%3E The question then is%3A Is that a %0D%0A%3E trigonometric proof of the Pythagorean %0D%0A%3E Theorem%3F%0D%0AIn all honesty I do not know. The reason you may call it trigonometric is the apparent reliance on the definition of the cosine. However%2C our ability to define cosine %28or perhaps use it%29 is conditioned on the properties of similar triangles. So the essential fact in your derivation is that the ratio of a leg to the hypotenuse in a right triangle depends only on the angle%2C not the size of the triangle. In other words%2C it is the same for all similar triangles. But from the similarity of triangles you get directly that the projection of leg a on the hypotenuse is a%26sup2%3B%2Fc and that of b is b%26sup2%3B%2Fc. So that the appearance of cosine is transitional and is not quite necessary. The proof is of the same variety as %23%23 6%2C 7%2C 8%2C 41.%0D%0A%0D%0ABut you are right in that some proofs only differ in triffle details. So that perhaps your proof deserves to be considered on its own merits as a proof. But think of it%2C what makes cosine a trigonometric function%3F Surely not a plain definition alone but rather a fact that this is a part of a whole branch of tools and activities.%0D%0A%0D%0AWhen I said that a trigonometric proof is impossible the idea was that the trigonometric relations heavily depend on the Pythagorean theorem and%2C for that reason%2C can%27t serve as a basis for a proof of the latter. But%2C unless you use a trigonometric relation%2C as opposed to using just a definition%2C should we call the derivation triginometric%3F I do not know. %0D%0A%0D%0AI can live with whatever the math community may decide. Ambiguity%2C too%2C won%27t impair my sleep.%0D%0A%0D%0A%0D%0A
3|2|2|||||RE%3A Pythagorean Theorem Proof -- Trig or Not%3F|Grapes||07:33:31|02/08/2008|%3EWhen I said that a trigonometric proof is impossible the %0D%0A%3Eidea was that the trigonometric relations heavily depend on %0D%0A%3Ethe Pythagorean theorem and%2C for that reason%2C can%27t serve as %0D%0A%3Ea basis for a proof of the latter. But%2C unless you use a %0D%0A%3Etrigonometric relation%2C as opposed to using just a %0D%0A%3Edefinition%2C should we call the derivation triginometric%3F I %0D%0A%3Edo not know. %0D%0A%0D%0AI got the impression that you were quoting or paraphrasing an old comment from someone else%2C too.  I thought it was interesting.%0D%0A%0D%0AIt%27s true the proof doesn%27t use identities %28and how could it%2C most of them take advantage of the pythagorean theorem at some point%29%2C but it does use basic trig relationships--similar triangles--even if we were to sidestep the definition of cosine.  %0D%0A%0D%0AThe discussion has been along the lines of what constitutes a trig proof%2C as opposed to algebraic or geometric--since they%27re pretty inter-related and each can be expressed in terms of the other%2C more or less.  Now%2C a %22graphical%22 proof is probably not too ambiguous%2C it%27s usually %22I know it when I see it.%22%0D%0A%0D%0AAnyway%2C what are the chances of adding it to the list%3F %3A%29
4|3|3|||||RE%3A Pythagorean Theorem Proof -- Trig or Not%3F|alexb||14:43:53|02/09/2008|%3E%3EWhen I said that a trigonometric proof is impossible the %0D%0A%3E%3Eidea was that the trigonometric relations heavily depend on %0D%0A%3E%3Ethe Pythagorean theorem and%2C for that reason%2C can%27t serve as %0D%0A%3E%3Ea basis for a proof of the latter. But%2C unless you use a %0D%0A%3E%3Etrigonometric relation%2C as opposed to using just a %0D%0A%3E%3Edefinition%2C should we call the derivation triginometric%3F I %0D%0A%3E%3Edo not know. %0D%0A%3E%0D%0A%3EI got the impression that you were quoting or paraphrasing %0D%0A%3Ean old comment from someone else%2C too.  I thought it was %0D%0A%3Einteresting. %0D%0A%0D%0AYes%2C Loomis mentions this in his book. %0D%0A%0D%0A%3EIt%27s true the proof doesn%27t use identities %28and how could %0D%0A%3Eit%2C most of them take advantage of the pythagorean theorem %0D%0A%3Eat some point%29%2C but it does use basic trig %0D%0A%3Erelationships--similar triangles--even if we were to %0D%0A%3Esidestep the definition of cosine. %0D%0A%0D%0ASo%2C this is not what the discussion is about. OK.%0D%0A%3E%0D%0A%3EThe discussion has been along the lines of what constitutes %0D%0A%3Ea trig proof%2C as opposed to algebraic or geometric--since %0D%0A%3Ethey%27re pretty inter-related and each can be expressed in %0D%0A%3Eterms of the other%2C more or less.  Now%2C a %22graphical%22 proof %0D%0A%3Eis probably not too ambiguous%2C it%27s usually %22I know it when %0D%0A%3EI see it.%22 %0D%0A%0D%0AIn my view%2C a proof deserves to be referred to as %22trigonometric%22 if without trigonometry it makes no sense. I do not believe your derivation passes this test. Instead of using cosine%2C state simply the proportions that follow from the similarity of triangles. That the ratios involved got honored with a notation used widely is irrelevant to the proof.%0D%0A%0D%0AAnother question to ask is what constitutes a distinct proof. %0D%0A%0D%0A%3EAnyway%2C what are the chances of adding it to the list%3F %3A%29 %0D%0A%0D%0AIf you refer to my list%2C zero.%0D%0A%0D%0A%0D%0A
5|4|4|||||RE%3A Pythagorean Theorem Proof -- Trig or Not%3F|Grapes||09:28:39|02/10/2008|%3EIn my view%2C a proof deserves to be referred to as %0D%0A%3E%22trigonometric%22 if without trigonometry it makes no sense. I %0D%0A%3Edo not believe your derivation passes this test. Instead of %0D%0A%3Eusing cosine%2C state simply the proportions that follow from %0D%0A%3Ethe similarity of triangles. That the ratios involved got %0D%0A%3Ehonored with a notation used widely is irrelevant to the %0D%0A%3Eproof. %0D%0A%3E%0D%0A%3EAnother question to ask is what constitutes a distinct %0D%0A%3Eproof. %0D%0A%0D%0AI%27m definitely not claiming it is a distinct proof.  Most of the proofs are interesting variations%2C though.%0D%0A%0D%0AI think it is most nearly the same as entry %236%2C but by using the cosine definition%2C it becomes more succinct.  %0D%0A%0D%0A%3EIf you refer to my list%2C zero.%0D%0A%0D%0AMaybe a note added to %236%3F  %3A%29
6|5|5|||||RE%3A Pythagorean Theorem Proof -- Trig or Not%3F|alexb||21:31:52|02/11/2008|%3EI think it is most nearly the same as entry %236%2C but by using %0D%0A%3Ethe cosine definition%2C it becomes more succinct. %0D%0A%0D%0AIt does acquire a distinct flavor. I can%27t make my mind about the taste%2C though.%0D%0A%0D%0A%3EMaybe a note added to %236%3F  %3A%29 %0D%0A%0D%0AThis would make sense%2C yes. Thank you.%0D%0A
8|6|6|||||RE%3A Pythagorean Theorem Proof -- Trig or Not%3F|Guoping Zeng|1|13:19:42|04/10/2008|I wote a note long time ago about the trignometric proof of Pythagorean theorem%2C which was similar to 1. I didn%27t know this post until I was directed by Prof. Bogomolny. I think we should change our statement %22There are no trignometric proofs of the Pythagorean Theorem%22 to something like %22There are no trigometric proofs of the Pythagorean theorem that use any trignometric identities such as the Pythagorean Identity%22. On the other hand%2C trignometry can be developed totally independent of the Pythagorean theorem.%0D%0A%0D%0A%0D%0A%0D%0A%3E%3EI think it is most nearly the same as entry %236%2C but by using %0D%0A%3E%3Ethe cosine definition%2C it becomes more succinct. %0D%0A%3E%0D%0A%3EIt does acquire a distinct flavor. I can%27t make my mind %0D%0A%3Eabout the taste%2C though. %0D%0A%3E%0D%0A%3E%3EMaybe a note added to %236%3F  %3A%29 %0D%0A%3E%0D%0A%3EThis would make sense%2C yes. Thank you. %0D%0A
9|7|8|||||RE%3A Pythagorean Theorem Proof -- Trig or Not%3F|alexb||13:49:10|04/10/2008|I believe that the designation %5Bb%5DTrigonometry%5B%2Fb%5D stands for or is a shorthand of %5Bb%5DTheory of Trigonmetry%5B%2Fb%5D. The latter has a more comprehensive feeling and clearly could not be meaningfully reduced to a few function definitions. Trigonometry is a theory. The essence of trigonometry%2C as that of any other theory%2C is derivation built on axioms and definitions. %0D%0A%0D%0AIn this particular case of a proof of the Pythagorean theorem%2C a simple proof based on the similarity of triangles is quite self-contained. Introducing cosine into the proof does not even simplify it. cosine just embodies the fact of similarity and nothing less%3B its use modifies the verbiage of the proof. There is nothing trigonometric about this change.