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0|0|0|||||Trigonometric proof of Pythagorean theorem %28proof 84%29.|Janez Lesjak|1|15:45:12|02/01/2011|What is actually meant by trigonometric proof of Pythagorean theorem%3F%0D%0ATrigonometric functions are defined in analysis by infinite series. All of their properties are then derived from them%2C including the number Pi and the %22Pythagorean relation%22 %28sin x%29%28sin x%29 %2B %28cos x%29%28cos x%29 %3D 1. The important question is how to relate the argument x to the Euclidean angle%3F
1|1|0|||||RE%3A Trigonometric proof of Pythagorean theorem %28proof 84%29.|alexb||19:01:11|02/01/2011|Far as I am concerned sine is defined as the ratio of a leg of a right triangle to its hypotenuse.%0D%0A%0D%0AI am curious why some function be called %22trigonometric%22 if they defined in analysis by infinite series.%0D%0A%0D%0AHowever%2C you asked a question. A trigonometric proof is a proof that uses trig functions in a significant way.
2|2|1|||||RE%3A Trigonometric proof of Pythagorean theorem %28proof 84%29.|Janez Lesjak|1|19:30:11|02/08/2011|OK%2C I know how trig functions are defined in geometry - by ratios of sides of a right angled triangle%2C but then the fundamental relation between sine and cosine requires Pythagorean theorem%2C the same holds for the addition theorems. So what was earlier - the egg or the hen%3F%0D%0A%0D%0AUsing the trig functions in a proof of a geometric theorem actually means using theorems %28and axioms%29 on similarity and congruence%2C so trig functions actually do not bring anything new to geometry. The term %22trigonometric%22 has of course a historical origin%2C as a strict mathematical definition of trig functions comes from analysis %28not necessarily by infinite series%2C there is a definition by inversion of algebraic integrals%29.%0D%0A
3|3|2|||||RE%3A Trigonometric proof of Pythagorean theorem %28proof 84%29.|alexb||19:41:27|02/08/2011|%3EOK%2C I know how trig functions are defined in geometry - by %0D%0A%3Eratios of sides of a right angled triangle%2C but then the %0D%0A%3Efundamental relation between sine and cosine requires %0D%0A%3EPythagorean theorem%2C the same holds for the addition %0D%0A%3Etheorems. So what was earlier - the egg or the hen%3F %0D%0A%0D%0AI think that in mathematics this is a wrong question. There is a multitude of sentences that could be derived from each other. There is no point in asking which came first%2C except historically or chronologically. But this is not what we are talking about here.%0D%0A%0D%0A%3EUsing the trig functions in a proof of a geometric theorem %0D%0A%3Eactually means using theorems %28and axioms%29 on similarity and %0D%0A%3Econgruence%2C so trig functions actually do not bring anything %0D%0A%3Enew to geometry. %0D%0A%0D%0AI believe I out forth an explanation as to why I think that proof desrves to be called %22trigonometric%22. It is based on some identities that%2C although could be written in terms of similarities%2C gain at least in clarity or owe their existence outright to their trigonometric interpretation. Do write the sibtraction formula for sine in %22geometric terms%22.%0D%0A%0D%0ASo I believe that that proof is made significantly more transparent through the use of trigonometric functions.%0D%0A%0D%0A%3EThe term %22trigonometric%22 has of course a %0D%0A%3Ehistorical origin%2C as a strict mathematical definition of %0D%0A%3Etrig functions comes from analysis %28not necessarily by %0D%0A%3Einfinite series%2C there is a definition by inversion of %0D%0A%3Ealgebraic integrals%29. %0D%0A%0D%0ANeah%2C can%27t believe that. Etimologically%2C %22trigonometry%22 means measuring a triangle.%0D%0A