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Subject: "Trigonometric Proof of the Pythagorean Theorem"

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C Reineke
Member since Jul-9-10

Jul-27-10, 09:29 AM (EST)

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"Trigonometric Proof of the Pythagorean Theorem"

https://www.cut-the-knot.org/pythagoras/TrigProof.shtml

Dear Alex,

it is very easy to derive the subtraction formulas from Euler’s formula.
(See the attached WORD-file, please).

But that would mean that you can derive the Pythagorean Theorem from
Euler’s formula.

Kind regards

Chris

Attachments
https://www.cut-the-knot.org/htdocs/dcforum/User_files/4c4edf1b522ee130.txt

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alexb
Charter Member
2764 posts

Jul-27-10, 10:16 AM (EST)

1. "RE: Trigonometric Proof of the Pythagorean Theorem"
In response to message #0

> But that would mean that you can derive the Pythagorean Theorem from Euler’s formula.

Yes, I suspect that Pythagorean theorem is necessary for the development of complex numbers and Euler's formula, in particular. Which does not invalidate the fact that Euler's formula is one way to derive addition and subtraction formulas for sine and cosine. It is actually a very convenient way.

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C Reineke
Member since Jul-9-10

Aug-02-10, 10:09 AM (EST)

2. "RE: Trigonometric Proof of the Pythagorean Theorem"
In response to message #1

Indeed, a derivation from Euler’s formula is possible:

https://mathworld.wolfram.com/PythagoreanTheorem.html
Formula (25), (26) and (27)

Let a+ib=ce^ix, then the complex conjugate is

a-ib=ce^-ix.

Multiplying both sides gives

a^2+b^2=c^2 (Machover 1996).

A very nice proof.

Kind regards

Chris

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alexb
Charter Member
2764 posts

Aug-02-10, 07:04 PM (EST)

3. "RE: Trigonometric Proof of the Pythagorean Theorem"
In response to message #2

As I mentioned before, I suspect that the Pythagorean theorem is necessary to make complex numbers work.

Using Eauler's formula shows that the definition is consistent but I believe proves nothing.

Curiously, I failed to locate Machover reference, although I do have online access to all MAA's publications, the AMM in particular.

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jmolokach
Member since Jan-11-11

Feb-15-11, 04:06 PM (EST)

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4. "RE: Trigonometric Proof of the Pythagorean Theorem"
In response to message #2

Chris have you been following the latest threads on complex numbers? I have enjoyed reading a whole new perspective on the axioms and thought this applied to this thread.

Regarding the trigonometric proof, I think I have developed a rather long and involved one stemming from the thread 'A question.' I have posted the paper on it in a new thread and hope you will comment on it.

All the best,

molokach

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