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Proving Pythagorean theorem using Trigonometry

I recently read that two students have proven the Pythagorean Theorem using Trigonometry.
I haven't looked at their proof, but the problem seems straight forward.
Consider a right triangle with sides a and b and hypotenuse c, and angle theta opposite side a.
Then the tan(theta) = a/b.
sine(theta) = a/c.
So c = a/sine(theta) = a/sine(arctan(a/b).
Using Mathematica:

a/Sin[ArcTan[a/b]] evaluates to Sqrt[1 + a^2/b^2] b

which is equivalent to Sqrt[a^2 +b^2]

    In[1]:= a/Sin[ArcTan[a/b]]

    Out[1]= Sqrt[1 + a^2/b^2] b

Am I missing something ?

POSTED BY: Frank Kampas
2 Replies
Posted 10 days ago

US Television show 60 Minutes recently broadcast a feature on the two high school students -- now first year students at university. The broadcast is now featured on a 13-minute video on YouTube; pertinent details start @1:58 into the video. The students had created two separate proofs of the Pythagorean Theorem. One is the "waffle cone" proof, which has gotten all the attention. The interview contained insufficient information on the second proof to understand it. The students said that they have 5 more proofs of the Pythagorean Theorem, and believe they have a framework for 5 more. No details were provided. One of the students is seeking a pharmacy major at Xavier; the other is pursuing an environmental engineering degree at LSU.

Most interesting was the brief conversation with their high school mathematics teacher Michelle Williams (@4:14), who proposed the bonus problem on the school-wide contest:

Q: And did you think anyone would solve it?

A: Well, I wasn't necessarily looking for a solve [...] I was looking for some ingenuity. [smile]

Hear, hear! I thought that Conrad Wolfram would be interested in this school's -- and this particular teacher's -- approach to maths.

POSTED BY: Phil Earnhardt
Posted 15 days ago

The problem is that the foundation for trigonometry is rooted in the Pythagorean Theorem and that any proof would seemingly be based on circular reasoning. Mathematicians thought any proof based on trigonometry was impossible, but the two High School students provided a proof.

The proof was annotated in a posting by Wolfram Alpha developer @Shenghui Yang. See New trigonometric proof of Pythagorean theorem via law of sines posted in the Wolfram Community discussions about a year ago.

POSTED BY: Phil Earnhardt
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