Sunday, November 1, 2009

y = x - (x - y): a Consequence Thereof

Last year I discovered a trick for helping myself to fall asleep at night. When I lie down in bed, I close my eyes and think about ways to prove the Pythagorean theorem. Usually I'm asleep in no time! It's very peaceful, watching those little triangles float past my mind's eye, like so many tricorn sheep leaping a stone wall.

The first two of these narcotic proofs turned out to exist already (as I noted here and here). Amusingly enough, however, the third seems to be new. Claims of novelty are never certain when it comes to the Pythagorean theorem, but in any case, you can see the proof at Forum Geometricorum, an open-access, peer-reviewed geometry journal with a recreational slant.

The idea for the proof is to use the subtraction formulas for sine and cosine to derive the trigonometric identity cos^2(x) + sin^2(x) = 1. From this identity the Pythagorean theorem follows immediately.

There are some details involved, but here is the crux of the argument. Given any x with 0 < x < 90, let y be any number with 0 < y < x < 90. Then x, y, and x-y are all strictly between 0 and 90 degrees, so we may straightforwardly apply the subtraction formulas cos(a-b) = cos(a)cos(b) + sin(a)sin(b) and sin(a-b) = sin(a)cos(b) - sin(b)cos(a). We note that:

cos(y) = cos(x-(x-y)) = cos(x)cos(x-y) + sin(x)sin(x-y) = cos(x)[cos(x)cos(y) + sin(x)sin(y)] + sin(x)[sin(x)cos(y) - sin(y)cos(x)] = cos(y)[cos^2(x) + sin^2(x)].

Hence cos^2(x) + sin^2(x) = 1. QED.

Strangely, it does not seem to have been recognized before that the identity cos^2(x) + sin^2(x) = 1 can be derived independently of the Pythagorean theorem using the subtraction formulas. (Of course, to be certain of this, one would have to comb through three centuries' worth of trigonometry textbooks!)

A note on the proof can be found at the recreational math website cut-the-knot.


The algebraic manipulation used above can also be represented diagrammatically using triangles; this yields a more traditional style proof, as shown here. The recursion that is apparent in the algebraic argument gets rendered diagrammatically by nesting a little proof of the subtraction formulas inside a bigger proof of the subtraction formulas.

I'm not sure whether I think of the diagrammatic version and the trigonometric version as being distinct. The style of reasoning is quite different in the two cases, although the two arguments are of course direct transcriptions of one another in some sense.

In any case, it is interesting to note that because the y parameter is arbitrary, the diagram is in a sense "flexible" - it can be drawn in infinitely many ways for any given initial right triangle. In that sense the basic argument spawns infinitely many proofs of the Pythagorean theorem. I made a movie in which each frame represents one of these proofs; you can see it on YouTube or here.

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