Sin 20 and Transcendental Numbers
Date: 29 Jun 1995 11:39:48 -0400 From: Sandra Wilkes Subject: sin 20 Hi Dr.Math, My name is Sandra and I am struggling with a problem in a workshop, Geometry and the Internet at Berry College. The problem is "what is the significance of sin 20 in geometry?" Hints are that it might have something to do with transcendental, rational, irrational. Dr. Vlach said sin 15, sin 10, cos 15, cos 10 could also be used. He also said it might have something to do with who discovered it was transcendental. Thanks for any ideas you have. Time is getting short. Sandra.
Date: 29 Jun 1995 19:20:48 -0400 From: Dr. Ken Subject: Re: sin 20 Hello there! I've spent a while thinking about your problem, and here's the most likely thing I can come up with. I don't want to spoil the whole answer, so I'll be a little vague about who did what and when it happened. You might try the following three URLs to find out some more: http://aleph0.clarku.edu/~djoyce/mathhist/mathhist.html http://www-groups.dcs.st-and.ac.uk:80/~history/ http://www.math.fsu.edu/Science/math.html These are three pages that are dedicated to math history. Most likely the askers of the question had in mind the area of compass-and-straightedge geometry. For a very long time, there was an open question in this field, "given any angle, construct an angle whose degree measure is one-third the given angle." In other words, trisect the angle. Well, it has been proven that the problem is unsolvable, and there are certain angles that can never be constructed. I believe that what the provers did was to first assert that if you could construct a certain angle, then you could also construct a segment whose length is equal to the sine of the angle. So if you could trisect a 60 degree angle (i.e. construct an angle with a measure of 20 degrees) you could construct a segment whose length is sin(20). Then the provers proved that you could only construct segments of certain lengths with reference to some unit length. In particular, all the lengths you can construct are _algebraic_ numbers (roots of a finite-degree polynomial with integer coefficients). I believe sin(20) was already know to be a transcendental (non-algebraic) number, so there was no hope of ever constructing a segment of length sin(20). Thus an angle of 20 degrees could never be constructed, so you could never trisect a 60-degree angle. At least, I think that's what they're talking about. Hope this helps! -K
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