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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
    
Associated Topics:
High School Constructions
High School Geometry

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