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### Sin 20 and Transcendental Numbers

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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.
```

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Date: 29 Jun 1995 19:20:48 -0400
From: Dr. Ken
Subject: Re: sin 20

Hello there!

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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