Re: Can I have fries and a calculator with that?

Peter Webb wrote:

> Indulge me here a little, this is a hobby (not a good start,
> I know).
>
> We know that sin (90 degrees) = 1, and we know that the value
> of sin(45 degrees) satisfies the equation 2x^2 = 1.
>
> We further know that the value of sin(1 degree) satisfies some
> equation sigma (choose(90,i)x^i) = 1 (or similar).
>
> Therefore sin(1 degree) is algebraic, and so is sin(1+1).
>
> We can therefore construct sin(n) or indeed sin(m/n) as algebraic.
>
> Therefore sin maps all rational degrees to algebraic numbers.
>
> The interesting (for me) is the fact that this doesn't occur
> for sin in radians, where sin(1) seems unobtainable within
> algebraics.

You might be interested in these two July 1999
alt.math.undergraduate posts of mine:

http://groups.google.com/group/alt.math.undergrad/msg/097f0dbd69a82f71
http://mathforum.org/kb/message.jspa?messageID=684452

http://groups.google.com/group/alt.math.undergrad/msg/077c6d6c8c71ccbf
http://mathforum.org/kb/message.jspa?messageID=684453

I now know of probably about 3 times as many references in
the literature for this result than I gave in the second post,
but I'm not going to look them up and type them into a usenet
post right now, not after the essay on rationalizing denominators
that I just wrote and posted.

Dave L. Renfro

.

Quantcast