Re: Simple, but a bit hard, Trigonometry problem.

On Thu, 24 May 2007 08:57:05 -0500, quasi <quasi@xxxxxxxx> wrote:

On Thu, 24 May 2007 08:48:21 -0500, quasi <quasi@xxxxxxxx> wrote:

On Thu, 24 May 2007 08:03:24 -0500, quasi <quasi@xxxxxxxx> wrote:

If:
a = Sin(5 degrees)
b = Sin(49 degrees)
c = Sin(87 degrees)

then prove that: Sin(73 degrees) = (a^2 - b^2 + a c) / ( 4 a (a^2 -
b^2 + a c) - (a-b+c) )

As a separate challenge, find a polynomial relationship between a,b,c.

Let me escalate the challenge.

Prove or disprove the following conjectures ...

Conjecture 1:

Let r,s,t be integers with gcd(r,s,t)=1 and let a=sin(r), b=sin(s),
c=sin(t). Then c _cannot_ be expressed as a polynomial in a,b.

Corrected statements:

Conjecture 1:

Let r,s,t be integers such that gcd(rs,t)=1, and let a=sin(r),
b=sin(s), c=sin(t). Then c _cannot_ be expressed as a polynomial in
a,b.

Conjecture 2:

Let r,s,t,u be integers such that gcd(rst,u)=1, and let a=sin(r),
b=sin(s), c=sin(t), d=sin(u) Then d _can_ be expressed as a polynomial
in a,b,c.

quasi

I posted the above challenge too quickly. I was on my way out and
posted it in a hurry without thinking it through. I just got back and
I see now that there are lots of things wrong with it, some minor,
some major. For example:

I neglected some easy counterexamples

I didn't get the gcd conditions right

I didn't specify the angle units

I said "polynomial" when I meant "rational function"

I accidentally snipped the reference to the OP

All in all, my prior post is a mess, so forget the conjectures I made
there. I'll replace it soon with a new, revised version. Stay tuned.

quasi
.

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