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

On 24 May 2007 04:52:53 -0700, gtsavdar@xxxxxxxxx wrote:


I've managed to solve it but only after a _massive_ number of
calculations.
Is there any other simpler solution? There must be!

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

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

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If the symbol of degree inside the Sin() does not appear in some
browsers then the problem is the follow:

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) )

Of course, it's easy to verify it numerically, at least up to an
acceptable accuracy, but I don't see an easy way to prove it
algebraically.

I can set it up as a horrible system of simultaneous nonlinear
polynomial equations which could be solved, in theory, using Grobner
bases. However it's clear that the degrees are too high for such a
method to be practical, at least for the simple way I'm
conceptualizing it.

So as of now, I don't see any practical way of doing it algebraically,
even with the help of Maple.

How did you actually do it? Maybe you could just outline the basic
strategy and key ideas. No need to show all the gory details.

As I see it, your problem calls out for generalization. It immediately
inspired some conjectures on my part, however the ones I proposed
earlier today in this thread were poorly thought out, so I withdrew
them. I may try to replace them, but first, I think it makes sense to
explore relationships with fewer angles. I have some simple questions
along these lines, but I'll pose them in a new thread.

quasi
.

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