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

On Mon, 28 May 2007 21:21:02 +0100, David Hartley <me9@xxxxxxxxxxx>
wrote:

In message <qtfm53l2nlpf05dq7r5b07moq1upe20cnv@xxxxxxx>, quasi
<quasi@xxxxxxxx> writes
On Mon, 28 May 2007 15:46:16 -0500, quasi <quasi@xxxxxxxx> wrote:

On Mon, 28 May 2007 15:30:21 -0500, quasi <quasi@xxxxxxxx> wrote:

On Mon, 28 May 2007 14:42:53 -0500, quasi <quasi@xxxxxxxx> wrote:

On Mon, 28 May 2007 14:27:57 -0500, quasi <quasi@xxxxxxxx> wrote:

Finding angles whose sines have a given relation is more problematic.

As a test question, do there exist 3 acute angles with integer degree
measures n1,n2,n3 such that

sin(n1)=sin(n2)*sin(n3)

Ok, that was too easy.

Let n1=30, n2=n3=45.

I'll try to think of a better one.

Ok, here's a "trig problem" ...

Consider the multiplicative group G generated by

{sin(1),...,sin(44)} union {sin(46),...,sin(89)}

where the angles above are in degrees.

Is G the free abelian group with 88 generators?

No, they're not independent.

sin (30) sin(2n) = sin(n) sin(90-n) for each n =1 to 44

Nice.

Ok, let's restrict the angles to less than 45 degrees.

Here's the revised problem.

Let G be the multiplicative group G generated by

{sin(1),...,sin(44)}

where the angles above are in degrees.

Is G the free abelian group with 44 generators?

quasi
.

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