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## Homework Statement

Prove that sin 1 (degree) is algebraic.

## Homework Equations

"The element [tex]a \in K[/tex] is said to be algebraic of degree n over F(a field) if it satisfies a nonzero polynomial over F of degree n but no nonzero polynomial of lower degree."

## The Attempt at a Solution

I thought it might work to try this formula:

[tex](cos x + i sin x)^n=cos nx + i sin nx[/tex]

then let x=sin 1

with the identity [tex] cos^2 x +sin^2 x =1[/tex]

I raised this to the 90th power because then cos disappears:

[tex](\sqrt{1-x^2} + ix)^{90} = sin 90=1[/tex]

then I put this into my TI-89 because it looks scary...and IT IS SCARY.

The coefficients all are integers (VERY LARGE INTEGERS).

The part that I don't know about is if it's the smallest nonzero polynomial that sin 1 degree satisfies and does anyboby know a neater way to go about finding this ploynomial? As it is, I can't even tell if eisenstein's criterion would apply. I'd have to go through a 90th degree polynomial and try to find a prime that would work. The coefficients look like this:

-7471375560,706252528630,-41604694413840...and there are some twice that size.

Any input would be appreciated.

CC

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