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Calculating $\sin(10^\circ)$ with a geometric method

Evaluate if $\sin10°$ be expressed in real surd form?

Thank you!

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Also see this for a surd form of sin(10°). I don't think it's what you were hoping for. –  Raskolnikov Mar 1 '12 at 11:01
    
Also see this thread –  bgins Mar 1 '12 at 13:55
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marked as duplicate by Hans Lundmark, lhf, Gerry Myerson, Aryabhata, Asaf Karagila Mar 1 '12 at 20:21

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Let $x$ be the sine of the $10$ degree angle. Note the identity $$\sin(3x)=3\sin x-4\sin^3 x.$$ This identity can be proved by using the addition laws for sine and cosin repeatedly, starting with $\sin(3x)=\sin(2x+x)=\sin 2x \cos x +\cos 2x\sin x$. The sine of the $30$ degree angle is $1/2$, Thus we obtain $$3x-4x^3=\frac{1}{2},$$ or equivalently $8x^2-6x+1=0.$

This is a cubic with integer coefficients. It is easy to verify that there is no rational root.

It is not hard to see that there are $3$ real roots, so we are in the casus irreducibilis of Cardano.

Using tools from Galois Theory, one can prove that in the casus irreducibilis, the roots of the cubic cannot be expressed in terms of "real surds." So the answer to your question is that we cannot express the sine of $10$ degrees with real surds only.

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