When I saw Gauss's original solution to $17$ sided gon, his method seemed all clever and tricky. I am wondering if there are some other ways to evaluate $$\cos\frac{2\pi}{17}$$
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Could you link Gauss' original solution? – Hubble May 26 '14 at 16:50
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@iHubble I read it in my book Higher Algebra by Jm Barnard and Child and I couldn't find his solution on internet. – Hashir Omer May 26 '14 at 16:52
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Are you familiar with Galois Theory? – Hubble May 26 '14 at 16:55
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No I haven't started it yet.. – Hashir Omer May 26 '14 at 16:56
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1This paper explains how to derive an expression for $\cos(2\pi/17)$ using Galois Theory. That's really the only other way I know unfortunately... – Hubble May 26 '14 at 16:59
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1See my manuscript A Detailed and Elementary Solution to $x^{17} = 1$, especially Chapters 3 and 4. – Dave L. Renfro May 27 '14 at 16:21
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$$ \cos\frac{2\pi}{17}=\frac{-1+\sqrt{17}+\sqrt{34-2\sqrt{17}}+2\sqrt{17+3\sqrt{17}-\sqrt{34-2\sqrt{17}}-2\sqrt{34+2\sqrt{17}}}}{16}. $$ – Tunk-Fey May 26 '14 at 17:03