How to find the exact value of $\cos(\frac{2\pi}{17})$ with WolframAlpha?

Question

I tried to find the exact value of $\cos(\frac{2\pi}{17})$ with WolframAlpha but only obtained a decimal approximation. Is there any way to find this exact value with WolframAlpha?

Answer 1

Wolfram give the results - see link below.

http://mathworld.wolfram.com/TrigonometryAnglesPi17.html

Answer 2

First, the value as Gauss wrote it, from Galois Theory by David A. Cox. Cox also shows how Gauss would have derived this, in exrecises in the previous chapter.

Instead of your number, it is easier to double it, and get a root of $$ x^8 + x^7 - 7 x^6 - 6 x^5 + 15 x^4 + 10 x^3 - 10 x^2 - 4 x + 1 $$

see page 18 of Reuschle (1875)

Note that $2 \cos \frac{2 \pi}{17} \approx 1.8649444588$