Is there nice radical expression for \(\displaystyle{\cos{{\left({\frac{{\pi}}{{{2}^{{k}}+{1}}}}\right)}}}\)

Brooks Barker

Brooks Barker

Answered question

2022-04-16

Is there nice radical expression for
cos(π2k+1)

Answer & Explanation

Ouhamiptkg

Ouhamiptkg

Beginner2022-04-17Added 18 answers

For the form p=2k+1, you can find an expression in real (not complex) radicals for,
cos(nπp)
and n a positive integer only when p is a Fermat ', hence
p=3,5,17,257,65537. For higher p there can be several ways to express it.
For p=17
cos2π17=14(1x+x(17+417)14)
x=14(117+(117)2+16)
or, from this post
1+17+34217
cos2π17+217+31734217234+21716
For p=257

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