OFFSET
0,1
COMMENTS
It is well known that the length sin 70° (cos 20°) is not constructible with ruler and compass, since it is a root of the irreducible polynomial 8x^3 - 6x - 1 and 3 fails to divide any power of 2. - Jean-François Alcover, Aug 10 2014 [cf. the Maxfield ref.]
A cubic number with denominator 2. - Charles R Greathouse IV, Aug 27 2017
From Peter Bala, Oct 21 2021: (Start)
The minimal polynomial of cos(Pi/9) is 8*x^3 - 6*x - 1 with discriminant (2^6)*(3^4), a square: hence the Galois group of the algebraic number field Q(sin(70°) over Q is the cyclic group of order 3.
REFERENCES
J. E. Maxfield and M. W. Maxfield, Abstract Algebra and Solution by Radicals, Dover Publications ISBN 0-486-67121-6, (1992), p. 197.
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Michael Penn, Proof that cos(20°) is irrational, YouTube video, 2022.
FORMULA
Equals cos(Pi/9) = (1/2)*A332437. - Peter Bala, Oct 21 2021
EXAMPLE
0.93969262...
MATHEMATICA
RealDigits[Sin[70 Degree], 10, 120][[1]] (* Harvey P. Dale, Aug 17 2012 *)
PROG
(PARI) cos(Pi/9) \\ Charles R Greathouse IV, Aug 27 2017
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
STATUS
approved