A019879 Decimal expansion of sine of 70 degrees.

9, 3, 9, 6, 9, 2, 6, 2, 0, 7, 8, 5, 9, 0, 8, 3, 8, 4, 0, 5, 4, 1, 0, 9, 2, 7, 7, 3, 2, 4, 7, 3, 1, 4, 6, 9, 9, 3, 6, 2, 0, 8, 1, 3, 4, 2, 6, 4, 4, 6, 4, 6, 3, 3, 0, 9, 0, 2, 8, 6, 6, 6, 2, 7, 7, 4, 2, 2, 1, 2, 1, 0, 9, 9, 5, 8, 8, 9, 4, 5, 8, 9, 4, 9, 7, 4, 5, 8, 8, 9, 8, 3, 7, 9, 4, 8, 0, 6, 7

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• cons
• changed

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.

The two other zeros of the minimal polynomial are cos(5*Pi/9) = - A019819 and cos(7*Pi/9) = - A019859. The mapping z -> 1 - 2*z^2 cyclically permutes the three zeros. The inverse permutation is given by the mapping z -> 2*z^2 - z - 1. (End)

Examples

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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.
• Index entries for algebraic numbers, degree 3

Crossrefs

Cf. A019819, A019844, A019859, A214778, A332437.

Programs

Mathematica

RealDigits[Sin[70 Degree],10,120][[1]] (* Harvey P. Dale, Aug 17 2012 *)

PARI

cos(Pi/9) \\ Charles R Greathouse IV, Aug 27 2017

Formula

Equals 2*A019844*A019864. - R. J. Mathar, Jan 17 2021

Equals cos(Pi/9) = (1/2)*A332437. - Peter Bala, Oct 21 2021

Equals 2F1(-1/6,1/6 ; 1/2; 3/4). - R. J. Mathar, Aug 31 2025