|
|
A019879
|
|
Decimal expansion of sine of 70 degrees.
|
|
9
|
|
|
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
(list;
constant;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
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.]
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)
|
|
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
|
|
|
FORMULA
|
|
|
EXAMPLE
|
0.93969262...
|
|
MATHEMATICA
|
RealDigits[Sin[70 Degree], 10, 120][[1]] (* Harvey P. Dale, Aug 17 2012 *)
|
|
PROG
|
|
|
CROSSREFS
|
|
|
KEYWORD
|
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|