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%I #45 Jul 10 2024 20:12:12
%S 0,1,7,4,5,2,4,0,6,4,3,7,2,8,3,5,1,2,8,1,9,4,1,8,9,7,8,5,1,6,3,1,6,1,
%T 9,2,4,7,2,2,5,2,7,2,0,3,0,7,1,3,9,6,4,2,6,8,3,6,1,2,4,2,7,6,4,0,5,9,
%U 7,3,8,4,2,0,3,9,2,8,0,7,0,0,4,2,0,0,1,9,2,6,7,9,1,0,2,1,3,4,6,9,1,4,4,8,8
%N Decimal expansion of sine of 1 degree.
%C An algebraic number of degree 48. - _Charles R Greathouse IV_, Apr 14 2014
%C This algebraic number has denominator 2, the least integer k > 0 such that k times the number is an algebraic integer. - _Charles R Greathouse IV_, Nov 12 2014
%C The Fifteenth Century Persian mathematician Jamshid Al-Kashi was the first to calculate the value of sine of one degree correct to ten sexagesimal places (17 decimal digits) in his Risala al-Watar wa'l Jaib. - _Mohammad K. Azarian_, Jan 14 2017
%C The minimal polynomial is 281474976710656 x^48 - 3377699720527872 x^46 + 18999560927969280 x^44 - 66568831992070144 x^42 + 162828875980603392 x^40 - 295364007592722432 x^38 + 411985976135516160 x^36 - 452180272956309504 x^34 + 396366279591591936 x^32 - 280058255978266624 x^30 + 160303703377575936 x^28 - 74448984852135936 x^26 + 28011510450094080 x^24 - 8500299631165440 x^22 + 2064791072931840 x^20 - 397107008634880 x^18 + 59570604933120 x^16 - 6832518856704 x^14 + 583456329728 x^12 - 35782471680 x^10 + 1497954816 x^8 - 39625728 x^6 + 579456 x^4 - 3456 x^2 + 1 (WolframAlpha). - _Rick L. Shepherd_, Apr 12 2017
%D Mohammad K. Azarian, Forty-Five Nested Equilateral Triangles and cosecant of 1 degree, Problem 813, College Mathematics Journal, Vol. 36, No. 5, November 2005, pp. 413-414. Solution published in Vol. 37, No. 5, November 2006, pp. 394-395.
%H Ivan Panchenko, <a href="/A019810/b019810.txt">Table of n, a(n) for n = 0..1000</a>
%H Mohammad K. Azarian, <a href="http://forumgeom.fau.edu/FG2015volume15/FG201523.pdf">A Study of Risa-la al-Watar wa'l Jaib ("The Treatise on the Chord and Sine")</a>, Forum Geometricorum, Volume 15 (2015) 229-242. Mathematical Reviews, MR 3418854 (Reviewed), Zentralblatt MATH, Zbl 1328.01015.
%H <a href="/index/Al#algebraic_48">Index entries for algebraic numbers, degree 48</a>
%F Equals sin(Pi/180) = cos(89*Pi/180) = (i^(89/90) - i^(91/90))/2 (the last from WolframAlpha, rearranged). - _Rick L. Shepherd_, Apr 12 2017
%e 0.01745240643728351281941897851631...
%t Join[{0},RealDigits[N[Sin[Pi/180],200]][[1]]] (* and/or *)
%t Join[{0},RealDigits[N[Sin[1 Degree],200]][[1]]] (* _Vladimir Joseph Stephan Orlovsky_, Feb 21 2011 *)
%o (PARI) sin(Pi/180)
%o (PARI) real((I^(89/90) - I^(91/90))/2) \\ (imaginary part is not exactly zero only because of finite precision) _Rick L. Shepherd_, Apr 12 2017
%Y Cf. A110937, A280188.
%K nonn,cons,easy
%O 0,3
%A _N. J. A. Sloane_
%E More terms from _James A. Sellers_, Jan 19 2000