%I #41 Jun 19 2024 14:48:01
%S 0,5,2,3,3,5,9,5,6,2,4,2,9,4,3,8,3,2,7,2,2,1,1,8,6,2,9,6,0,9,0,7,8,4,
%T 1,8,7,3,1,0,1,8,2,5,3,9,4,0,1,6,4,9,2,0,4,8,3,5,0,9,3,8,1,5,9,9,8,5,
%U 7,1,0,4,6,4,1,7,5,4,5,4,6,8,6,4,4,6,4,5,9,8,8,1,1,8,8,6,9,3,9,8
%N Decimal expansion of sine of 3 degrees.
%C An algebraic number of degree 16 and denominator 2. - _Charles R Greathouse IV_, Nov 02 2012
%C The minimal polynomial is p(x) = 65536*x^16 - 262144*x^14 + 430080*x^12 - 372736*x^10 + 182784*x^8 - 50176*x^6 + 7040*x^4 - 384*x^2 + 1 and p(x/2) = x^16 - 16*x^14 + 105*x^12 - 364*x^10 + 714*x^8 - 784*x^6 + 440*x^4 - 96*x^2 + 1. - _Joerg Arndt_, Apr 23 2024
%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) from sine of 3 degrees in his Risala al-Watar wa'l Jaib. - _Mohammad K. Azarian_,Jan 14 2017
%H Ivan Panchenko, <a href="/A019812/b019812.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 Dr. Rob, The Math Forum at Drexel, <a href="http://web.archive.org/web/20180423114114/http://mathforum.org/library/drmath/view/54149.html">The Exact Value of the Sine of 1 Degree</a>
%H Scott Surgent, <a href="http://math.la.asu.edu/~surgent/mat170/Exact_Trig_Values.pdf">Exact values of the sine and cosine functions in increments of 3 degrees</a>, 2012
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Exact_trigonometric_values">Exact trigonometric values</a>
%H <a href="/index/Al#algebraic_16">Index entries for algebraic numbers, degree 16</a>
%F Equals cos(87 degrees) = cos(29*Pi/60) = sin(Pi/60) = sqrt(8-sqrt(3)-sqrt(15)-sqrt(10-2*sqrt(5)))/4 (an intermediate calculation by Dr. Rob - see Math Forum link). - _Rick L. Shepherd_, Jul 03 2006
%F Equals A019811*A019898 + A019810*A019897. - _R. J. Mathar_, Jan 27 2021
%e 0.052335956242943832722118629609...
%t First[RealDigits[Sin[Pi/60], 10, 100, -1]] (* _Paolo Xausa_, Jun 19 2024 *)
%o (PARI) sin(Pi/60) \\ _Charles R Greathouse IV_, Aug 27 2017
%Y Cf. A019896 (cosine), A019901 (tangent), A019985 (cotangent): for 3 degrees.
%Y Cf. A019810, A019811, A019897, A019898.
%K nonn,cons
%O 0,2
%A _N. J. A. Sloane_