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%I #66 Dec 31 2020 05:55:44
%S 8,4,1,4,7,0,9,8,4,8,0,7,8,9,6,5,0,6,6,5,2,5,0,2,3,2,1,6,3,0,2,9,8,9,
%T 9,9,6,2,2,5,6,3,0,6,0,7,9,8,3,7,1,0,6,5,6,7,2,7,5,1,7,0,9,9,9,1,9,1,
%U 0,4,0,4,3,9,1,2,3,9,6,6,8,9,4,8,6,3,9,7,4,3,5,4,3,0,5,2,6,9,5
%N Decimal expansion of sin(1).
%C Also, decimal expansion of the imaginary part of e^i. - _Bruno Berselli_, Feb 08 2013
%C By the Lindemann-Weierstrass theorem, this constant is transcendental. - _Charles R Greathouse IV_, May 12 2019
%H Muniru A Asiru, <a href="/A049469/b049469.txt">Table of n, a(n) for n = 0..2000</a>
%H Mohammad K. Azarian, <a href="http://www.jstor.org/stable/30044897">Forty-Five Nested Equilateral Triangles and cosecant of 1 degree, Problem 813</a>, College Mathematics Journal, Vol. 36, No. 5, November 2005, p. 413-414.
%H Mohammad K. Azarian, <a href="http://www.jstor.org/stable/27646393">Solution of Forty-Five Nested Equilateral Triangles and cosecant of 1 degree, Problem 813</a>, College Mathematics Journal, Vol. 37, No. 5, November 2006, pp. 394-395.
%H I. S. Gradsteyn, I. M. Ryzhik, <a href="http://mathtable.com/gr/index.html">Table of integrals, series and products</a>, (1980), page 10 (formula 0.245.8).
%H Simon Plouffe, <a href="http://www.worldwideschool.org/library/books/sci/math/MiscellaneousMathematicalConstants/chap82.html">sin(1)</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/FactorialSums.html">Factorial Sums</a>
%H <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>
%F Continued fraction representation: sin(1) = 1 - 1/(6 + 6/(19 + 20/(41 + ... + (2*n - 1)*(2*n - 2)/((4*n^2 + 2*n - 1) + ... )))). See A074790 for details. - _Peter Bala_, Jan 30 2015
%F Equals Sum_{k > 0} (-1)^(k-1)/((2k-1)!) = Sum_{k > 0} (-1)^(k-1)/A009445(k-1) [See Gradshteyn and Ryzhik]. - _A.H.M. Smeets_, Sep 22 2018
%F Equals Product{k>=1} cos(1/2^k). - _Amiram Eldar_, Aug 20 2020
%e 0.8414709848078965...
%p evalf(sin(1)); # _Altug Alkan_, Sep 22 2018
%t RealDigits[N[Sin[1], 110]] [[1]]
%o (PARI) sin(1) \\ _Charles R Greathouse IV_, Aug 20 2012
%o (PARI) sumalt(n=0, (-1)^(n%2)/(2*n+1)!) \\ _Gheorghe Coserea_, Sep 23 2018
%Y Cf. A049470 (real part of e^i), A211883 (real part of -(i^e)), A211884 (imaginary part of -(i^e)). - _Bruno Berselli_, Feb 08 2013
%Y Cf. A074790.
%K cons,easy,nonn
%O 0,1
%A Albert du Toit (dutwa(AT)intekom.co.za), _N. J. A. Sloane_
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