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%I #62 Jan 23 2023 19:53:33
%S 5,4,0,3,0,2,3,0,5,8,6,8,1,3,9,7,1,7,4,0,0,9,3,6,6,0,7,4,4,2,9,7,6,6,
%T 0,3,7,3,2,3,1,0,4,2,0,6,1,7,9,2,2,2,2,7,6,7,0,0,9,7,2,5,5,3,8,1,1,0,
%U 0,3,9,4,7,7,4,4,7,1,7,6,4,5,1,7,9,5,1,8,5,6,0,8,7,1,8,3,0,8,9
%N Decimal expansion of cos(1).
%C Also, decimal expansion of the real part of e^i. - _Bruno Berselli_, Feb 08 2013
%C By the Lindemann-Weierstrass theorem, this constant is transcendental. - _Charles R Greathouse IV_, May 13 2019
%H Muniru A Asiru, <a href="/A049470/b049470.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 0.245.7).
%H Simon Plouffe, <a href="http://www.worldwideschool.org/library/books/sci/math/MiscellaneousMathematicalConstants/chap14.html">cos(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: cos(1) = 1/(1 + 1/(1 + 2/(11 + 12/(29 + ... + (2*n - 2)*(2*n - 3)/((4*n^2 - 2*n - 1) + ... ))))). See A275651 for proof. Cf. A073743. - _Peter Bala_, Sep 02 2016
%F Equals Sum_{k >= 0} (-1)^k/A010050(k), where A010050(k) = (2k)! [See Gradshteyn and Ryzhik]. - _A.H.M. Smeets_, Sep 22 2018
%F Equals 1/A073448. - _Alois P. Heinz_, Jan 23 2023
%e 0.5403023058681397...
%p evalf(cos(1)); # _Altug Alkan_, Sep 22 2018
%t RealDigits[Cos[1], 10, 110] [[1]]
%o (PARI) cos(1) \\ _Charles R Greathouse IV_, Jan 04 2016
%Y Cf. A049469 (imaginary part of e^i), A211883 (real part of -(i^e)), A211884 (imaginary part of -(i^e)). - _Bruno Berselli_, Feb 08 2013
%Y Cf. A073743 ( cosh(1) ), A073448, A275651.
%Y Cf. A068985, A346441, A346440, A346439, A346438, A346437, A346436, A346435, A196498.
%K cons,easy,nonn
%O 0,1
%A Albert du Toit (dutwa(AT)intekom.co.za), _N. J. A. Sloane_
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