A049470 - OEIS

A049470

Decimal expansion of cos(1).

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, 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, 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

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OFFSET

0,1

COMMENTS

Also, decimal expansion of the real part of e^i. - Bruno Berselli, Feb 08 2013

By the Lindemann-Weierstrass theorem, this constant is transcendental. - Charles R Greathouse IV, May 13 2019

LINKS

Muniru A Asiru, Table of n, a(n) for n = 0..2000

Mohammad K. Azarian, Forty-Five Nested Equilateral Triangles and cosecant of 1 degree, Problem 813, College Mathematics Journal, Vol. 36, No. 5, November 2005, p. 413-414.

Mohammad K. Azarian, Solution of Forty-Five Nested Equilateral Triangles and cosecant of 1 degree, Problem 813, College Mathematics Journal, Vol. 37, No. 5, November 2006, pp. 394-395.

I. S. Gradsteyn and I. M. Ryzhik, Table of integrals, series and products, (1980), page 10 (formula 0.245.7).

Simon Plouffe, cos(1)

Eric Weisstein's World of Mathematics, Factorial Sums

Index entries for transcendental numbers

FORMULA

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

Equals Sum_{k >= 0} (-1)^k/A010050(k), where A010050(k) = (2k)! [See Gradshteyn and Ryzhik]. - A.H.M. Smeets, Sep 22 2018

Equals 1/A073448. - Alois P. Heinz, Jan 23 2023

From Gerry Martens, May 04 2024: (Start)

Equals (4*(cos(1/4)^4 + sin(1/4)^4) - 3).

Equals (16*(cos(1/4)^6 + sin(1/4)^6) - 10)/6. (End)

EXAMPLE

0.5403023058681397...

MAPLE

evalf(cos(1)); # Altug Alkan, Sep 22 2018

MATHEMATICA

RealDigits[Cos[1], 10, 110] [[1]]

PROG