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

Offset: 0

Author

Albert du Toit (dutwa(AT)intekom.co.za), N. J. A. Sloane

Keywords

• cons
• easy
• nonn

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

Examples

			0.5403023058681397...
		

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

Crossrefs

Cf. A049469 (imaginary part of e^i), A211883 (real part of -(i^e)), A211884 (imaginary part of -(i^e)). - Bruno Berselli, Feb 08 2013

Cf. A073743 ( cosh(1) ), A073448, A275651.

Cf. A068985, A346441, A346440, A346439, A346438, A346437, A346436, A346435, A196498.

Programs

Maple

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

Mathematica

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

PARI

cos(1) \\ Charles R Greathouse IV, Jan 04 2016

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)