A211883 -id:A211883 - cp's OEIS frontend

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

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

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

			0.5403023058681397...

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

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.

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

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

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

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)

A049469 Decimal expansion of sin(1).

Original entry on oeis.org

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, 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, 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

Offset: 0

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

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

By the Lindemann-Weierstrass theorem, this constant is transcendental. - Charles R Greathouse IV, May 12 2019

			0.8414709848078965...

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, I. M. Ryzhik, Table of integrals, series and products, (1980), page 10 (formula 0.245.8).
Paul J. Nahin, Inside interesting integrals, Undergrad. Lecture Notes in Physics, Springer (2020), chapter 1.5
Simon Plouffe, sin(1)
Eric Weisstein's World of Mathematics, Factorial Sums
Index entries for transcendental numbers

Cf. A049470 (real part of e^i), A211883 (real part of -(i^e)), A211884 (imaginary part of -(i^e)). - Bruno Berselli, Feb 08 2013

Cf. A074790.

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

Mathematica
```
RealDigits[N[Sin[1], 110]] [[1]]
```

sin(1) \\ Charles R Greathouse IV, Aug 20 2012

sumalt(n=0, (-1)^(n%2)/(2*n+1)!) \\ Gheorghe Coserea, Sep 23 2018

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
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
Equals Product{k>=1} cos(1/2^k). - Amiram Eldar, Aug 20 2020
Equals Integral_{x=-1..1} cos(x)/[exp(1/x)+1] dx. [Nahin]. - R. J. Mathar, May 16 2024

A211884 Decimal expansion of the imaginary part of -(i^e).

Original entry on oeis.org

9, 0, 3, 6, 7, 4, 6, 2, 3, 7, 7, 6, 3, 9, 5, 5, 3, 6, 6, 0, 0, 8, 5, 3, 4, 5, 9, 3, 3, 4, 3, 4, 2, 5, 4, 1, 9, 4, 1, 8, 0, 0, 3, 0, 3, 4, 6, 3, 2, 2, 0, 2, 0, 5, 8, 9, 7, 4, 3, 7, 5, 9, 9, 9, 1, 3, 1, 2, 1, 6, 5, 3, 9, 9, 0, 8, 5, 8, 9, 5, 2, 8, 7, 8, 4, 9, 3

Offset: 0

Stanislav Sykora, Apr 24 2012

Also, decimal expansion of -sin(Pi*e/2).

The decimal expansion of the real part of -i^e is in A211883.

			0.90367462377639553660085345933434254194180030346322020589743759991312...

Cf. A211883.

RealDigits[Im[-I^E], 10, 87][[1]] (* Bruno Berselli, Apr 24 2012 *)

A222128 Decimal expansion of the real part of 1/i^Pi, where i=sqrt(-1).

Original entry on oeis.org

2, 2, 0, 5, 8, 4, 0, 4, 0, 7, 4, 9, 6, 9, 8, 0, 8, 8, 6, 6, 8, 9, 4, 5, 9, 1, 3, 2, 5, 5, 7, 8, 7, 5, 1, 0, 4, 5, 8, 8, 4, 8, 0, 3, 8, 1, 5, 9, 4, 1, 0, 6, 7, 2, 3, 7, 0, 0, 4, 8, 8, 7, 3, 2, 2, 4, 8, 3, 3, 5, 5, 1, 2, 5, 0, 5, 9, 5, 6, 3, 9, 7, 2, 7, 1, 1, 3

Offset: 0

Bruno Berselli, Feb 08 2013

Also, decimal expansion of the real part of i^Pi.

			0.22058404074969808866894591325578751045884803815941067237004887322...

Vincenzo Librandi, Table of n, a(n) for n = 0..5000

Cf. A102753, A211883, A222129 (imaginary part of 1/i^Pi).

RealDigits[Re[1/I^Pi], 10, 90][[1]] (* or *) RealDigits[Cos[Pi^2/2], 10, 90][[1]]

fpprec:90; ev(bfloat(realpart(1/%i^%pi)));

Equals cos(Pi^2/2).

cp's OEIS Frontend

A049470 Decimal expansion of cos(1).

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A049469 Decimal expansion of sin(1).

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A211884 Decimal expansion of the imaginary part of -(i^e).

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A222128 Decimal expansion of the real part of 1/i^Pi, where i=sqrt(-1).

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