A233527 -id:A233527 - cp's OEIS frontend

A233528 Decimal expansion of arctan(2*Pi): adjacent angle for a right triangle of equal area to a disk.

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

Offset: 1

John W. Nicholson, Dec 11 2013

In radians, this constant is the arctan(base / height) = arctan(Adjacent / Opposite) = arctan(circumference / radius) for a unit circle is arctan(A019692), where A019692 = 2*A000796.

"Archimedes used the tools of Euclidean geometry to show that the area inside a circle is equal to that of a right triangle whose base has the length of the circle's circumference and whose height equals the circle's radius in his book Measurement of a Circle." quote from Wikipedia link.

			1.412965136506737759063712949856932518493513459088501850071914328940...

Wikipedia, Area of a circle: Triangle proof

Cf. A019692: 2*Pi; A232273: arctan(Pi); A233527: arctan(1/(2*Pi)).

RealDigits[ArcTan[2 Pi], 10, 110][[1]] (* Bruno Berselli, Dec 16 2013 *)

PARI
```
atan(2*Pi)
```

Equals A019669 - A233527. [Bruno Berselli, Dec 16 2013]

A233700 Decimal expansion of 1/sin(arctan(1/t)) or t/sin(arctan(t)) where t = 2*Pi: hypotenuse for a right triangle of equal area to a disk.

Original entry on oeis.org

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

Offset: 1

John W. Nicholson, Dec 16 2013

"The great mathematician Archimedes used the tools of Euclidean geometry to show that the area inside a circle is equal to that of a right triangle whose base has the length of the circle's circumference and whose height equals the circle's radius in his book Measurement of a Circle." (Quote from Wikipedia link)

			6.362265131567328393691245440586804410699714985138989686582041617045998587331...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Wikipedia, Area of a disk: Triangle method

Cf. A233527, A019692, A233528.

using Nemo
RR = RealField(310)
t = const_pi(RR) + const_pi(RR)
t/sin(atan(t)) |> println # Peter Luschny, Mar 13 2018

C := ComplexField(); Sqrt(1 + 4*Pi(C)^2) // G. C. Greubel, Jan 08 2018

Magma

R:=RealField(110); SetDefaultRealField(R); n:=Sqrt(1+4*Pi(R)^2); Reverse(Intseq(Floor(10^108*n))); // Bruno Berselli, Mar 13 2018

Mathematica

RealDigits[(2*Pi)/Sin[ArcTan[2*Pi]],10,120][[1]] (* Harvey P. Dale, Jul 12 2014 *) RealDigits[ Sqrt[1 + 4*Pi^2], 10, 111][[1]] (* Robert G. Wilson v, Mar 12 2015 *)

PARI

sqrt(1+(2*Pi)^2)

Formula

Equals sqrt(1+(2*Pi)^2) = sqrt(1 + (A019692)^2) = sqrt(1 + A212002) = 1/sin(A233527) = A019692/sin(A233528) = 1/cos(A233528) = A019692/cos(A233527).

cp's OEIS Frontend

A233528 Decimal expansion of arctan(2*Pi): adjacent angle for a right triangle of equal area to a disk.

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