A232273 -id:A232273 - cp's OEIS frontend

A232272 Decimal expansion of arctan(1/Pi).

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

Offset: 0

Bruno Berselli, Nov 22 2013

			0.30816907111598493578699960803405309858963702926316931351366234390168...

Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 35, page 338.

Cf. A019669, A049541, A232247, A232273.

Mathematica
```
RealDigits[ArcTan[1/Pi], 10, 90][[1]]
```

atan(1/Pi) \\ Charles R Greathouse IV, Nov 26 2024

Equals A019669 - A232273.
Equals Sum_{i>=0} (-1)^k/((1+2*k)*Pi^(1+2*k)).
From Wolfe Padawer, Feb 16 2023: (Start)
Equals arccot(Pi).
Equals arcsin(1/sqrt(Pi^2 + 1)).
Equals arccos(1/sqrt(1 + 1/Pi^2)). (End)

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

Original entry on oeis.org

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]

A360700 Decimal expansion of arcsec(Pi).

Original entry on oeis.org

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

Offset: 1

Wolfe Padawer, Feb 16 2023

			1.246850219862915899250368861010977606425828574215431584950410917894...

Michael I. Shamos, A catalog of the real numbers (2011).

Cf. A232272, A232273, A360701.

Mathematica
```
RealDigits[ArcSec[Pi], 10, 105][[1]]
```

Equals Pi/2 - arccsc(Pi) = Pi/2 - A360701.
Equals arcsin(sqrt(1 - 1/Pi^2)).
Equals arctan(sqrt(Pi^2 - 1)).
Equals |arcsech(Pi)|.
Equals arccos(1/Pi).

A360701 Decimal expansion of arccsc(Pi).

Original entry on oeis.org

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

Offset: 0

Wolfe Padawer, Feb 16 2023

			0.323946106931980719980952830628773835672756125472121325537061378259...

Michael I. Shamos, A catalog of the real numbers (2011).

Cf. A232272, A232273, A360700.

Mathematica
```
RealDigits[ArcCsc[Pi], 10, 105][[1]]
```

Equals Pi/2 - arcsec(Pi).
Equals arccos(sqrt(1 - 1/Pi^2)).
Equals arctan(1/(sqrt(Pi^2 - 1))).

cp's OEIS Frontend

A232272 Decimal expansion of arctan(1/Pi).

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

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A360700 Decimal expansion of arcsec(Pi).

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A360701 Decimal expansion of arccsc(Pi).

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