A243445 -id:A243445 - cp's OEIS frontend

A195696 Decimal expansion of arccos(sqrt(1/3)) and of arcsin(sqrt(2/3)) and arctan(sqrt(2)).

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

Offset: 0

Clark Kimberling, Sep 23 2011

Angle (in radians) between an edge and (the normal of) a face of the regular tetrahedron. - R. J. Mathar, Feb 23 2012
Also known as magic angle; root of P_2(cos(theta)), with P_2(x) being second-order Legendre polynomial. - Stanislav Sykora, May 25 2012
From Stanislav Sykora, Nov 14 2013: (Start)
Also the angle between the body diagonal of a cube and an incident edge, and therefore the polar angle of the cone circumscribed to a cube from one of its vertices.
Also half of the tetrahedral angle (A156546).
In nuclear magnetic resonance, the angle, with respect to the direction of the main magnetic field, under which a solid sample needs to be spun in order to average to zero unwanted dipole-dipole spin interactions (the magic angle spinning, or MAS, technique). (End)
Also <3_2> in Conway et al. (1999). - Eric W. Weisstein, Nov 06 2024

			0.9553166181245092781638571025157577... (= 54.73561031... degrees).

G. C. Greubel, Table of n, a(n) for n = 0..10000
John H. Conway, Charles Radin, and Lorenzo Sadun, On Angles Whose Squared Trigonometric Functions are Rational, arXiv:math-ph/9812019, 1998. See also Discr. Computat. Geom. (1999) Vol. 22, 321-332.
H. B. Dwight, Tables of Integrals and other Mathematical Data. 507.13, 507.22 in Inverse trigonometric functions. New York: Macmillan Publishing, p. 120, 1961.
Steven R. Finch, Errata and Addenda to Mathematical Constants, arXiv:2001.00578 [math.HO], 2020-2022; p. 53.
C. O. Horgan and J. G. Murphy, On an angle with magical properties, Notices Amer. Math. Soc., 69:1 (2022), 22-25.
G. Jacob Martens, Rational right triangles and the Congruent Number Problem, arXiv:2112.09553 [math.GM], 2021, see section 2.6.2 The Trinity vectors, the magic angle, equation (34).
Stanislav Sýkora, Magnetic Resonance on OEIS, Stan's NMR Blog (Dec 31, 2014), Retrieved Nov 12, 2019.
Eric Weisstein's World of Mathematics, Dehn Invariant.
Wikipedia, Tetrahedron
Wikipedia, Magic angle
Index entries for transcendental numbers

Cf. A156546, A195695, A197739, A210974 (in degrees), A243445.

[Arccos(Sqrt(1/3))]; // G. C. Greubel, Nov 18 2017

RealDigits[ArcTan[Sqrt[2]],10,120][[1]] (* Harvey P. Dale, Dec 13 2014 *)

atan(sqrt(2)) \\ G. C. Greubel, Jul 05 2017

Equals i*log(sqrt(1/3) - i*sqrt(2/3)). - Andrea Pinos, Nov 03 2023

Equals A156546/2 = 2*A197739. - Hugo Pfoertner, Nov 06 2024

A195695 Decimal expansion of arcsin(sqrt(1/3)) and of arccos(sqrt(2/3)).

Original entry on oeis.org

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

Offset: 0

Clark Kimberling, Sep 23 2011

The complementary magic angle, that is, Pi/2 - A195696. The angle between the body-diagonal and a congruent face-diagonal of a cube. And also the polar angle of the cone circumscribed to a regular tetrahedron from one of its vertices. - Stanislav Sykora, Nov 21 2013

This is the value of the angle of the circular cone to the axis, that maximizes the volume of the cone enclosed by a given area. See the +plus link. - Michel Marcus, Aug 27 2017

			arcsin(sqrt(1/3)) = 0.61547970867038734106746458912399...

G. C. Greubel, Table of n, a(n) for n = 0..5000
John D. Barrow, Outer space: Archimedean ice cream cones, +plus magazine.
Wikipedia, Polyhedron, and further links therein.
Index entries for transcendental numbers

Cf. A195696 (magic angle), A195698, A020760, A157697, A243445.

[Arcsin(Sqrt(1/3))]; // G. C. Greubel, Nov 18 2017

r = Sqrt[1/3];
N[ArcSin[r], 100]
RealDigits[%]  (* A195695 *)
N[ArcCos[r], 100]
RealDigits[%]  (* A195696 *)
N[ArcTan[r], 100]
RealDigits[%]  (* A019673 *)
N[ArcCos[-r], 100]
RealDigits[%]  (* A195698 *)

atan(1/sqrt(2)) \\ Michel Marcus, Aug 27 2017

Also equals arctan(1/sqrt(2)). - Michel Marcus, Aug 27 2017

A195723 Decimal expansion of arctan(golden ratio).

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Sep 23 2011

The polar angle, in radians, of the cone circumscribed to a regular icosahedron from one of its vertices. - Stanislav Sykora, Feb 15 2014

The angle between the diagonal and the shorter side of a golden rectangle. - Amiram Eldar, May 18 2021

			arctan((1+sqrt(5))/2) = 1.0172219678978513677227...

Stanislav Sykora, Table of n, a(n) for n = 1..2000
Index entries for transcendental numbers

Cf. A001622, A195693, A243445.

SetDefaultRealField(RealField(100)); Arctan((1+Sqrt(5))/2); // G. C. Greubel, Aug 20 2018

r=GoldenRatio; N[ArcTan[r],100]
RealDigits[%] (* A195723 *)

atan((1+sqrt(5))/2) \\ G. C. Greubel, Aug 20 2018

Equals arccos(sqrt((5-sqrt(5))/10)). - Stanislav Sykora, Feb 15 2014

Equals Pi/2 - A195693. - Amiram Eldar, May 18 2021

cp's OEIS Frontend

A195696 Decimal expansion of arccos(sqrt(1/3)) and of arcsin(sqrt(2/3)) and arctan(sqrt(2)).

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A195695 Decimal expansion of arcsin(sqrt(1/3)) and of arccos(sqrt(2/3)).

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Magma

Mathematica

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A195723 Decimal expansion of arctan(golden ratio).

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Magma

Mathematica

PARI

Formula