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

0,1

Comments

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)

Links

G. C. Greubel, Table of n, a(n) for n = 0..10000

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.

Wikipedia, Tetrahedron

Wikipedia, Magic angle

Formula

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

Example

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

Mathematica

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

Prog

(PARI) atan(sqrt(2)) \\ G. C. Greubel, Jul 05 2017
(Magma) [Arccos(Sqrt(1/3))]; // G. C. Greubel, Nov 18 2017

Crossrefs

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

Sequence in context: A089491 A199792 A193960 * A362000 A197838 A155534

Adjacent sequences: A195693 A195694 A195695 * A195697 A195698 A195699

Keyword

nonn,cons

Author

Clark Kimberling, Sep 23 2011

Status

approved