A137914 Decimal expansion of arccos(1/3).

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

Offset: 1

Author

Rick L. Shepherd, Feb 22 2008

Keywords

• cons
• nonn

Comments

Dihedral angle in radians of regular tetrahedron.

Arccos(1/3) is the central angle of a cube, made by the center and two neighboring vertices. - Clark Kimberling, Feb 10 2009

Also the complementary tetrahedral angle, Pi-A156546, and therefore related to the magic angle (Pi-2*A195696). - Stanislav Sykora, Jan 23 2014

Polar angle (or apex angle) of the cone that subtends exactly one third of the full solid angle. - Stanislav Sykora, Feb 20 2014

Also the acute angle in the rhombi and isosceles trapezoids in the trapezo-rhombic dodecahedron. - Eric W. Weisstein, Jan 09 2019

Also the angle between the tangent lines to the curves y = sin(x) at y = cos(x) at the points of intersection. - David Radcliffe, Jan 17 2023

Examples

			1.2309594173407746821349291782479873757103400093550948390555483336639923144...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..10000
• Steven R. Finch, Errata and Addenda to Mathematical Constants, p. 58.
• Eric Weisstein's World of Mathematics, Tetrahedron.
• Eric Weisstein's World of Mathematics, Dihedral Angle.
• Eric Weisstein's World of Mathematics, Trapezo-Rhombic Dodecahedron.
• Index entries for transcendental numbers

Crossrefs

Cf. A137915 (same in degrees), A019670, A195695, A195696, A238238, Platonic solids dihedral angles: A156546 (octahedron), A019669 (cube), A236367 (icosahedron), A137218 (dodecahedron).

Programs

Magma

SetDefaultRealField(RealField(100)); Arccos(1/3); // G. C. Greubel, Aug 20 2018

Mathematica

RealDigits[ArcCos[1/3], 10, 120][[1]] (* Harvey P. Dale, Jul 06 2018 *)
RealDigits[ArcSec[3], 10, 120][[1]] (* Eric W. Weisstein, Jan 09 2019 *)

PARI

acos(1/3)

Formula

arccos(1/3) = arctan(2*sqrt(2)) = 2*arcsin(sqrt(3)/3) = arcsin(2*sqrt(2)/3).

Equals sqrt(2)*Sum_{k>=0} (-1)^k/(2^k*(2*k+1)). - Davide Rotondo, Jun 07 2025

Equals 2*A195695. - Hugo Pfoertner, Jun 07 2025