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

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

Author

Clark Kimberling, Sep 23 2011

Keywords

• nonn
• cons

Comments

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

Examples

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

Links

• 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

Crossrefs

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

Programs

Magma

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

Mathematica

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 *)

PARI

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

Formula

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