A195692 Decimal expansion of arccos(1/phi), where phi = (1+sqrt(5))/2 (the golden ratio).

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

Offset

0,1

Comments

Every cyclic quadrilateral all of whose angles are greater than arccos((sqrt(5)-1)/2) admits a 3 × 1 grid dissection into three cyclic quadrilaterals [Thm. 2.3 in Choi et al. p. 2]. - Michel Marcus, Aug 13 2019

The base angle of the isosceles triangle of smallest perimeter which circumscribes a semicircle (DeTemple, 1992). - Amiram Eldar, Jan 22 2022

Smallest positive root of the equation sin(x) = cot(x). - Wolfe Padawer, Apr 11 2023

Links

Table of n, a(n) for n=0..104.

Erica Choi, Dan Ismailescu, Jiho Lee and Joonsoo Lee, Grid dissections of tangential quadrilaterals, arXiv:1908.02251 [math.MG], 2019.

Duane W. DeTemple, The Triangle of Smallest Perimeter which Circumscribes a Semicircle, The Fibonacci Quarterly, Vol. 30, No. 3 (1992), p. 274.

Formula

From Amiram Eldar, Feb 07 2022: (Start)

Equals Pi/2 - A175288.

Equals arcsin(1/sqrt(phi)).

Equals arctan(sqrt(phi)). (End)

Example

arccos(1/phi) = 0.904556894302381364127316795661958721...
cos(0.904556894302381364127316795661958721...) = 1/(golden ratio) = 0.618...
sec(0.904556894302381364127316795661958721...) = (golden ratio) = 1.618...

Mathematica

r = 1/GoldenRatio;
N[ArcCos[r], 100]
RealDigits[%]

Crossrefs

Cf. A001622, A019669, A139339, A175288, A195693, A195694, A197762.

Sequence in context: A215141 A248951 A198226 * A021088 A021529 A196398

Adjacent sequences: A195689 A195690 A195691 * A195693 A195694 A195695

Keyword

nonn,cons

Author

Clark Kimberling, Sep 22 2011

Extensions

Terms replaced with intended terms by Rick L. Shepherd, Jan 30 2013

Status

approved