login
A195692
Decimal expansion of arccos(1/phi), where phi = (1+sqrt(5))/2 (the golden ratio).
4
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
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[%]
KEYWORD
nonn,cons
AUTHOR
Clark Kimberling, Sep 22 2011
EXTENSIONS
Terms replaced with intended terms by Rick L. Shepherd, Jan 30 2013
STATUS
approved