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

Author

Clark Kimberling, Sep 22 2011

Keywords

• nonn
• cons
• changed

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

Examples

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

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.
• Index entries for transcendental numbers

Crossrefs

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

Programs

Mathematica

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

PARI

acos(2/(sqrt(5)+1)) \\ Charles R Greathouse IV, Nov 21 2024

Formula

From Amiram Eldar, Feb 07 2022: (Start)

Equals Pi/2 - A175288.

Equals arcsin(1/sqrt(phi)).

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

Extensions

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