OFFSET
0,1
COMMENTS
This is the angle (in radians) at which the modified loop curve x^4=x^2*y-y^2 returns to the origin. Writing the curve in (r,phi) circular coordinates, r = sin(phi) * (cos^2(phi)-sin(phi)) /cos^4(phi), the two values of r=0 are phi=0 and the value of phi defined here. The equivalent angle of the Bow curve is Pi/4.
Also the minimum positive solution to tan(x) = cos(x). - Franklin T. Adams-Watters, Jun 17 2014
LINKS
Eric Weisstein's World of Mathematics, Bow.
FORMULA
From Amiram Eldar, Feb 07 2022: (Start)
Equals Pi/2 - A195692.
Equals arccos(1/sqrt(phi)).
Equals arctan(1/sqrt(phi)) = arccot(sqrt(phi)). (End)
Root of the equation cos(x) = tan(x). - Vaclav Kotesovec, Mar 06 2022
EXAMPLE
x = 0.66623943.. = 38.1727076... degrees.
MATHEMATICA
r = 1/GoldenRatio;
N[ArcSin[r], 100]
RealDigits[%] (* A175288 *)
RealDigits[x/.FindRoot[Cos[x]^2==Sin[x], {x, .6}, WorkingPrecision->120]] [[1]] (* Harvey P. Dale, Nov 08 2011 *)
RealDigits[ ArcCos[ Sqrt[ (Sqrt[5] - 1)/2]], 10, 105] // First (* Jean-François Alcover, Feb 19 2013 *)
CROSSREFS
KEYWORD
AUTHOR
R. J. Mathar, Mar 23 2010, Mar 29 2010
EXTENSIONS
Disambiguated the curve here from the Mathworld bow curve - R. J. Mathar, Mar 29 2010
STATUS
approved