cp's OEIS Frontend

A175288 Decimal expansion of the minimal positive constant x satisfying (cos(x))^2 = sin(x).

%I A175288 #30 Jul 05 2025 05:20:32
%S A175288 6,6,6,2,3,9,4,3,2,4,9,2,5,1,5,2,5,5,1,0,4,0,0,4,8,9,5,9,7,7,7,9,2,7,
%T A175288 2,0,6,6,7,4,9,0,1,3,8,7,2,5,9,4,7,8,4,2,8,3,1,4,7,3,8,4,2,8,0,3,9,7,
%U A175288 8,9,8,9,3,7,9,0,5,9,2,8,1,7,0,7,9,0,6,8,3,1,1,6,9,5,8,1,1,3,5,2,5,9,7,7,6
%N A175288 Decimal expansion of the minimal positive constant x satisfying (cos(x))^2 = sin(x).
%C A175288 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.
%C A175288 Also the minimum positive solution to tan(x) = cos(x). - _Franklin T. Adams-Watters_, Jun 17 2014
%H A175288 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Bow.html">Bow</a>.
%F A175288 x = arcsin(A094214). cos(x)^2 = sin(x) = 0.618033988... = A094214.
%F A175288 From _Amiram Eldar_, Feb 07 2022: (Start)
%F A175288 Equals Pi/2 - A195692.
%F A175288 Equals arccos(1/sqrt(phi)).
%F A175288 Equals arctan(1/sqrt(phi)) = arccot(sqrt(phi)). (End)
%F A175288 Root of the equation cos(x) = tan(x). - _Vaclav Kotesovec_, Mar 06 2022
%e A175288 x = 0.66623943.. = 38.1727076... degrees.
%t A175288 r = 1/GoldenRatio;
%t A175288 N[ArcSin[r], 100]
%t A175288 RealDigits[%]  (* A175288 *)
%t A175288 RealDigits[x/.FindRoot[Cos[x]^2==Sin[x],{x,.6}, WorkingPrecision->120]] [[1]] (* _Harvey P. Dale_, Nov 08 2011 *)
%t A175288 RealDigits[ ArcCos[ Sqrt[ (Sqrt[5] - 1)/2]], 10, 105] // First (* _Jean-François Alcover_, Feb 19 2013 *)
%Y A175288 Cf. A019669, A094214, A001622, A195692, A352151.
%K A175288 cons,easy,nonn
%O A175288 0,1
%A A175288 _R. J. Mathar_, Mar 23 2010, Mar 29 2010
%E A175288 Disambiguated the curve here from the Mathworld bow curve - _R. J. Mathar_, Mar 29 2010