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

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, 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, 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

Offset: 0

Author

R. J. Mathar, Mar 23 2010, Mar 29 2010

Keywords

• cons
• easy
• nonn

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

Examples

			x = 0.66623943.. = 38.1727076... degrees.
		

Links

• Eric Weisstein's World of Mathematics, Bow.

Crossrefs

Cf. A019669, A094214, A001622, A195692, A352151.

Programs

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 *)

Formula

x = arcsin(A094214). cos(x)^2 = sin(x) = 0.618033988... = A094214.

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

Extensions

Disambiguated the curve here from the Mathworld bow curve - R. J. Mathar, Mar 29 2010