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A175288 Decimal expansion of the constant x satisfying (cos(x))^2 = sin(x). 4
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 (list; constant; graph; refs; listen; history; text; internal format)
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

Table of n, a(n) for n=0..104.

Eric Weisstein, Bow, MathWorld.

FORMULA

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

EXAMPLE

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

MATHEMATICA

r = 1/GoldenRatio;

N[ArcSin[r], 100]

RealDigits[%]  (* A175288 *)

N[ArcCos[r], 100]

RealDigits[%]  (* A195692 *)

N[ArcTan[r], 100]

RealDigits[%]  (* A195693 *)

N[ArcCos[-r], 100]

RealDigits[%]  (* A195694 *)

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-Fran├žois Alcover, Feb 19 2013 *)

CROSSREFS

Cf. A195692.

Sequence in context: A019103 A272619 A172360 * A153509 A248093 A292091

Adjacent sequences:  A175285 A175286 A175287 * A175289 A175290 A175291

KEYWORD

cons,easy,nonn

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

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Last modified December 19 10:10 EST 2018. Contains 318246 sequences. (Running on oeis4.)