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A175288
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Decimal expansion of the constant x satisfying (cos(x))^2 = sin(x).
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5
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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
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OFFSET
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0,1
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COMMENTS
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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.
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LINKS
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Eric Weisstein's World of Mathematics, Bow.
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FORMULA
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Equals arccos(1/sqrt(phi)).
Equals arctan(1/sqrt(phi)) = arccot(sqrt(phi)). (End)
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EXAMPLE
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x = 0.66623943.. = 38.1727076... degrees.
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MATHEMATICA
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r = 1/GoldenRatio;
N[ArcSin[r], 100]
RealDigits[x/.FindRoot[Cos[x]^2==Sin[x], {x, .6}, WorkingPrecision->120]] [[1]] (* Harvey P. Dale, Nov 08 2011 *)
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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Disambiguated the curve here from the Mathworld bow curve - R. J. Mathar, Mar 29 2010
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STATUS
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approved
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