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A197021
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Decimal expansion of the radius of the circle tangent to the curve y=cos(3x) at points (x,y) and (-x,y), where 0<x<1.
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2
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3, 7, 1, 8, 1, 1, 0, 4, 1, 7, 3, 6, 1, 7, 2, 1, 8, 4, 0, 1, 9, 5, 6, 4, 7, 3, 5, 1, 5, 8, 8, 5, 7, 9, 0, 2, 8, 9, 7, 0, 6, 2, 6, 3, 9, 2, 8, 8, 3, 6, 4, 8, 1, 7, 8, 7, 7, 3, 4, 1, 4, 7, 3, 3, 1, 8, 5, 2, 8, 8, 2, 0, 5, 1, 3, 1, 2, 7, 3, 1, 4, 2, 0, 5, 9, 8, 0, 8, 0, 0, 1, 2, 2, 6, 8, 5, 7, 4, 2
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OFFSET
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0,1
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COMMENTS
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Let (x,y) denote the point of tangency. Then
x=0.346818914654599529577486037538498433565584415464...
y=0.505826306518745297430716717373078359704411629139...
slope=-2.5879060509806840663013781941932136174746999...
(The Mathematica program includes a graph.)
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LINKS
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EXAMPLE
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radius=0.3718110417361721840195647351588579028970626...
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MATHEMATICA
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r = .371; c = 3;
Show[Plot[Cos[c*x], {x, -0.5, 0.5}],
ContourPlot[x^2 + (y - r)^2 == r^2, {x, -1, 1}, {y, -1, 1}], PlotRange -> All, AspectRatio -> Automatic]
t = x /. FindRoot[c*Sin[c*x] Cos[c*x] - x == x*Sqrt[1 + (c*Sin[c*x])^2], {x, .25, .55}, WorkingPrecision -> 100]
RealDigits[t] (* x coordinate of tangency point *)
y = Cos[c*t] (* y coordinate of tangency point *)
radius = Cos[c*t] - t/(c*Sin[c*t])
slope = -c*Sin[c*t] (* slope at tangency point *)
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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