A197016 Decimal expansion of the radius of the circle tangent to the curve y=cos(x) and to the positive x and y axes.

4, 2, 8, 7, 7, 8, 5, 3, 6, 0, 3, 0, 6, 1, 2, 8, 6, 3, 6, 1, 3, 6, 9, 1, 7, 4, 1, 0, 4, 8, 9, 9, 9, 7, 0, 4, 9, 0, 6, 0, 5, 8, 9, 3, 6, 1, 5, 2, 0, 2, 6, 8, 5, 1, 9, 9, 3, 7, 8, 8, 2, 4, 6, 9, 8, 4, 7, 1, 3, 9, 3, 2, 2, 8, 8, 8, 9, 7, 9, 4, 8, 6, 0, 3, 5, 1, 0, 1, 5, 5, 4, 3, 3, 2, 3, 1, 2, 3, 6

Offset: 0

Author

Clark Kimberling, Oct 08 2011

Keywords

• nonn
• cons

Comments

Let (x,y) denote the point of tangency. Then

x=0.65099256993050253383380179140902527170294599...

y=0.79548271667012269646991174255794794798663548...

slope=-0.6059762763335882427824587356062000...

(The Mathematica program includes a graph.)

Examples

			radius=0.428778536030612863613691741048999...
		

Crossrefs

Cf. A197017, A197018, A197019, A197020.

Programs

Mathematica

r = .428;
Show[Plot[Cos[x], {x, 0, Pi}],
ContourPlot[(x - r)^2 + (y - r)^2 == r^2, {x, -1, 1}, {y, -1, 1}], PlotRange -> All, AspectRatio -> Automatic]
f[x_] := (x - Sin[x] Cos[x])/(1 - Sin[x]);
t = x /.FindRoot[Cos[x] == f[x] + Sqrt[2*f[x]*x - x^2], {x, .5, 1}, WorkingPrecision -> 100]
x1 = Re[t]  (* x coordinate of tangency point *)
y = Cos[x1] (* y coordinate of tangency point *)
radius = f[x1]
RealDigits[radius] (* A197016 *)
slope = -Sin[x1]   (* slope at tangency point *)