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Decimal expansion of xo, where P=(xo,yo) is the point nearest O=(0,0) in which a line y=mx meets the curve y=cos(x+Pi/3) orthogonally.
2

%I #8 Apr 10 2021 02:04:49

%S 2,5,5,4,6,5,2,8,6,1,0,3,8,5,3,5,9,6,6,9,5,8,8,2,6,9,6,6,1,3,3,2,0,2,

%T 7,2,6,5,4,7,8,8,3,5,5,9,5,3,7,0,8,5,2,8,9,3,0,2,5,2,6,7,6,7,2,9,7,6,

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

%N Decimal expansion of xo, where P=(xo,yo) is the point nearest O=(0,0) in which a line y=mx meets the curve y=cos(x+Pi/3) orthogonally.

%C See the Mathematica program for a graph.

%C xo=0.255465286103853596695882696613320272654788...

%C yo=0.264932084602776862434116494762571068650190...

%C m=1.0370570837365150046614795837584277605222343...

%C |OP|=0.3680373919265496189530095416155881110455...

%t c = Pi/3;

%t xo = x /. FindRoot[x == Sin[x + c] Cos[x + c], {x, .8, 1.2}, WorkingPrecision -> 100]

%t RealDigits[xo] (* A197004 *)

%t m = 1/Sin[xo + c]

%t RealDigits[m] (* A197005 *)

%t yo = m*xo

%t d = Sqrt[xo^2 + yo^2]

%t Show[Plot[{Cos[x + c], yo - (1/m) (x - xo)}, {x, -Pi/4, Pi/2}],

%t ContourPlot[{y == m*x}, {x, 0, Pi}, {y, 0, 1}], PlotRange -> All,

%t AspectRatio -> Automatic, AxesOrigin -> Automatic]

%Y Cf. A197002, A196996, A197000.

%K nonn,cons

%O 0,1

%A _Clark Kimberling_, Oct 10 2011