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%I #17 Jun 16 2021 15:26:59
%S 1,0,9,3,1,6,9,7,4,4,9,8,5,0,1,6,9,2,2,0,8,8,1,5,3,2,1,4,1,6,0,5,7,9,
%T 7,1,4,4,0,4,8,9,0,6,5,9,2,9,4,8,9,8,8,8,3,5,6,3,5,1,7,5,1,3,3,2,4,9,
%U 6,0,5,3,7,6,7,0,9,4,4,7,3,6,8,3,7,6,7,0,6,7,9,9,3,4,8,1,7,9,3,4,2
%N Decimal expansion of the slope of the line y=mx which meets the curve y=cos(x+Pi/4) orthogonally over the interval [0, 2*Pi] (as in A197002).
%C See the Mathematica program for a graph.
%C xo=0.3695425666075803208276560438369...
%C yo=0.4039727532995172093189617400663...
%C m=1.09316974498501692208815321416057...
%C |OP|=0.54749949218543621432520415035...
%F Equals sqrt(2-2*sqrt(1-d^2))/d where d = A003957. - _Gleb Koloskov_, Jun 16 2021
%t c = Pi/4;
%t xo = x /. FindRoot[x == Sin[x + c] Cos[x + c], {x, .8, 1.2}, WorkingPrecision -> 100]
%t RealDigits[xo] (* A197002 *)
%t m = 1/Sin[xo + c]
%t RealDigits[m] (* A197003 *)
%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, 1}], ContourPlot[{y == m*x}, {x, 0, Pi}, {y, 0, 1}], PlotRange -> All, AspectRatio -> Automatic, AxesOrigin -> Automatic]
%o (PARI) my(d=solve(x=0,1,cos(x)-x)); sqrt(2-2*sqrt(1-d^2))/d \\ _Gleb Koloskov_, Jun 16 2021
%Y Cf. A003957, A197002, A196996, A197000.
%K nonn,cons
%O 1,3
%A _Clark Kimberling_, Oct 09 2011
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