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A197002 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/4) orthogonally.

%I A197002 #19 Feb 03 2025 09:57:29
%S A197002 3,6,9,5,4,2,5,6,6,6,0,7,5,8,0,3,2,0,8,2,7,6,5,6,0,4,3,8,3,6,9,3,6,7,
%T A197002 0,2,0,0,6,7,0,5,8,7,9,4,5,0,3,7,8,7,3,2,4,8,2,8,4,0,3,1,7,8,8,6,6,4,
%U A197002 2,3,2,7,4,4,1,7,7,3,7,9,7,2,9,9,6,8,8,0,5,3,4,6,5,8,8,3,2,6,5,9
%N A197002 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/4) orthogonally.
%C A197002 See the Mathematica program for a graph.
%C A197002 xo=0.36954256660758032082765604383693...
%C A197002 yo=0.40397275329951720931896174006631...
%C A197002 m=1.093169744985016922088153214160579...
%C A197002 |OP|=0.547499492185436214325204150357...
%H A197002 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.
%F A197002 Equals d/2 = A003957/2, where d is the Dottie number. - _Gleb Koloskov_, Jun 16 2021
%p A197002 evalf(solve(cos(x)=x,x)/2, 140);  # _Alois P. Heinz_, Feb 20 2024
%t A197002 c = Pi/4;
%t A197002 xo = x /. FindRoot[x == Sin[x + c] Cos[x + c], {x, .8, 1.2}, WorkingPrecision -> 100]
%t A197002 RealDigits[xo] (* A197002 *)
%t A197002 m = 1/Sin[xo + c]
%t A197002 RealDigits[m]  (* A197003 *)
%t A197002 yo = m*xo
%t A197002 d = Sqrt[xo^2 + yo^2]
%t A197002 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 A197002 (PARI) solve(x=0,1,cos(x)-x)/2  \\ _Gleb Koloskov_, Jun 16 2021
%Y A197002 Cf. A197003, A196996, A197000, A003957.
%K A197002 nonn,cons
%O A197002 0,1
%A A197002 _Clark Kimberling_, Oct 09 2011