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%I #20 Jun 16 2021 15:27:20
%S 1,0,9,4,9,9,8,9,8,4,3,7,0,8,7,2,4,2,8,6,5,0,4,0,8,3,0,0,7,1,5,5,2,4,
%T 6,7,1,2,9,1,0,5,1,4,0,6,0,7,0,5,4,3,6,0,2,0,6,5,8,0,3,3,4,2,9,5,5,1,
%U 8,7,5,4,4,9,6,2,2,1,4,0,5,4,1,3,0,7
%N Decimal expansion of the maximal normal distance between sine and cosine; see Comments
%C Let S and C denote the graphs of y = sin x and y = cos x. For each point (u, sin u) on S, let S(u) be the line normal to S at (u, sin u), and let (snc u, cos(snc u) be the point of intersection of S(u) and C. Let d(u) be the distance from (u,sin u) to (snc u, cos(snc u)). We call d(u) the u-normal distance from S to C and note that in [0,Pi], there is a unique number u' such that d(u') > d(u) for all real numbers u except those of the form u' + k*Pi. We call d(u') the maximal normal distance between sine and cosine, and we call snc the sine-normal-to-cosine function.
%C For each (v, cos v) on C, let C(u) be the line normal to C at (v, cos v), and let (cns v, sin(cns v)) be the point of intersection of C(u) and S. Let e(v) be the distance from (v, cos v) to (cns v, sin(cns v)). We call d(v) the v-normal distance from C to S and note that there exists a unique number v' that maximizes e, and e(v') = d(u'). We call cns the cosine-normal-to-sine function. The numbers u' and v' are given in A332501 and A332503.
%C Note that the maximal normal distance (see Example) exceeds the normal distance from (Pi/2,1) in sine to (Pi/2,0) in cosine - possibly a surprise!
%F d(u') = 2*sqrt((u - 3 Pi/4)^2 + (sin u)^2).
%F Equals sqrt(d^2+2-2*sqrt(1-d^2)) where d = A003957. - _Gleb Koloskov_, Jun 16 2021
%e 2.72573705679992524967463858129656... = the number u in [0,2 Pi] such that the line normal to S at (u, sin u) passes through the point (3 Pi/4,0); cf. A332501.
%e 0.4039727532995172093189617400663... = sin u; cf. A086751.
%e 1.0949989843708724286504083007155... = maximal normal distance between sine and cosine.
%e 1.9866519235847646080193264936226... = snc u; cf A332503.
%t Plot[{Sin[x], Cos[x]}, {x, -Pi, 3 Pi}, AspectRatio -> Automatic,
%t ImageSize -> 600, PlotLabel -> "sine and cosine"]
%t t = Table[x = x /. FindRoot[Cos[x] == -x Sec[u] + u Sec[u] + Sin[u], {x, 0}], {u, -2 Pi, 2 Pi, Pi/101}];
%t ListPlot[t, PlotLabel -> "y \[Equal] snc(x)"]
%t ListPlot[Cos[t], PlotLabel -> "y \[Equal] cos(snc (x))"]
%t t = Table[x = x /. FindRoot[Sin[x] == x Csc[u] - u Csc[u] + Cos[u], {x, 0.1}], {u, -2 Pi + .01, 2 Pi - .01, Pi/101}];
%t ListPlot[t, PlotLabel -> "y \[Equal] cns(x)"]
%t ListPlot[Sin[t], PlotLabel -> "y \[Equal] sin(cns(x)"]
%t u = u /. FindRoot[0 == (-3 Pi/4) Sec[u] + u Sec[u] + Sin[u], {u, 1}, WorkingPrecision ->120] (* A332501 *)
%t y = Sin[u] (* A086751 *)
%t d = 2*Sqrt[(u - 3 Pi/4)^2 + y^2] (* A332500 *)
%t RealDigits[u][[1]] (* A332501 *)
%t RealDigits[y][[1]] (* A086751 *)
%t RealDigits[d][[1]] (* A332500 *)
%o (PARI) my(d=solve(x=0,1,cos(x)-x)); sqrt(d^2+2-2*sqrt(1-d^2)) \\ _Gleb Koloskov_, Jun 16 2021
%Y Cf. A086751, A332501, A332502, A332503, A332504, A332505, A332506, A332507, A003957.
%K nonn,cons
%O 1,3
%A _Clark Kimberling_, May 05 2020
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