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A332500 Decimal expansion of the maximal normal distance between sine and cosine; see Comments.

%I A332500 #30 Jun 27 2025 10:55:07
%S A332500 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 A332500 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 A332500 8,7,5,4,4,9,6,2,2,1,4,0,5,4,1,3,0,7
%N A332500 Decimal expansion of the maximal normal distance between sine and cosine; see Comments.
%C A332500 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 A332500 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 A332500 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!
%H A332500 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.
%F A332500 d(u') = 2*sqrt((u - 3 Pi/4)^2 + (sin u)^2).
%F A332500 Equals sqrt(d^2+2-2*sqrt(1-d^2)) where d = A003957. - _Gleb Koloskov_, Jun 16 2021
%e A332500 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 A332500 0.4039727532995172093189617400663... = sin u; cf. A086751.
%e A332500 1.0949989843708724286504083007155... = maximal normal distance between sine and cosine.
%e A332500 1.9866519235847646080193264936226... = snc u; cf A332503.
%t A332500 Plot[{Sin[x], Cos[x]}, {x, -Pi, 3 Pi}, AspectRatio -> Automatic,
%t A332500 ImageSize -> 600, PlotLabel -> "sine and cosine"]
%t A332500 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 A332500 ListPlot[t, PlotLabel -> "y \[Equal] snc(x)"]
%t A332500 ListPlot[Cos[t], PlotLabel -> "y \[Equal] cos(snc(x))"]
%t A332500 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 A332500 ListPlot[t, PlotLabel -> "y \[Equal] cns(x)"]
%t A332500 ListPlot[Sin[t], PlotLabel -> "y \[Equal] sin(cns(x))"]
%t A332500 u = u /. FindRoot[0 == (-3 Pi/4) Sec[u] + u Sec[u] + Sin[u], {u, 1}, WorkingPrecision ->120]  (* A332501 *)
%t A332500 y = Sin[u]  (* A086751 *)
%t A332500 d = 2*Sqrt[(u - 3 Pi/4)^2 + y^2]  (* A332500 *)
%t A332500 RealDigits[u][[1]]  (* A332501 *)
%t A332500 RealDigits[y][[1]]  (* A086751 *)
%t A332500 RealDigits[d][[1]]  (* A332500 *)
%o A332500 (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 A332500 Cf. A086751, A332501, A332502, A332503, A332504, A332505, A332506, A332507, A003957.
%K A332500 nonn,cons
%O A332500 1,3
%A A332500 _Clark Kimberling_, May 05 2020