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A332503 Decimal expansion of the number v such that the maximal normal distance between sine and cosine is the distance between (u, sin u) and (v, sin v), where u is the number u' in A332501; see Comments.

%I A332503 #8 Mar 20 2025 04:15:30
%S A332503 2,7,4,3,6,8,4,8,9,7,4,0,3,8,6,6,6,5,1,8,2,6,2,7,9,2,8,8,6,9,4,8,2,8,
%T A332503 1,2,0,4,8,6,3,4,6,3,8,0,4,5,9,4,1,1,0,2,1,0,3,8,9,7,1,2,7,6,0,4,2,7,
%U A332503 3,3,4,6,1,4,8,0,3,6,2,6,2,4,9,9,4,9
%N A332503 Decimal expansion of the number v such that the maximal normal distance between sine and cosine is the distance between (u, sin u) and (v, sin v), where u is the number u' in A332501; see Comments.
%C A332503 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. See A332500.
%F A332503 v = 3Pi/4 - u, where u is given by A332501.
%e A332503 v = 1.96870408298082320586768578622442620...
%t A332503 u = u /. FindRoot[u - 3 Pi/4 == Sin[u], {u, 1}, WorkingPrecision -> 120]  (* A332501 *)
%t A332503 v = 3 Pi/2 - u   (* A332503 *)
%t A332503 RealDigits[v][[1]]
%Y A332503 Cf. A332500, A332501, A086751.
%K A332503 nonn,cons
%O A332503 1,1
%A A332503 _Clark Kimberling_, May 05 2020