A345644 - OEIS

%I #10 Jul 01 2021 07:36:48

%S 6,4,2,7,0,7,8,7,2,5,4,6,5,3,2,4,4,5,7,7,9,2,1,1,7,7,8,4,6,8,6,0,7,9,

%T 1,8,2,8,5,0,4,7,8,2,4,0,8,1,4,6,3,0,3,9,8,5,3,3,1,5,0,7,9,4,6,4,4,9,

%U 0,0,0,9,9,3,4,6,5,2,5,4,5,3,1,3,3,8,2,4,4,2,8,0,9,7,2,7,3,7,8

%N Decimal expansion of the radius of the circle tangent to the curves y=cos(x), y=-cos(x) and to the y-axis for x in [0,Pi/2].

%C Let r and (x,y) denote the radius of the circle and the point of tangency in the first quadrant, respectively.

%C Then r in [0,1] is the root of equation cos(r+sqrt(r^2-1+sqrt(1-r^2)))^2 = 1-sqrt(1-r^2),

%C r = 0.642707872546532445779211778468607918285047824...,

%C x = r+sqrt(r^2-1+sqrt(1-r^2)) = 1.066010072972971718857583783392083793389510385...,

%C y = sqrt(1-sqrt(1-r^2)) = 0.483620364074368181073730094271148302685427120...

%e 0.642707872546532445779211778468607918285047824...

%t r = r /. FindRoot[Cos[r + Sqrt[-1 + r^2 + Sqrt[1 - r^2]]]^2 == 1 - Sqrt[1 - r^2], {r, 1/2}]; Show[Plot[Cos[x], {x, 0, Pi/2}], Plot[-Cos[x], {x, 0, Pi/2}], Graphics[Circle[{r, 0}, r]], PlotRange -> All, AspectRatio -> Automatic] (* _Vaclav Kotesovec_, Jul 01 2021 *)

%o (PARI) solve(r=0,1,cos(r+sqrt(r^2-1+sqrt(1-r^2)))^2-1+sqrt(1-r^2))

%Y Cf. A197016, A197017, A197018, A197019, A197020.

%K nonn,cons

%O 0,1

%A _Gleb Koloskov_, Jun 21 2021