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A345644 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]. 0
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, 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, 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 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET
0,1
COMMENTS
Let r and (x,y) denote the radius of the circle and the point of tangency in the first quadrant, respectively.
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),
r = 0.642707872546532445779211778468607918285047824...,
x = r+sqrt(r^2-1+sqrt(1-r^2)) = 1.066010072972971718857583783392083793389510385...,
y = sqrt(1-sqrt(1-r^2)) = 0.483620364074368181073730094271148302685427120...
LINKS
EXAMPLE
0.642707872546532445779211778468607918285047824...
MATHEMATICA
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 *)
PROG
(PARI) solve(r=0, 1, cos(r+sqrt(r^2-1+sqrt(1-r^2)))^2-1+sqrt(1-r^2))
CROSSREFS
Sequence in context: A145011 A173625 A086036 * A019849 A244685 A118421
KEYWORD
nonn,cons
AUTHOR
Gleb Koloskov, Jun 21 2021
STATUS
approved

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Last modified February 21 07:54 EST 2024. Contains 370219 sequences. (Running on oeis4.)