A197739 Decimal expansion of least x>0 having sin(2x)=3*sin(6x).

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

Offset: 0

Author

Clark Kimberling, Oct 18 2011

Keywords

• nonn
• cons

Comments

This constant is the least x>0 for which the function f(x)=(sin(x))^2+(cos(3x))^2 has its maximal value. Least positive solutions of the equations f(x)=m/2, f(x)=m/3, f(x)=1, and f(x)=1/2 are given by sequences shown in the guide below.

In general, suppose that b and c are distinct positive real numbers. Let f(x)=(sin(bx))^2+cos((cx))^2. The extrema of f are the solutions of b*sin(2bx)=c*sin(2cx).

In the following guide, constants x given by the sequences (or explicit number) listed for each b,c are, in this order:

(1) least x>0 such that f(x)=(its maximum, m)

(2) m, the maximum of f

(3) least x>0 having f(x)=m/2

(4) least x>0 having f(x)=m/3

(5) least x>0 having f(x)=1

(6) least x>0 having f(x)=1/2

...

(b,c)=(1,2): A195700, x=25/16, A197589, A197591,

A019670, A197592

(b,c)=(1,3): A197739, A197588, A197590, A197755,

A003881, A197488

(b,c)=(1,4): A197758, A197759, A197760, A197761,

A019692 (x=pi/5), A003881

(b,c)=(1,pi): A197821, A197822, A197823, A197824,

A197726, A197826

(b,c)=(1,2*pi): A197827, A197828, A197829, A197830,

A197700, A197832

(b,c)=(1,3*pi): A197833, A197834, A197835, A197836,

A197837, A197838

Examples

			0.47765830906225463908192855125787887712170734750500...
		

Links

• Index entries for transcendental numbers.

Crossrefs

Cf. A197739, A197588.

Programs

Mathematica

b = 1; c = 3;
f[x_] := Cos[b*x]^2; g[x_] := Sin[c*x]^2; s[x_] := f[x] + g[x];
r = x /. FindRoot[b*Sin[2 b*x] == c*Sin[2 c*x], {x, .47, .48}, WorkingPrecision -> 110]
RealDigits[r]  (* A197739 *)
m = s[r]
RealDigits[m]  (* A197588 *)
Plot[{b*Sin[2 b*x], c*Sin[2 c*x]}, {x, 0, Pi}]
d = m/2; t = x /. FindRoot[s[x] == d, {x, 0.7, 0.8}, WorkingPrecision -> 110]
RealDigits[t]  (* A197590 *)
Plot[{s[x], d}, {x, 0, Pi}, AxesOrigin -> {0, 0}]
d = m/3; t = x /. FindRoot[s[x] == d, {x, 0.8, 0.9}, WorkingPrecision -> 110]
RealDigits[t]  (* A197755 *)
Plot[{s[x], d}, {x, 0, Pi}, AxesOrigin -> {0, 0}]
d = 1; t = x /. FindRoot[s[x] == d, {x, 0.7, 0.8}, WorkingPrecision -> 110]
RealDigits[t]  (* A003881 *)
Plot[{s[x], d}, {x, 0, Pi}, AxesOrigin -> {0, 0}]
d = 1/2; t = x /. FindRoot[s[x] == d, {x, .9, .93}, WorkingPrecision -> 110]
RealDigits[t]  (* A197488 *)
Plot[{s[x], d}, {x, 0, Pi}, AxesOrigin -> {0, 0}]
RealDigits[ ArcTan[ Sqrt[ 2-Sqrt[3] ] ], 10, 99] // First (* Jean-François Alcover, Feb 27 2013 *)