A197682 Decimal expansion of Pi/(2 + 2*Pi).

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

Offset: 0

Author

Clark Kimberling, Oct 17 2011

Keywords

• nonn
• cons

Comments

The number Pi/(2 + 2*Pi) is the least x > 0 such that sin(x) = cos(Pi*x).

If b and c are distinct real numbers, the solutions of sin(bx) = cos(cx) are x = (k - 1/2)*Pi/(b + c), where k runs through the integers. Thus, if b > 0 and c > 0, the least solution x > 0 is Pi/(2*b + 2*c), so that this is also the least x > 0 for which sin(c*x) = cos(b*x). Related sequences, each with a Mathematica program which includes a graph:

...

b.....c.......sequence........x

1.....2.......A019673........ x = Pi/6

1.....3.......A019678........ x = Pi/8

1.....4.......(A000796)/10... x = Pi/10

1.....Pi......A197682........ x = Pi/(2+2*Pi)

1.....2*Pi....A197683........ x = Pi/(2+4*Pi)

1.....1/Pi....A197684........ x = Pi^2/(2+2*Pi)

1.....2/Pi....A197685........ x = Pi^2/(4+2*Pi)

1.....Pi/2....A197686........ x = Pi/(2+Pi)

1.....Pi/3....A197687........ x = 3*Pi/(6+2*Pi)

1.....Pi/4....A197688........ x = 2*Pi/(4+Pi)

1.....Pi/6....A197689........ x = 3*Pi/(6+Pi)

2.....3.......(A000796)/10... x = Pi/10

2.....Pi......A197690........ x = Pi/(4+2*Pi)

2.....2*Pi....A197691........ x = Pi/(4+4*Pi)

2.....1/Pi....A197692........ x = Pi^2/(2+4*Pi)

2.....2/Pi....A197693........ x = Pi^2/(4+4*Pi)

2.....Pi/2....A197694........ x = Pi/(4+Pi)

3.....Pi......A197695........ x = Pi/(2+2*Pi)

3.....2*Pi....A197696........ x = Pi/(6+4*Pi)

3.....1/Pi....A197697........ x = Pi^2/(2+6*Pi)

3.....2/Pi....A197698........ x = Pi^2/(4+6*Pi)

3.....Pi/2....A197699........ x = Pi/(6+Pi)

1/2...Pi......A197700........ x = Pi/(1+2*Pi)

1/2...2*Pi....A197701........ x = Pi/(1+4*Pi)

1/2...1/Pi....A197724........ x = Pi^2/(2+Pi)

1/2...2/Pi....A197725........ x = Pi^2/(4+Pi)

1/2...Pi/2....A197726........ x = Pi/(1+Pi)

1/2...Pi/4....A197727........ x = 2*Pi/(2+Pi)

1/3...Pi/3....A197728........ x = 3*Pi/(2+2*Pi)

1/3...Pi/6....A197729........ x = 3*Pi/(2+Pi)

2/3...Pi/6....A197730........ x = 3*Pi/(4+Pi)

1/4...Pi......A197731........ x = 2*Pi/(1+4*Pi)

1/4...Pi/2....A197732........ x = 2*Pi/(1+2*Pi)

1/4...Pi/4....A197733........ x = 2*Pi/(1+Pi)

1/5...Pi/5....10*A197691..... x = 5*Pi/(2+2*Pi)

1/6...Pi/6....A197735........ x = 3*Pi/(1+Pi)

1/8...Pi/8....A197736........ x = 4*Pi/(1+Pi)

Examples

			0.37927349649738807267221534452244643...
		

Links

• Index entries for transcendental numbers

Crossrefs

Cf. A197683.

Programs

Mathematica

b = 1; c = Pi;
t = x /. FindRoot[Sin[b*x] == Cos[c*x], {x, .3, .4}]
N[Pi/(2*b + 2*c), 110]
RealDigits[%]  (* A197682 *)
Simplify[Pi/(2*b + 2*c)]
Plot[{Sin[b*x], Cos[c*x]}, {x, 0, Pi}]

PARI

1/(2/Pi+2) \\ Charles R Greathouse IV, Sep 27 2022