A195703 Decimal expansion of arccos(sqrt(3/8)) and of arcsin(sqrt(5/8)).

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

Offset: 0

Author

Clark Kimberling, Sep 23 2011

Keywords

• nonn
• cons

Comments

From Roman Witula, Oct 31 2012: (Start)

It can be deduced that cos(a)*cos(3*a) = min {cos(x)*cos(3*x): x in R} = -9/16 = -0,5625. Moreover we have 1 = cos(Pi)*cos(3*Pi) = max {cos(x)*cos(3*x): x in R}.

Note that also cot(3*a) = -3*cot(a) since d(-cos(x)*cos(3*x))/dx = sin(x)*cos(3*x) + 3*cos(x)*sin(3*x). Moreover we have max{F(x): x in R} = -2*sqrt(2)*cos(a)*cos(3*a) = 9*sqrt(2)/8, where F(x) := cos(x) + sin(x) - sqrt(2)*sin(2*x). Indeed, we obtain F(x) = sqrt(2)*cos(x)*(sin(Pi/4) - sin(x)) + sqrt(2)*sin(x)*(cos(Pi/4) - cos(x)) = 2*sqrt(2)*sin(Pi/8 - x/2)*(cos(x)*cos(Pi/8 + x/2) - sin(x)*sin(Pi/8 + x/2)) = 2*sqrt(2)*sin(Pi/8 - x/2)*cos(3*x/2 + Pi/8) = -2*sqrt(2)*cos((x - 5*Pi/4)/2)*cos(3*(x - 5*Pi/4)/2).

(End)

Examples

			arccos(sqrt(3/8)) = 0.91173829096848763635848956431...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..5000
• Index entries for transcendental numbers

Crossrefs

Cf. A195700.

Programs

Magma

[Arccos(Sqrt(3/8))]; // G. C. Greubel, Nov 18 2017

Mathematica

r = Sqrt[3/8];
N[ArcSin[r], 100]
RealDigits[%]  (* A195700 *)
N[ArcCos[r], 100]
RealDigits[%]  (* A195703 *)
N[ArcTan[r], 100]
RealDigits[%]  (* A195705 *)
N[ArcCos[-r], 100]
RealDigits[%]  (* A195706 *)

PARI

acos(sqrt(3/8)) \\ G. C. Greubel, Nov 18 2017

Formula

Equals arctan(sqrt(5/3)). - Amiram Eldar, Jul 09 2023