OFFSET
0,1
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)
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..5000
FORMULA
Equals arctan(sqrt(5/3)). - Amiram Eldar, Jul 09 2023
EXAMPLE
arccos(sqrt(3/8)) = 0.91173829096848763635848956431...
MATHEMATICA
PROG
(PARI) acos(sqrt(3/8)) \\ G. C. Greubel, Nov 18 2017
(Magma) [Arccos(Sqrt(3/8))]; // G. C. Greubel, Nov 18 2017
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Clark Kimberling, Sep 23 2011
STATUS
approved