... with a little help from Desmos and Wolfram Alpha
Desmos graphical approximation: appears to be about 1.932
The 4 and the 3 reminded me that there is a trig identity that looks like that. The title helped as well. But I had to look it up:
cos(3t) = 4*(cos(t))^3  3*cos(t)
Let x = cos(a) and y = cos(b)
We need to maximize cos(a) + cos(b), so it cannot be greater than 2.
(cos(3a))^2 + (cos(3b))^2 = 1
let u = 3a and v = 3b
cos(u)^2 + cos(v)^2 = 1
cos(u)^2 = 1  cos(v)^2 = sin(v)^2
cos(u) = ± sin(v)
if plus: v = pi/2  u
if minus: v = u  pi/2
Take the first one:
x + y = cos(a) + cos(b)
= cos(u/3) + cos(v/3)
= cos(u/3) + cos((pi/2  u)/3)
= cos(u/3) + cos(pi/6  u/3)
= cos(u/3) + cos(pi/6)cos(u/3) + sin(pi/6)sin(u/3)
f(u) = cos(u/3) + (√3/2)cos(u/3) + (1/2)sin(u/3)
take the derivative
f'(u) = (1/3)sin(u/3)  (√3/6)sin(u/3) + (1/6)cos(u/3) = 0
At this point, I let Wolfram do the heavy lifting.
Wolfram Alpha has the maximum as:
√(2 + √3) or approx 1.93185165258

Posted by Larry
on 20231112 13:33:40 