sum of cosines

Quote:

Originally Posted by makenqau1

Show that in any triangle ABC the following inequality holds:cosA+cos B+cos C is less than equal to 3/2 . What is the minimal value of the sum?

We are trying to maximize $f(a,b,c)=\cos a+\cos b+\cos c$ subject to the constraint $a+b+c=\pi$ .

So $\langle -\sin a,-\sin b,-\sin c\rangle=\lambda\langle 1,1,1\rangle$

$\sin a=-\lambda$
$\sin b=-\lambda$
$\sin c=-\lambda$
$a+b+c=\pi$

Solve these four equations for $a,b,c$ .

Spoiler:

$3\arcsin(-\lambda)=\pi\implies\lambda=-\sin(\pi/3)\implies\lambda=-\frac{\sqrt{3}}{2}\implies a=b=c=\frac{\pi}{3}$

So the maximum value is $3\cos(\pi/3)=\frac{3}{2}$ .

For the minimum value, there isn't really one unless we allow $a=b=0$ and $c=\pi$ . (Note that this, coupled with $\lambda=0$ , also generates a valid solution to the above system of equations.) Regardless, there is an infimum, and it is $1$ .