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 $\displaystyle f(a,b,c)=\cos a+\cos b+\cos c$ subject to the constraint $\displaystyle a+b+c=\pi$.
So $\displaystyle \langle -\sin a,-\sin b,-\sin c\rangle=\lambda\langle 1,1,1\rangle$
$\displaystyle \sin a=-\lambda$
$\displaystyle \sin b=-\lambda$
$\displaystyle \sin c=-\lambda$
$\displaystyle a+b+c=\pi$
Solve these four equations for $\displaystyle a,b,c$.
Spoiler: