1. ## sum of cosines

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?

2. 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 $\displaystyle f(a,b,c)=\cos a+\cos b+\cos c$ 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:
$\displaystyle 3\arcsin(-\lambda)=\pi\implies\lambda=-\sin(\pi/3)\implies\lambda=-\frac{\sqrt{3}}{2}\implies a=b=c=\frac{\pi}{3}$

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

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