sum of cosines

**makenqau1** · October 31st 2009, 07:51 AM

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?

**redsoxfan325** · October 31st 2009, 04:40 PM

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: