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?

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- Oct 31st 2009, 07:51 AMmakenqau1sum of cosinesShow 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?
- Oct 31st 2009, 04:40 PMredsoxfan325
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 $\displaystyle a,b,c$.

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