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Thread: sum of cosines

  1. #1
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    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?
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  2. #2
    Super Member redsoxfan325's Avatar
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    Quote Originally Posted by makenqau1 View Post
    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 $\displaystyle 1$.
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