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Math Help - sum of cosines

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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 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:
    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 3\cos(\pi/3)=\frac{3}{2}.

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