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Thread: Triangles with maximum product of sin of all angles

  1. #1
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    Triangles with maximum product of sin of all angles

    I need to find triangles for which is sinx*siny*sinz maximum when x,y,z are angles. Are those ones with all three angles 60?
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  2. #2
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    Re: Triangles with maximum product of sin of all angles

    True

    Proof:

    $\displaystyle \sqrt[3]{\sin x *\sin y* \sin z}\leq \frac{\sin x + \sin y + \sin z}{3}\leq \sin \left(\frac{x+y+z}{3}\right)$

    the first inequality: the geometric average is less than the arithmetic average

    the second one you can prove using the fact that the graph of sin is concave down between 0 and pi

    in our situation x + y + z = pi, so

    $\displaystyle \sin x *\sin y* \sin z\leq \left(\left.\sqrt{3}\right/2\right)^3$
    Last edited by Idea; Jan 30th 2015 at 09:50 AM.
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  3. #3
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    Re: Triangles with maximum product of sin of all angles

    Hi,
    Idea's proof is very clever, but it is "advanced". You need to know the inequality about means (standard, but no obvious proof) and some basic knowledge about convex sets. Assuming you know elementary calculus, you can solve the problem easily with Lagrange multipliers:
    Maximize $F(x,y,z)=\sin(x)\cdot \sin(y)\cdot \sin(z)$ subject to the constraint $x+y+z=\pi$ where each of $x,y,z$ are in $[0,\pi]$.
    Furthermore this method finds the solution for you.
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