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Math Help - Find real solutions

  1. #1
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    Find real solutions

    Find real solutions of the system

    sin x + 2 sin(x+y+z) = 0,
    sin y + 3 sin(x+y+z) = 0,
    sin z + 4 sin(x+y+z) = 0.
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  2. #2
    MHF Contributor red_dog's Avatar
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    We have \displaystyle \sin(x+y+z)=-\frac{\sin x}{2}=-\frac{\sin y}{3}=-\frac{\sin z}{4}.
    Adding the first two equations and substracting the third, we have:
    \sin x+\sin y-\sin z+\sin(x+y+z)=0
    \displaystyle 2\sin\frac{x+y}{2}\cos\frac{x-y}{2}+2\sin\frac{x+y}{2}\cos\frac{x+y+2z}{2}=0
    \displaystyle 2\sin\frac{x+y}{2}\left(\cos\frac{x-y}{2}+\cos\frac{x+y+2z}{2}\right)=0
    \displaystyle4\sin\frac{x+y}{2}\cos\frac{x+z}{2}\c  os\frac{y+z}{2}=0 (1)

    If \displaystyle\sin\frac{x+y}{2}=0\Rightarrow x+y=2k\pi\Rightarrow y=2k\pi-x
    Then \displaystyle\frac{\sin x}{2}=-\frac{\sin x}{3}\Rightarrow\sin x=0\Rightarrow x=l\pi\Rightarrow y=m\pi\Rightarrow z=n\pi

    In the same way if one of the others two factors from (1) is 0.
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