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Math Help - Inequality . HELP me

  1. #1
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    Inequality . HELP me

    If you can, please with this trigonometric equation Worried)
    2./




    THANKS YOU VERY MUCH
    Last edited by mr fantastic; July 21st 2009 at 06:54 PM. Reason: Deleted duplicate question
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  2. #2
    MHF Contributor red_dog's Avatar
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    2) 2\cos x-1\neq 0\Rightarrow x\neq\pm\frac{\pi}{3}+2k\pi, \ k\in\mathbb{Z}

    The left side member is:

    \frac{4\sin x\cos x+2\cos x-2\sin x-1}{2\cos x-1}=\frac{(2\cos x-1)(2\sin x+1)}{2\cos x-1}=2\sin x+1

    Then the equation becomes

    2\sin x+1=\cos 2x+\sqrt{3}(\sin x+1)\Rightarrow 2\sin^2x+(2-\sqrt{3})\sin x-\sqrt{3}=0

    \sin x=-1\Rightarrow x=-\frac{\pi}{2}+2k\pi, \ k\in\mathbb{Z}

    \sin x=\frac{\sqrt{3}}{2}\Rightarrow x=-\frac{\pi}{3}+(2k+1)\pi, \ k\in\mathbb{Z}
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