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Thread: solve equation

  1. #1
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    solve equation

    1.solve for x in $\displaystyle sinx tanx+ tanx-2sinx+cosx=0$ for 0<orequalx<or equal $\displaystyle 2\pi$ rads

    2. find the exact value of $\displaystyle (\frac{7\pi}{6})$
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  2. #2
    MHF Contributor red_dog's Avatar
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    Here is the graph of the function in the left side of the equation.

    The equation has one solution between 3 and 4.
    Attached Thumbnails Attached Thumbnails solve equation-equation.jpg  
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  3. #3
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    my bad
    Last edited by mathaddict; Jul 29th 2009 at 11:43 PM.
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  4. #4
    MHF Contributor red_dog's Avatar
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    Quote Originally Posted by mathaddict View Post
    (1)
    sin^2 x +cos^2 x + sin x -2 sin x cos x =0

    sin x (1- 2 cos x )=0


    $\displaystyle \sin x(1-2\cos x)=-1$
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  5. #5
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    More information please

    Hello william

    What do you mean? This doesn't make sense.
    Quote Originally Posted by william View Post
    ...
    2. find the exact value of $\displaystyle (\frac{7\pi}{6})$
    If you mean 'Find the exact value of $\displaystyle \sin(\tfrac{7\pi}{6})$', then the answer is that $\displaystyle \sin(\pi+A) = -\sin A$, and $\displaystyle \sin(\tfrac{\pi}{6}) = \tfrac12$. So...?

    Grandad
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  6. #6
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    Quote Originally Posted by red_dog View Post
    Here is the graph of the function in the left side of the equation.

    The equation has one solution between 3 and 4.
    By a numerical method, I got $\displaystyle x=3.4968$, correct to 4 d.p. Are you sure we have the correct question here?

    Grandad
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  7. #7
    MHF Contributor red_dog's Avatar
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    I tried another way, but that doesn't lead to an exact solution.

    $\displaystyle \sin x-2\sin x\cos x+1=0$

    $\displaystyle 2\sin\frac{x}{2}\cos\frac{x}{2}\left(\sin^2\frac{x }{2}+\cos^2\frac{x}{2}\right)-$

    $\displaystyle -4\sin\frac{x}{2}\cos\frac{x}{2}\left(\cos^2\frac{x }{2}-\sin^2\frac{x}{2}\right)+\left(\sin^2\frac{x}{2}+\ cos^2\frac{x}{2}\right)^2=0$

    $\displaystyle \sin^4\frac{x}{2}+6\sin^3\frac{x}{2}\cos\frac{x}{2 }+2\sin^2\frac{x}{2}\cos^2\frac{x}{2}-2\sin\frac{x}{2}\cos^3\frac{x}{2}+\cos^4\frac{x}{2 }=0$

    Divide by $\displaystyle \cos^4\frac{x}{2}$ and let $\displaystyle \tan\frac{x}{2}=t$

    $\displaystyle t^4+6t^3+2t^2-2t+1=0$

    $\displaystyle (t+1)(t^3+5t^2-3t+1)=0$

    $\displaystyle t=-1$ which is not good because $\displaystyle x\neq\frac{(2k+1)\pi}{2}$

    $\displaystyle t^3+5t^2-3t+1=0$ which has an irational solution.
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