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Math Help - find expression for an angle

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    find expression for one line segment

    Attachment 25106


    Link to image:

    In this geometric setup, I'm trying to find an expression for A dependent on \phi, given that R and \theta are known.
    My take on this is to write the two expressions for the two sides DL and DR

    1)  DL^2 = A^2 + R^2 - 2 A R \cos\left(\frac{1}{2}\pi - \phi\right)
    2)  DR^2 = A^2 + R^2 - 2 A R \cos\left(\frac{1}{2}\pi + \phi\right) = A^2 + R^2 + 2 A R \cos\left(\frac{1}{2}\pi - \phi\right)

    and an expression for DS = 2A

    3)  DS^2 = DL^2 + DR^2 - 2 DL \cdot DR \cos\left(\theta\right)

    Then it seems that I could achieve my goal by eliminating DL and DR in the equation for DS.

    Summation of 1) and 2) gives

    4)  DL^2 + DR^2= 2A^2 + 2R^2

    and multiplication of 1) and 2) gives

    5) DL^2\cdot DR^2 = \left(A^2 + R^2 - 2 A R \cos\left(\frac{1}{2}\pi - \phi\right)\right)\left(A^2 + R^2 + 2 A R \cos\left(\frac{1}{2}\pi - \phi\right) \right)
    = \left(A^2 + R^2 \right)^2 - 4 A^2 R^2 \cos^2\left(\frac{1}{2}\pi -  \phi\right)

    Inserting 4) into 3) and squaring provides

    6)  DL^2 \cdot DR^2 = \frac{(R^2 - A^2)^2}{\cos^2\left(\theta\right)}

    Equating 5) and 6) gives an equation dependent on only A and \phi

    7) \left(A^2 + R^2 \right)^2 - 4 A^2 R^2 \cos^2\left(\frac{1}{2}\pi -  \phi\right) = \frac{(R^2 - A^2)^2}{\cos^2\left(\theta\right)}

    Rearranging

    8) \cos^2\left(\theta\right)\left(A^4 + R^4 + 2 A^2R^2\right) - 4 A^2 R^2 \cos^2\left(\frac{1}{2}\pi -   \phi\right)\cos^2\left(\theta\right) = A^4 + R^4 - 2 A^2R^2

    9)  A^4(1-\cos^2\left(\theta\right)) - A^2(2 R^2 + 2 R^2\cos^2\left(\theta\right) - 4 R^2 \cos^2\left(\frac{1}{2}\pi -   \phi\right)\cos^2\left(\theta\right)) + R^4(1-\cos^2\left(\theta\right)) = 0

    10)  A^4 - A^2 \frac{2 R^2(1 + \cos^2\left(\theta\right) - 2 \cos^2\left(\frac{1}{2}\pi -    \phi\right)\cos^2\left(\theta\right)^2)}{1-\cos^2\left(\theta\right)} + R^4 = 0

    I have a hard time believing this expression to be true because of the following reasoning:
    For \phi=0 then A = R\tan\left(\frac{1}{2}\theta\right)
    As \phi is increased I would assume that A should also increase (relative to R\tan\left(\frac{1}{2}\theta\right)). However, for the numerical values that I have used to test the expression in 10) I find that A decreases.

    Looking back, I can't find out where I go wrong.
    Last edited by niaren; October 8th 2012 at 07:32 AM. Reason: Found possible error in derivation
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