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Math Help - Arctangent Simplification

  1. #1
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    Sep 2009
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    Arctangent Simplification

    Hello!

    I am stuck with this challenge:

    I want to derive an expression for angle BAC, using angle ABC, distances BC and AB.


    Translating the problem to mathematics, It looks like this:
    <br />
BAC=tan^{-1} {\lgroup {\gamma \cdot sin(\alpha) \over 1-\gamma \cdot cos(\alpha)} \rgroup}<br />

    Where

    <br />
\gamma \equiv {\overline {BC} \over \overline {AB}}<br />

    After some manipulation, trying to simplify the expression in order to get the angle directly (since that sine and cosine are converting an angle to cartesian, an then the arctangent goes backward, to an angle again - seems redundant)

    I could only manage to get this:

    <br />
BAC=tan^{-1} \lgroup { {2 \cdot \gamma \cdot tan({\alpha \over 2})} \over {(1-\gamma)+(1+\gamma) \cdot tan^2({\alpha \over 2}))}} \rgroup<br /> <br />

    I can't simplify more than that - what am I missing? Can it be so ugly?

    Greetings, and thanks in advance!

    Manuel
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  2. #2
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    Joined
    Sep 2009
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    I forgot to include the image!
    Last edited by Manuel; September 7th 2009 at 03:08 PM. Reason: Typographic mistake
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