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Math Help - Simplify arctan expression

  1. #1
    Senior Member Spec's Avatar
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    Simplify arctan expression

    Simplify:

    3arctan(2) - arctan(2/11)

    I'd know how to do it if it weren't for the 3 infront of arctan(2).

    EDIT: I guess I could do it like this:

    arctan(2) + arctan(2) + arctan(2) - arctan(2/11)

    That seems a bit over the top though. I'm still looking for an easier solution.


    Also, I've never encountered three arctan terms before:

    arctan(2) + arctan(3) + arctan(4)

    I'm guessing I need to first simplify arctan(2) + arctan(3), and then use that to get (arctan(2) + arctan(3)) + arctan(4) by using the tan(u+v) rule again. And of course make sure it's in the range of \pi/2<arctan(2)+arctan(3)+arctan(4)<3\pi/2
    Last edited by Spec; September 15th 2007 at 12:50 PM.
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  2. #2
    Math Engineering Student
    Krizalid's Avatar
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    Call \alpha=3\arctan2 & \beta=\arctan\frac2{11}

    Some identity could be useful for this.

    The answer is \pi
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  3. #3
    Senior Member Spec's Avatar
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    I already knew the answer and I've found a way to calculate it, so now I'm just looking for an easier solution than the one I proposed.
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  4. #4
    MHF Contributor
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    Quote Originally Posted by Spec View Post
    Simplify:

    3arctan(2) - arctan(2/11)

    I'd know how to do it if it weren't for the 3 infront of arctan(2).
    I am just curious. Can you please show us how to simplify
    arctan(2) -arctan(2/11)
    then?
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  5. #5
    Senior Member Spec's Avatar
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    tan(arctan(2)-arctan(\frac{2}{11})) = \frac{tan(arctan(2)) - tan(arctan(\frac{2}{11}))}{1 + tan(arctan(2)) * tan(arctan(\frac{2}{11}))} =
    \frac{2 - \frac{2}{11}}{1 + 2 * \frac{2}{11}} = \frac{20}{15} = \frac{4}{3}

    Therefore,
    arctan(2) - arctan(\frac{2}{11}) = arctan(\frac{4}{3}) + n\pi

    Note that's there's only one n for which the solution is within the defined range.
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  6. #6
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    Quote Originally Posted by Spec View Post
    tan(arctan(2)-arctan(\frac{2}{11})) = \frac{tan(arctan(2)) - tan(arctan(\frac{2}{11}))}{1 + tan(arctan(2)) * tan(arctan(\frac{2}{11}))} =
    \frac{2 - \frac{2}{11}}{1 + 2 * \frac{2}{11}} = \frac{20}{15} = \frac{4}{3}

    Therefore,
    arctan(2) - arctan(\frac{2}{11}) = arctan(\frac{4}{3}) + n\pi

    Note that's there's only one n for which the solution is within the defined range.
    Umm, I see.

    Then 3arctan(2) -arctan(2/11)
    = [arctan(2) +arctan(2)] +[arctan(2) -arctan(2/11)]
    ===> -4/3 +4/3
    = 0

    Hence,
    3arctan(2) -arctan(2/11)
    = arctan(0)
    = 0 or pi
    But pi is the answer.

    You don't like that long solution?
    Last edited by ticbol; September 15th 2007 at 04:36 PM.
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