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Math Help - Trig equation with tangents

  1. #1
    Member wiseguy's Avatar
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    Trig equation with tangents

    Here's the problem: tan^2x-3tanx-4=0, x [0, 3pi]

    What I've worked out so far:

    (tanx+1)(tanx-4)=0
    tan(x)=-1, tan(x)=4
    arctan(-1)=x, arctan(4)=x
    -0.7853982=x, 1.325818=x

    I'm aware the period is 360 degrees... do I simply add 360 to both solutions and then check to see which ones are correct? How do I use the "tangent is positive in the first and fourth quadrants" to solve?
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  2. #2
    A Plied Mathematician
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    Be careful with radians versus degrees there. Which mode was your calculator in?
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  3. #3
    Member wiseguy's Avatar
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    Radians - I double checked
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  4. #4
    Member Mathelogician's Avatar
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    The period for the tangent is 180 degrees!
    And the tangent is positive in the 1st and the 3rd quadrant...
    Use the general solution for the tangent: tgx=tga => x=k.pi+a and find out for what values of k the solution is in the given interval.
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  5. #5
    A Plied Mathematician
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    Then you wouldn't want to add 360 degrees, but how much? (This is a multi-faceted question, because the tangent function is periodic with what period?)
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  6. #6
    Member Mathelogician's Avatar
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    If radian, then add pi instead of 180 degrees...
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  7. #7
    Member wiseguy's Avatar
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    Okay, got it, so adding pi to -0.7853 and 1.3258 will bring the answers?
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  8. #8
    A Plied Mathematician
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    You'll need to add it several times, until you get outside the allowed region.
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  9. #9
    Member Mathelogician's Avatar
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    Sure! Just check the obtained x is in the given iterval.
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  10. #10
    Member wiseguy's Avatar
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    So essentially this:

    x=-0.785399+π, x=-0.785399+2π, x=-0.785399+3π; x=1.32582, x=1.32582+π, x=1.32582+2π
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  11. #11
    A Plied Mathematician
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    Looks correct to me. Always double-check your answers, though.
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  12. #12
    Member wiseguy's Avatar
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    So 6* answers is typical of tan? Thanks so much I just want to be prepared for my test...
    Last edited by wiseguy; July 19th 2010 at 11:02 AM. Reason: *meant to say 6 not 8
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  13. #13
    Member Mathelogician's Avatar
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    Of course it is!
    In this special case it has 8. For anothrer equations it may have less or more.
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  14. #14
    A Plied Mathematician
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    For that region of interest, you got 6 answers. That would be typical of your original equation that is quadratic in tan(x), if the roots are distinct and real.
    Last edited by Ackbeet; July 19th 2010 at 10:59 AM. Reason: Clearer expression.
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