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Math Help - determine arctan(r) +arctan(s) if r and s are routes of a quadratic equation

  1. #1
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    determine arctan(r) +arctan(s) if r and s are routes of a quadratic equation

    Hi.

    I've encountered this problem and have no idea where to start:

    Let r and s be the two solutions to the equation x^2 - 2003x + 2004 = 0. Determine arctan(r) + arctan(s). One is not allowed to use a computer or calculator when solving this, as it was once in a past math competition. Any ideas on how to solve it?
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  2. #2
    GJA
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    Re: determine arctan(r) +arctan(s) if r and s are routes of a quadratic equation

    Hi ShadowKnight8702,

    This is a tough problem to simply give a hint on, but I'll give it a shot.

    Instead of looking directly at \arctan(r)+\arctan(s), I would suggest looking at \tan\left(\arctan(r)+\arctan(s)\right). Start here and also google "properties of roots of quadratic equations." I have debated whether or not to mention the specific result from this search which will be relevant and have decided to leave it out for now; that said, the fact needed will be one of the first results.

    This is a bit vague, but it sounded like you wanted to get this one for yourself. Let me know if you need a bit more, or if anything is unclear. Good luck!
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    Re: determine arctan(r) +arctan(s) if r and s are routes of a quadratic equation

    tan(arctan(r) + arctan(s)) was a big enough hint. Thanks! One last question though. After simplifying, it is easy to see that the answer is arctan(-1), but how does one decide whether the answer will be 3pi/4 or 7pi/4?
    Last edited by ShadowKnight8702; June 4th 2013 at 07:20 PM.
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