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

1. ## 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?

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

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!

3. ## 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?