AIME - Trigonometry

Find a positive integer n such that:



Keep in mind that, since this is a non calculator test, frantically entering this into your calculator wont help you, as its miles away in the Mojave Desert being cocooned by military spiders. So you best prepare to figure it out on your own.

First off, here's a tip:

Whenever you see inverse trig functions, try to get rid of them by applying trig functions to the equation.

With that said, lets take the tangent of both sides:




The LHS looks like it can be simplified using our tangent sum formula. If you don't know it (and you should), it's:

However, we don't have two terms, we have four. So the best thing to do is split the sum of the four into a sum of two. Just add some parenthesis in to see what I mean:



This can start to get a little messy, so lets split it into two variables:




Calculate each using the sum formula:
















Now plug these back in:






(Here I made the substitution right away, rather than type out that whole ugly expression. Resubstitute in our original expressions for x and y if you want to see what I was avoiding.)



The rest is simply the algebra of solving for n:














Tadaa. If you're practicing outside of the AIME, you can plug that into your calculator and check your answer.

Now, what would've we done if we didn't get a positive integer for an answer? Well, it's the AIME, so if they say solve for n, you can be sure it's a positive integer. Not to mention they right out say it, too. You should go back and check your work for answers. Work your away up instead of starting at the top, checking for arithmetic errors first. If you still cant find it, then check your logic for errors too. If you STILL cant find it, well then, I'd say skip that one for a bit and attempt it again later.