# 2006 STEP 3 Q4

For the last part,

I let $T(x) = t(tan x)$ and I showed that $T(x) + T(y) = T(x+y)$

hence using the first part I concluded $T(x) = Ax = t(tanx)$

to work out $t(u)$ I let $u=tanx$ and hence $x = arctanu + n\pi$

therefore $t(u) = Ax = A (arctan u +n\pi)$

But the solutions say $t(u) = A arctan u$ - to find the most general solution don't you need to add on the $n\pi$ ?

I would be grateful for a quick reply since the STEP exams are this week.

Thanks for all the help so far

### Don't worry

I know this kinda sucks, but I think what's going on is they don't want you to stress about details of arctan and the fact that there are multiple values of $u$ which give the same $x$. I know they said find the most general, but there would still be some fiddling to do once you've added your $n\pi$ term (see here: http://math.stackexchange.com/questions/502189/why-is-arctan-fracxy1-xy-...) which isn't actually the point of the question.
How can you tell it's not the point of the question? It's hard to say, but a rule of thumb is that when you're looking for values of $\theta$ that satisfy certain equations, you need to be careful about the fact that $\sin(x)=\sin(y)$ doesn't imply $x=y$, etc., while when you're talking about functions you can be a little less careful (so here, you can interpret the arctan in the answer as taking whichever value it needs to, sort of).
I hope that makes sort of sense and I'm sorry I can't be more definitive. Like I said though, try not to worry about it - I doubt your answer would have lost any marks, so just be careful not to let questions like this waste too much of your time (my strategy was always to write an answer that didn't deal with messy bits like extra $\pi$'s, then come back and try to work out what happened in the other cases if I had time at the end of the exam).

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