Ok how can I turn this trig function into a function that doesn't use any trig functions?
$\displaystyle f(x)=sin(\arctan(x))$
Draw a right triangle with sides 1, x and $\displaystyle \sqrt(1+x^2)$.
Can you find the angle whose tangent is x (hint - opposite over adjacent = x/1)?
That's arctan(x). Now look at the triangle. What is the sine of that angle? (hint = opposite / hypotenuse). That's what you're looking for.