Math Help - Finding inverse of an arctan problem

1. Finding inverse of an arctan problem

Hi everyone. My calc II final is coming up, and this was one of the problems I missed on a test during the semester. I just can't seem to figure it out. I think I'm doing it right, but I always get the wrong answer.

so the problem is:
f(x) = sin(arctan(x)). Find f-1(1/2).

If anyone could help that'd be great. thank you!

2. so the problem is: f(x) = sin(arctan(x)). Find f-1(1/2).
Okay. Let $y=f(x)$.
So we have $y=sin(arctan(x))$. The idea is to isolate x and have it in function of y. The x variable is affected by arctan and also by sin. But as sin also affects arctan, we should start "taking it off" first. We can rewrite the equation into the following one which is equivalent : $arcsin(y)=arctan(x)$. Now taking off the arctan affecting x : $tan(arcsin(y))=x$. Done, since $f^{-1}(x)$ is precisely $tan(arcsin(x))$. Just evaluate this function with $x=\frac{1}{2}$.

3. I'm confused as to your explanation. Is there a better way to break this down for me?

4. tan, sin, arctan and arcsin are continuous functions, so:

$y = sin(arctan(x)) \implies$

$arcsin(y) = arcsin(sin(arctan(x))) = arctan(x) \implies$

$tan(arcsin(y)) = tan(arctan(x)) = x\implies$

So we have that $x = tan(arcsin(y))$

This is because $tan(arctan(v)) = v$ and $arctan(tan(v)) = v$

The same goes for all trigonometic functions.

5. As far as I know, the method I gave you is the easiest way to get the inverse of a function. I'll try to be more precise. Example : You have a function f, so that $f(x)=x+3$. What is the inverse function of f? That is, $f^{-1}(x)$. The way to get $f^{-1}(x)$ is first to put $f(x)=y$. We had $f(x)=x+3$, so now we have $y=x+3$. What to do with this? The idea is to isolate x. From $y=x+3$, subtract 3 from each sides of the equation. You get $y-3=x$. You have x in terms of y, and that was the idea. The equation can be rewritten as $x(y)=y-3$. But what is $x(y)$? It's $f^{-1}(y)$. We wanted $f^{-1}(x)$, so you just have to exchange the x for the y, so you get $f^{-1}(x)=x-3$. This is what we were looking for.
Hope you will understand what you've missed on your test. Tell us where you don't understand.

6. Originally Posted by etha
f(x) = sin(arctan(x)). Find f-1(1/2).
Here's a simple-minded approach to this problem.

If $x=f^{-1}(1/2)$ then $f(x) = 1/2$. So we want a solution to the equation $\sin(\arctan x) = 1/2$. But $\sin(30^{\circ}) = 1/2$. Therefore $\sin(\arctan x) = \sin(30^{\circ})$. So let $\arctan x = 30^{\circ}$, or in other words $x = \tan(30^{\circ}) = 1/\sqrt3$.