# Show that arcsin is the inverse of sin

1. Sep 19, 2009

### gymko

1. The problem statement, all variables and given/known data

I have a problem, I don't know to substantiate, why arcsin(sin(x)) = sin(arcsin(x)) = x ?
Thank you very much for each advice.

2. Relevant equations
3. The attempt at a solution

2. Sep 19, 2009

### praharmitra

Re: arcsin(sin(x))

i didnt exactly get your question. Are you trying to find the range of values of x for which the above eq is valid?

what are u asking? can u be clearer.

3. Sep 19, 2009

### gymko

Re: arcsin(sin(x))

My question is: why arcsin(sin(x)) is possible to regulate for x. Why arcsin(sin(x)) = x?

Why graph for y = arcsin(sin(x)) is y = x?

Thank you.

4. Sep 19, 2009

### Bohrok

Re: arcsin(sin(x))

Do you know about the composition of inverse functions?
Also, arcsin(sin(x)) = sin(arcsin(x)) = x only for -1≤x≤1 for all three.

5. Sep 19, 2009

### g_edgar

Re: arcsin(sin(x))

$\sin(\arcsin(x)) = x$ for all real $x$ but $\arcsin(\sin(x))$ only for $-\pi/2 \le x \le \pi/2$ . That's with one standard way of defining $\arcsin$

6. Sep 19, 2009

### tnutty

Re: arcsin(sin(x))

well look at it this way :

sin( asin(x ) )

let x = 1;

the asin(1) = P;
then sin(P) = 1;

7. Sep 19, 2009

### Hurkyl

Staff Emeritus
Re: arcsin(sin(x))

Did you mean all x in [-1,1]? (Or are you making an assertion about the complex Arcsin function?)

8. Sep 20, 2009

### gymko

Re: arcsin(sin(x))

Thank you very much for all. :)

9. Sep 20, 2009

### Gregg

Re: arcsin(sin(x))

y=arcsin(sin(x))

sin(y) = sin(x)

y = x

10. Sep 20, 2009

### Hurkyl

Staff Emeritus
Re: arcsin(sin(x))

11. Sep 20, 2009

### g_edgar

Re: arcsin(sin(x))

You are right. For my equation you have to use complex arcsin. Wherever arcsin is defined, we have sin(arcsin(x)) = x , that is what we mean by arcsin.

12. Sep 20, 2009

### Gregg

Re: arcsin(sin(x))

y=arcsin[sin[x]] implies sin[y] = sin[x] and y=x. Forgetting the intervals, I don't the question had much to do with the interval which this valid for. It was just how does the graph look like y=x.