Show that arcsin is the inverse of sin

[gymko]

Homework Statement

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.

Homework Equations

The Attempt at a Solution

 

[praharmitra]

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.

 

[gymko]

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.

 

[Bohrok]

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.

 

[g_edgar]

$\sin(\arcsin(x)) = x$ 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 $\arcsin$

 

[tnutty]

well look at it this way :

sin( asin(x ) )

let x = 1;

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

 

[Hurkyl]

$\sin(\arcsin(x)) = x$ for all real $x$

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

 

[gymko]

Thank you very much for all. :)

 

[Gregg]

y=arcsin(sin(x))

sin(y) = sin(x)

y = x

 

[Hurkyl]

y=arcsin(sin(x))

sin(y) = sin(x)

y = x

 

[g_edgar]

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

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.

 

[Gregg]

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.

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.