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Show that arcsin is the inverse of sin

by gymko
Tags: arcsinsinx
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gymko
#1
Sep19-09, 05:17 AM
P: 4
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
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praharmitra
#2
Sep19-09, 07:24 AM
P: 311
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
#3
Sep19-09, 08:22 AM
P: 4
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
#4
Sep19-09, 03:25 PM
P: 867
Show that arcsin is the inverse of sin

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
#5
Sep19-09, 05:21 PM
P: 607
[itex]\sin(\arcsin(x)) = x[/itex] for all real [itex]x[/itex] but [itex]\arcsin(\sin(x))[/itex] only for [itex]-\pi/2 \le x \le \pi/2[/itex] . That's with one standard way of defining [itex]\arcsin[/itex]
tnutty
#6
Sep19-09, 06:42 PM
P: 328
well look at it this way :

sin( asin(x ) )

let x = 1;

the asin(1) = P;
then sin(P) = 1;
Hurkyl
#7
Sep19-09, 06:54 PM
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Quote Quote by g_edgar View Post
[itex]\sin(\arcsin(x)) = x[/itex] for all real [itex]x[/itex]
Did you mean all x in [-1,1]? (Or are you making an assertion about the complex Arcsin function?)
gymko
#8
Sep20-09, 04:23 AM
P: 4
Thank you very much for all. :)
Gregg
#9
Sep20-09, 06:26 AM
P: 463
y=arcsin(sin(x))

sin(y) = sin(x)

y = x
Hurkyl
#10
Sep20-09, 06:40 AM
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Quote Quote by Gregg View Post
y=arcsin(sin(x))

sin(y) = sin(x)

y = x
g_edgar
#11
Sep20-09, 07:57 AM
P: 607
Quote Quote by Hurkyl View Post
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
#12
Sep20-09, 08:55 AM
P: 463
Quote Quote by Hurkyl View Post
Quote Quote by gymko View Post
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.


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