Show that arcsin is the inverse of sin

In summary, the conversation discusses the concept of arcsin and its relationship with the sin function. It is mentioned that for certain values of x, arcsin(sin(x)) is equal to x. The conversation also touches on the graph of y = arcsin(sin(x)) and how it resembles the line y = x. It is also noted that this relationship is only valid for certain intervals and may involve the use of complex arcsin.
  • #1
gymko
4
0

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

 
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  • #2


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


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


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


[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]
 
  • #6


well look at it this way :

sin( asin(x ) )

let x = 1;

the asin(1) = P;
then sin(P) = 1;
 
  • #7


g_edgar said:
[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?)
 
  • #8


Thank you very much for all. :)
 
  • #9


y=arcsin(sin(x))

sin(y) = sin(x)

y = x
 
  • #10


Gregg said:
y=arcsin(sin(x))

sin(y) = sin(x)

y = x
:confused:
 
  • #11


Hurkyl said:
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.
 
  • #12


Hurkyl said:
:confused:

gymko said:
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.
 

What is the definition of inverse?

The inverse of a function is a mathematical operation that undoes the original function. It essentially reverses the process of the original function.

What is the inverse of sin?

The inverse of sin is arcsin, also known as inverse sine or sin-1. It is the inverse function of sin, and is used to solve for the angle when given the sine value of that angle.

How do you show that arcsin is the inverse of sin?

To show that arcsin is the inverse of sin, we must prove that the composition of the two functions, sin(arcsin(x)) and arcsin(sin(x)), results in the identity function f(x) = x. This can be done algebraically or graphically.

What is the domain and range of sin and arcsin?

The domain of sin is all real numbers, while the range is between -1 and 1. The domain of arcsin is also all real numbers, but the range is between -π/2 and π/2.

Why is it important to understand the concept of inverse functions?

Inverse functions are important because they allow us to solve for unknown values in equations and to perform operations that would otherwise be impossible. Inverse functions are also used in various areas of mathematics and science, such as trigonometry, calculus, and physics.

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