Difference between arcsin and inverse sine.

Question

I first learned that arcsin and inverse sine are two ways of saying the same thing.

But then I was thinking about the inverse sine function being a function, so it must be limited in it's range from -1 to +1.

What would you call the sine function reflected through the line y=x that is not limited in range ?

Sure, it would not be a function but what would you call this graph ?

Could it be that arcsin is not a function and has infinite solutions whereas inverse sine is a function and has only one solution, e.g. $\arcsin (0.5) = \frac{\pi}{6}+2n\pi,n \in\mathbb Z, \sin^{-1}(0.5)=\frac{\pi}{6}$ ?

If not and they both have only one solution then how would you express the graph that has infinite solutions ?

I found this:

https://www.dummies.com/education/math/trigonometry/how-to-distinguish-between-trigonometry-functions-and-relations/

It seems a convention of capital/small letters are used to distinguish functions and relations.

Answer 1

these are in fact the same function. $\arcsin$ is the better notation I feel, because many students think that they are equal $$\sin^{-1}(x) \neq \frac{1}{\sin(x)}$$

when in fact $\arcsin(x)$ (or $\sin^{-1}(x)$) is the unique function so that:

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

Now to address your other question. $\arcsin(x)$ IS a function. If you wish to know it's domain and range, first consider the domain and range of $\sin(x)$

This is a bounded periodic function whose domain is all of the real numbers and whose range is from -1 to 1 inclusive. Recall that the domain and range of inverses switch. Therefore the domain of $\arcsin$ is from -1 to 1 with vertical asymptotes at -1 and 1 and the range is all real numbers:

Answer 2

If I understand your question correctly, you want to ask the following:

• I have a function which is not injective; this means that there exists two values $x_1$ and $x_2$ with $x_1\neq x_2$ and $f(x_1)=f(x_2)$

  In your case: $f(x)=\sin(x)$

• Now I'm drawing the graph $x=f(y)$

  Because the function is not injective the graph contains multiple points with the same $x$ coordinate.

• Does a certain term (word, name) exist for such kinds of graphs?
• Is there a special term for some kind of function $g(y)=\{x:f(x)=y\}=\{x_1,x_2,...\}$?

I doubt that there is a certain term (word, name) for such graphs and/or functions in general.

Especially in the case of the "inverse function" you'll have to keep in mind that a function has exactly one function value for a given function argument.

This means that: $f(x_1)=f(x_2)\Rightarrow g(f(x_1))=g(f(x_2))$.

It is not possible to define a function with $\sin^*(0)=0$ and $\sin^*(0)=\pi$ the same time!

You could however define a function whose function value is a set rather then a number.

Example:

$f(x)=x+5\\ f^{-1}(y)=y-5\\ f^*(y)=\{y-5\}$

Note that $f^{-1}(y)\neq f^*(y)$ because $f^{-1}(y)$ is a number and $f^*(y)$ is a set!

You might now define $sin^*(y)=\{x:sin(x)=y\}$ so $sin^*(0)=\{0,\pm\pi,\pm 2\pi,\pm 3\pi...\}$.

However I doubt that a name of such a function exists.

Answer 3

For a relation $R$, usually $R^{-1}$ denotes the converse relation, i.e. $R^{-1}=\{(x,y)\mid (y,x)\in R\}$. Some relations are functions, and if the relation $f$ is a function and its converse relation $f^{-1}$ is also a function we call it the inverse function of $f$, their compositions are then identities.

The function $\sin:\mathbb{R}\to[-1,1]$ is not injective, therefore its converse relation $\sin^{-1}\subset[-1,1]\times\mathbb{R}$ is not a function. $\sin$ is however surjective and therefore has right-inverses, we pick one of them and call it $\arcsin$, usually $\arcsin:=\left(\sin|_{\left[-\frac{\pi}2,\frac{\pi}2\right]}\right)^{-1}$. This is a function, but not the inverse function of $\sin$ (which doesn't exist) but a right-inverse function of $\sin$, i.e. $\sin\circ\arcsin=\operatorname{id}_{[-1,1]}$ but $\arcsin\circ\sin\neq\operatorname{id}_{\mathbb{R}}$.