# How would the arcsin function look like without a restricted range?

tahayassen
I need help. I'm totally puzzled. How would the arcsin function look like without a restricted range? Can anyone post an image?

http://img94.imageshack.us/img94/9886/arcsin.png [Broken]

Last edited by a moderator:

## Answers and Replies

Homework Helper

Why do you think the range is not R, but it is restricted ? Usually plotting computer software knows its maths...

tahayassen

Why do you think the range is not R, but it is restricted ? Usually plotting computer software knows its maths...

Hmm? I understand that arcsin is just the inverse of the sin function which is restricted between negative pi/2 and positive pi/2. But I want a graph of the function that is the inverse of the sin function without any domain restrictions on the sin function. It wouldn't be a function though, because it wouldn't pass the vertical line test.

Homework Helper

So you want the subset of $\mathbb{R}\oplus\mathbb{R}$ made up of

$$S= \{(x,\arcsin x)| x\in [-1,1]\}$$

and the values of arcsine are 'copied' from $[-\pi/2,\pi/2]$ into $[-\pi/2 +n\pi ,\pi/2 + n \pi]$ and n can take any integer value ?

So it's just an infinite multiplication of the plot of the standard arcsine with a shift along Oy axis of \pi.

tahayassen

So you want the subset of $\mathbb{R}\oplus\mathbb{R}$ made up of

$$S= \{(x,\arcsin x)| x\in [-1,1]\}$$

and the values of arcsine are 'copied' from $[-\pi/2,\pi/2]$ into $[-\pi/2 +n\pi ,\pi/2 + n \pi]$ and n can take any integer value ?

So it's just an infinite multiplication of the plot of the standard arcsine with a shift along Oy axis of \pi.

I really wish I understood what a subset is. We're learning the inverse trigonometric functions in the trigonometry unit of my precalculus class, so I haven't really gotten into more complex stuff.

What do you mean by infinite multiplication of the plot of the standard arcsine? What's Oy and \pi?

Homework Helper

Multiplication means repeating, copying the points in the plot of arcsin x, where x ranges from -1 to 1. Oy is the Y axis.

tahayassen

So it would look like this: http://img819.imageshack.us/img819/5679/62578051.png [Broken]

Last edited by a moderator:
tahayassen

Actually, no, that's wrong. I understand the arcsin's with vertical shifts of 2pi, but sin(x + pi) = -sin(x), so wouldn't the arcsin's with vertical shifts of npi (where n is every odd integer) need to be horizontally reflected?

Homework Helper

Without restricting the domain, the graph of arcsine would look exactly like the graph of sine but along the y-axis, not the x-axis. Of course, that is not the graph of a function since one value of x would give infinitely many values of y.

tahayassen

Thanks everyone. It makes sense now.