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]

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dextercioby

Homework Helper
Re: Arcsin

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

tahayassen

Re: Arcsin

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.

dextercioby

Homework Helper
Re: Arcsin

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

Re: Arcsin

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?

dextercioby

Homework Helper
Re: Arcsin

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

Re: Arcsin

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

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tahayassen

Re: Arcsin

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?

HallsofIvy

Homework Helper
Re: Arcsin

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

Re: Arcsin

Thanks everyone. It makes sense now.

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