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

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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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  • #2
dextercioby
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Why do you think the range is not R, but it is restricted ? Usually plotting computer software knows its maths...
 
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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.
 
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dextercioby
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So you want the subset of [itex] \mathbb{R}\oplus\mathbb{R} [/itex] made up of

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

and the values of arcsine are 'copied' from [itex] [-\pi/2,\pi/2] [/itex] into [itex] [-\pi/2 +n\pi ,\pi/2 + n \pi] [/itex] 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.
 
  • #5
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So you want the subset of [itex] \mathbb{R}\oplus\mathbb{R} [/itex] made up of

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

and the values of arcsine are 'copied' from [itex] [-\pi/2,\pi/2] [/itex] into [itex] [-\pi/2 +n\pi ,\pi/2 + n \pi] [/itex] 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?
 
  • #6
dextercioby
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Multiplication means repeating, copying the points in the plot of arcsin x, where x ranges from -1 to 1. Oy is the Y axis.
 
  • #7
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So it would look like this: http://img819.imageshack.us/img819/5679/62578051.png [Broken]
 
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  • #8
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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?
 
  • #9
HallsofIvy
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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.
 
  • #10
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Thanks everyone. It makes sense now.
 

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