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Re: Is it possible to express sin(xy) in terms of sin(x) and sin(y)?
Posted:
Sep 25, 2002 5:33 PM
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David C. Ullrich <ullrich@math.okstate.edu> wrote in message news:<fte3pu4aigrnnh2nh7gqlpmksf124im1ur@4ax.com>...
> On 25 Sep 2002 06:13:44 -0700, neil.fitzgerald@ic.ac.uk (Midianight)
> wrote:
>
> >mathma18@hotmail.com (Narasimham G.L.) wrote in message news:<676dc11a.0209220350.24732dd3@posting.google.com>...
> >> Is it possible to express or expand sin(x*y) in terms of sin(x) and sin(y)?
> >> If so what is the expansion?
> >> [ sin (x+y) is elementary, but not sin(x*y)]
> >
> >
> >Hi everybody (in voice of Dr. Nick),
> >
> >It seems to me that all the other posters to this thread have been
> >missing the point.
> >
> >Fair enough, they've shown that there cannot exist a function f of two
> >variables such that
> >
> >sin(xy) = f(sin(x), sin(y)), for all x and y
> >
> >but I don't think that's exactly what the original poster was after.
> >
> >I don't know enough theory to be able to phrase what I want to say
> >properly, but let's define a "pseudo-function" f : X -> Y to be
> >something that takes elements of X and returns non-empty subsets of Y
> >that are at most countable (say).
> >
> >So then we have pseudo-functions such as natural logarithm, inverse
> >trigonometric functions (which are really the same thing, of course)
> >and pseudo-functions to solve polynomial equations.
> >
> >Now consider the following pseudo-function:
> >
> >f(x, y) = x.sqrt(1 - y^2) + y.sqrt(1 - x^2) (*)
> >
> >(Ok, I know that, strictly speaking, I should say precisely what I
> >mean by "sqrt" in this formula, but isn't it obvious?)
> >
> >This has the property that it "expresses sin(x + y) in terms of sin(x)
> >and sin(y)". Furthermore, notice that it is expressed without
> >mentioning sines at all, unlike, say
> >
> >f(x, y) = sin(sin^-1(x) + sin^-1(y))
> >
> >
> >I think that what the original poster wants is something like (*)
> >above, that is expressed preferably using only arithmetic operations
> >and "root extraction" pseudo-functions.
> >
> >Would anyone like to have a stab at this (or prove (as I suspect) that
> >no such pseudofunction exists)?
>
> It's obvious that such a pseudofunction exists. Define
>
> f(x,y) = {all real numbers}
>
> for all x, y.
>
> David C. Ullrich
Please at least read my posts before replying
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