Virgil
Posts:
3,521
Registered:
12/6/04
|
|
Re: Is it possible to express sin(xy) in terms of sin(x) and sin(y)?
Posted:
Sep 22, 2002 3:33 PM
|
|
In article <emtrouk76p9nu630qqqrr406skg2k38251@4ax.com>,
Angus Rodgers <angus_prune@bigfoot.com> wrote:
> On Sun, 22 Sep 2002 16:31:12 +0100, "David R. MacIver"
> <davidr.maciver@virgin.net> wrote:
>
> > Narasimham G.L. wrote:
>
> > > 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)]
>
> > Not in the same way, no. Certainly sin(x,y) cannot be
> > written as p(Sin(x), Cos(x), Sin(y), Cos(y)) where P is
> > a polynomial function. If it could then it would have a
> > period which was a rational multiple of Pi in each of x
> > and y. However, consider what would happen when x=Sqrt(2)
> > for example. Sin(xy) has a y-period of Pi * Sqrt(2),
> > which is not a rational multiple of Pi. [...]
>
> Let me just rephrase this argument, in the spirit of
> _Mathematics Made Difficult_:
>
> There's a very fundamental little lemma in set theory
> which says that if you have functions f:A->B, g:A->C,
> then there exists a function h:B->C such that g = h o f
> if and only if, for all u, v in A, if f(u) = f(v) then
> g(u) = g(v).
>
> All the trigonometric functions are in a sense 'really'
> defined on the circle S = R/2piZ (a topological group),
> so any putative formula for sin(xy), in terms of any
> of the trigonometric functions of x and y, would have
> to be defined on the torus T = SxS, i.e there would
> have to be a function h:T->R making the diagram below
> commute (where the unlabelled arrows are the 'natural
> projections'):
>
> xy
> RxR ----> R ----> S
> | |
> | | sin
> | |
> v v
> T .............> R
> h?
>
> The fundamental lemma shows that there exists such a
> function h if and only if, for all x, y, x', y' in R,
> if x - x' \in 2piZ and y - y' in 2piZ then sin(xy) =
> sin(x'y'). But this is not true, as David's example
> shows (e.g. taking x = x' = sqrt(2), y = 2pi, y' = 0,
> we would have to have sin(2sqrt(2)pi) = 0, which is
> false).
>
> If you wonder why on Earth I bothered to obfuscate a
> perfectly good argument in this fashion, it's because
> I wanted to see if there was any way to justify (in
> 'pure maths' terms) my feeling that such an identity
> would be "dimensionally incorrect", because (so to
> speak) "angles cannot be multiplied". One possible
> way is to interpret "angle" to mean an element of S,
> and then the argument above shows that there is no
> function SxS->S making the diagram
>
> xy
> RxR ----> R
> | |
> | |
> | |
> v v
> SxS ....> S
> ?
>
> commute.
>
> If charged with pointless pedantry, I'll plead guilty.
I prefer to view it as excessive elegance.
|
|
