Re: really easy trig question


matt271829-news@xxxxxxxxxxx wrote:
> . wrote:
> > <matt271829-news@xxxxxxxxxxx> wrote in message
> > news:1125486982.505779.162180@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
> > >
> > > . wrote:
> > >> Hi All,
> > >>
> > >> I used to know this, but my brain has got rusty with age.
> > >>
> > >> Why is it that sin (Pi/3) = (root 3)/2 ?
> > >>
> > >> Thanks for your help
> > >>
> > >> Michael
> > >
> > > Draw an equilateral triangle, and drop a perpendicular from the apex to
> > > the base. Then a bit of Pythagoras and you should be done.
> > >
> >
> > Sorry, I must have worded my question badly.
> >
> > If I do as you have said for a unit equalateral triangle I get ~0.866 ,
> > which ~(root 3)/2
> > But if I didn't know 0.866 ~(root 3)/2, how would I come up with the idea of
> > trying (root 3)/2 ?
>
> I don't entirely understand how you got 0.866... without knowing how
> you got it. Unless you did this with physical measurement?? I didn't
> mean to do that.
>
> Take an equilateral triangle ABC of side length 1. Drop a perpendicular
> from A to BC, to intersect BC at point D. ADC is a right-angled
> triangle. Angle ACD is pi/3.
>
> We have that AC = 1, and, by symmetry, DC = 1/2. By Pythagoras'
> theorem, AD^2 + DC^2 = AC^2, so AD = sqrt(AC^2 - DC^2) = sqrt(1 - 1/4)
> = sqrt(3)/2.
>
> Then from the definition of the sine, sin(pi/3) = AD/AC = sqrt(3)/2.
>
> (This shows that sin(pi/3) IS equal to sqrt(3)/2, but whether it shows
> WHY sin(pi/3) is equal to sqrt(3)/2 is another matter!)

The WHY is easy:
Sin (pi/3)=sqrt(3)/2 because of the definition of the sine function.
If you want to know why sine was defined in such a way that you get
this result, then you need to ask the one who originally devised it
(probably some ancient egyptian, greek, babylonian, or mayan; maybe
more than one of them :-)

.

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