Trigonometry fun

I thought I would have some fun taking the log-base-2 of sines and
cosines of different angles and see what I came up with. Not a lot,
however, here are some obvious ones: (using ln x/ln(2)for log base 2)

ln(sin(45))/ln 2=-0.5, -600 cents = 600 cents
ln(sin(30))/ln 2==-1, -1200 cents = 0 cents
ln(sin(15))/ln 2=-1.9499, =~-2340 cents =~60 cents

And in general, ln(sin(theta))+ln(cos(theta))/ln(2) + 1 (octave)
=ln(sin(2*theta))/ln(2) which I discovered by accident on my
calculator.

Also interesting is that ln(sin(15))+ln(cos(15))/ln(2)=-2 which
follows from the above, as being one less than ln(sin(30)/ln(2).

Not much else-- ln(tan(15))/ln(2)*1200 is about 120 cents, does
all this point to a use for 20-et?

Paul

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<paul.hjelmstad@u...> wrote:
> I thought I would have some fun taking the log-base-2 of sines and
> cosines of different angles and see what I came up with. Not a lot,
> however, here are some obvious ones: (using ln x/ln(2)for log base
2)
>
> ln(sin(45))/ln 2=-0.5, -600 cents = 600 cents
> ln(sin(30))/ln 2==-1, -1200 cents = 0 cents
> ln(sin(15))/ln 2=-1.9499, =~-2340 cents =~60 cents
>
> And in general, ln(sin(theta))+ln(cos(theta))/ln(2) + 1 (octave)
> =ln(sin(2*theta))/ln(2) which I discovered by accident on my
> calculator.
>
> Also interesting is that ln(sin(15))+ln(cos(15))/ln(2)=-2 which
> follows from the above, as being one less than ln(sin(30)/ln(2).
>
> Not much else-- (ln(tan(15))/ln(2))*1200 is about 120 cents, does
> all this point to a use for 20-et?
>
> Paul

Also (ln(sin(15))/ln(2))*1200=~-2340 cents=~60 cents and (ln(cos
(15))/ln(2))*1200=~-60 cents, the only angle I know of that has this
property
Its the only angle that has this property

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<paul.hjelmstad@u...> wrote:
> --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
> <paul.hjelmstad@u...> wrote:
> > I thought I would have some fun taking the log-base-2 of sines
and
> > cosines of different angles and see what I came up with. Not a
lot,
> > however, here are some obvious ones: (using ln x/ln(2)for log
base
> 2)
> >
> > ln(sin(45))/ln 2=-0.5, -600 cents = 600 cents
> > ln(sin(30))/ln 2==-1, -1200 cents = 0 cents
> > ln(sin(15))/ln 2=-1.9499, =~-2340 cents =~60 cents
> >
> > And in general, ln(sin(theta))+ln(cos(theta))/ln(2) + 1 (octave)
> > =ln(sin(2*theta))/ln(2) which I discovered by accident on my
> > calculator.

Caveat: Only for 0 < theta < 90 degrees
> >
> > Also interesting is that ln(sin(15))+ln(cos(15))/ln(2)=-2 which
> > follows from the above, as being one less than ln(sin(30)/ln(2).
> >
> > Not much else-- (ln(tan(15))/ln(2))*1200 is about 120 cents, does
> > all this point to a use for 20-et?
> >
> > Paul
>
> Also (ln(sin(15))/ln(2))*1200=~-2340 cents=~60 cents and (ln(cos
> (15))/ln(2))*1200=~-60 cents, the only angle I know of that has
this
> property
> Its the only angle that has this property