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From: don on 21 Jul 2010 15:25
Why is the chord of a 40 degree angle ALWAYS less than double the chord of
a 20 degree angle?


From: Chip Eastham on 21 Jul 2010 15:49
On Jul 21, 3:25 pm, "don" <d...(a)panix.com> wrote:
> Why  is the chord of a 40 degree angle ALWAYS
> less than double the chord of a 20 degree angle?

The central angle that subtends a chord is
proportional to the curved length of the
circular arc subtended by that angle. I'll
assume you mean to compare chords of a common
circle (or at least of circles with equal
radii, which amounts to the same thing), and
that you understand a circle's chords that
subtend 20 degree angles are all the same
length.

So one way to look at the comparison is to
take the arc subtended by a chord of a 40
degree angle and draw two new chords from
the midpoint of the arc to the endpoints
of that original chord.

These new chords subtend a 20 degree angle.
Clearly the path they form between endpoints
is longer (because not straight) than the
original chord subtending a 40 degree angle.

In other words the chord of the 40 degree
angle is less (in length) than twice the
chord of a 20 degree angle.

regards, chip
From: don on 21 Jul 2010 16:24

<"Chip Eastham" <hardmath(a)gmail.com> wrote in message
news:adf56d56-0b3f-4425-9c1a-

<.......
<In other words the chord of the 40 degree
<angle is less (in length) than twice the
<chord of a 20 degree angle.

<regards, chip

Thank-you for that very clear explanation - I"m using autocad to try and
draw and follow what you just said.........


From: Chip Eastham on 21 Jul 2010 18:47
On Jul 21, 4:24 pm, "don" <d...(a)panix.com> wrote:
> <"Chip Eastham" <hardm...(a)gmail.com> wrote in message
>
> news:adf56d56-0b3f-4425-9c1a-
>
> <.......
> <In other words the chord of the 40 degree
> <angle is less (in length) than twice the
> <chord of a 20 degree angle.
>
> <regards, chip
>
> Thank-you for that very clear explanation - I"m using autocad to try and
> draw and follow what you just said.........

That was a "geometry" explanation. From an
algebra point of view, the length of the chord
subtended by angle A is 2*sin(A/2)*R, which can
be seen by drawing a perpendicular bisector of
the chord from the circle's center, R being the
circle's radius. For simplicity we will set R
to 1 and forget about it.

So, expressing angles in degrees for consistency
with the original post, the chord of a 40 degree
angle is:

2*sin(20 deg)

while twice the chord of a 20 degree angle is:

4*sin(10 deg)

Using the double angle identity:

sin(2A) = 2 sin(A) cos(A)

we can rewrite the first expression, chord length
for 40 degrees, as:

4*sin(10 deg)*cos(10 deg)

Compare that to twice the chord length of a 20
degree angle, and you'll see that we can account
for why the chord of 40 degrees is less than
twice the chord of 20 degrees by the factor
cos(10 deg). The cosine function at 10 degrees
is less than 1, proving what we wanted to show.

regards, chip
From: Bill on 22 Jul 2010 04:18
don wrote:
> Why is the chord of a 40 degree angle ALWAYS less than double the chord of
> a 20 degree angle?
>
>

The triangle inequality?

Bill
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