Least Angle

[Robust]

My kids (2 generations of them) knowing how astute I am at the maths, get some kind of weird enjoyment watching me battle my way through these postings. I tell them to sign-up and join in if they're so smart. Anyway, the eldest did come up with a legitimate question which, if she won't I can answer.

 

"You give a formula for the least distance between each angular degree. What formula do you have for determining the chord length to that distance on the arc?"

 

I'm not that easy of a mark, so I suggested in return that one must first determine the least right angle. I have to say "Good on her"...she did come up with it, giving the formula of diameter/180 degrees!....What say y'all?

[C1ay]

My kids (2 generations of them) knowing how astute I am at the maths, get some kind of weird enjoyment watching me battle my way through these postings. I tell them to sign-up and join in if they're so smart. Anyway, the eldest did come up with a legitimate question which, if she won't I can answer.

 

"You give a formula for the least distance between each angular degree. What formula do you have for determining the chord length to that distance on the arc?"

 

I'm not that easy of a mark, so I suggested in return that one must first determine the least right angle. I have to say "Good on her"...she did come up with it, giving the formula of diameter/180 degrees!....What say y'all?

Your post doesn't make much sense. One can find the chord with the Law of Cosines for any angle, i.e. c=√(2r²-2r²cos(theta))

[Robust]

Your post doesn't make much sense. One can find the chord with the Law of Cosines for any angle, i.e. c=√(2r²-2r²cos(γ))

Clay, You know darn well that I can't do all that cosine jazz. Just come right out and tell us what the least angle is . You do that and I'll promise to educate myself on cosines.

Damn....I just knew this place was going to cause me a lot work!

[C1ay]

Clay, You know darn well that I can't do all that cosine jazz. Just come right out and tell us what the least angle is . You do that and I'll promise to educate myself on cosines.

Damn....I just knew this place was going to cause me a lot work!

Maybe this will help....

[Robust]

Maybe this will help....

 

I got the picture, Clay.The question is: What is the angle - a ri ght angle - giving that chord which defines on the arc the least possible distance between two adjacent angular degrees. We are navigating the Cosmos now. I tthink it a most salient point that we be able to define the parallax of lines involved in such effort to an exacting measure.

 

The girl gave us the answer: diameter/180 degrees - taking into account however that by diameter is meant the E/W coordinate of the circumference....if I see it correctly. Hey, Clay!!....can I be your navigator?

[C1ay]

I got the picture, Clay.The question is: What is the angle - a ri ght angle - giving that chord which defines on the arc the least possible distance between two adjacent angular degrees. We are navigating the Cosmos now. I tthink it a most salient point that we be able to define the parallax of lines involved in such effort to an exacting measure.

 

The girl gave us the answer: diameter/180 degrees - taking into account however that by diameter is meant the E/W coordinate of the circumference....if I see it correctly. Hey, Clay!!....can I be your navigator?

The distance of the chord bridging one degree is the same for all one degree increments and is given by the law of cosines already quoted. It also doesn't matter if you use a right angle or an isosceles angle, the chord length of one degree is the same either way.

[Robust]

The distance of the chord bridging one degree is the same for all one degree increments and is given by the law of cosines already quoted. It also doesn't matter if you use a right angle or an isosceles angle, the chord length of one degree is the same either way.

Yes, I understand that, Clay. The question is: What is that distance of the chord ?

[C1ay]

Yes, I understand that, Clay. The question is: What is that distance of the chord ?

Oh sorry, I didn't realize you couldn't use a calculator either. For a unit circle it is approximately 0.017453070996747869929776427947169

[Qfwfq]

The girl gave us the answer: diameter/180 degrees - taking into account however that by diameter is meant the E/W coordinate of the circumference....if I see it correctly. Hey, Clay!!....can I be your navigator?

Neither you, nor the girl, will ever be my navigator!!!

 

The chord can be given by multiplying the radius by twice the cosine of half the angle. It is not proportional to the angle; if you multiply the angle by a number n, the chord won't be n times the previous value. The chord for 1 degree can't be given by the above ratio.

 

The diameter is the chord for 180 degrees. It is not 180 times the chord for 1 degree.

[Robust]

Neither you, nor the girl, will ever be my navigator!!!

 

The chord can be given by multiplying the radius by twice the cosine of half the angle. It is not proportional to the angle; if you multiply the angle by a number n, the chord won't be n times the previous value. The chord for 1 degree can't be given by the above ratio.

 

The diameter is the chord for 180 degrees. It is not 180 times the chord for 1 degree.

I can only take your word for what you just say, but you do not give the result. What is the unit length of that chord which defines the least distance between 2 angular degrees?

[Robust]

I can only take your word for what you just say, but you do not give the result. What is the unit length of that chord which defines the least distance between 2 angular degrees?

The unit length I get is 0.0707106.....

[Qfwfq]

The unit length I get is 0.0707106.....

Why do you call that a unit of length? Oh, no, of course, your odd choice of terms... forget the question!

 

I notice I posted cosine instead of sine, just due to hurried posting. For angles in degrees:

 

1) 2sin(1/2) = 0,017453070996747869929776427947169

 

not much different from:

 

2) sin(1) = 0,017452406437283512819418978516316

 

or even from:

 

3) pi/180 = 0,017453292519943295769236907684886

 

but very different from your value.

 

2) and 3) are small angle approximations.

[Robust]

Why do you call that a unit of length? Oh, no, of course, your odd choice of terms... forget the question!

 

I notice I posted cosine instead of sine, just due to hurried posting. For angles in degrees:

 

1) 2sin(1/2) = 0,017453070996747869929776427947169

 

not much different from:

 

2) sin(1) = 0,017452406437283512819418978516316

 

or even from:

 

3) pi/180 = 0,017453292519943295769236907684886

 

but very different from your value.

 

2) and 3) are small angle approximations.

I call it a unit of length because that's what it is: 0.0707....unit chord length between 2 angular degrees 0.07901234567....unit apart.

[C1ay]

The unit length I get is 0.0707106.....

:Alien: Sounds to me like your calculator needs new batteries or else there is a malfunction between the keyboard and the chair.

[Robust]

:circle: Sounds to me like your calculator needs new batteries or else there is a malfunction between the keyboard and the chair.

Then please enlighten us, Clay. What figure does your calculator come up with?

[Robust]

In reviewing the posts it appears that I neglected to give the complete formulae for determining the least angle. Here it is now then, bearing in mind that by diameter is meant the E/W coordinate of the 4 quadrants:

 

1). Diameter/180-degrees = 0.05 as leg of right-angle triangle;

2). A^2 + B^2 = C^2 = hypotenuse;

3). Sqrt hypotenuse = 0.0707....chord length between each adjacent angular degree.

4). Chord length*pi/4*sqrt 2 = minimal degree-distance possible.

 

There appears to be no compromise with these figures. I consider them a major step forward navigationally speaking as regards our space programs.

[Rincewind]

1). Diameter/180-degrees = 0.05 as leg of right-angle triangle;

What is it the diameter of, why are you dividing it by 180°, and what is the right angled triangle that you have calculated one side of?