Is there a relationship between the ratios of arc to chord between the angles ?

[Ahatmose]

Hi I have been trying to determine if there is any rhyme or reason to the ratios or arc to chords for various angles. For example if I knew the ratio for 180 degrees between the arc and the chord (which we do know as Pi/2) is there any formula devised that would give me the ratio for say 121 degrees ?

180 = 1.5707963268

121 = 1.2132099582

and that would work for all other angles.

Thanks in advance

db

 

[Dr.Peterson]

Hi I have been trying to determine if there is any rhyme or reason to the ratios o[f] arc to chords for various angles. For example if I knew the ratio for 180 degrees between the arc and the chord (which we do know as Pi/2) is there any formula devised that would give me the ratio for say 121 degrees ?

180 = 1.5707963268

121 = 1.2132099582

and that would work for all other angles.

Thanks in advance

db

There's a reason, but no rhyme!

That is, arcs and chords don't have any sort of proportionality; rather, the relationship has to be described in terms of a special function that ancient mathematicians and astronomers had to calculate in tables.

In modern terms, the relationship is [math]c = 2r\sin\left(\frac{\theta}{2}\right)[/math] where r is the radius, [imath]\theta[/imath] is the central angle of the arc, and c is the length of the chord. Equivalently, letting [imath]s=r\theta[/imath] be the length of the arc, [math]c = 2r\sin\left(\frac{s}{2r}\right)[/math]

 

[Ahatmose]

Just a note the ratios go from 1.00000000 for 0 degrres to Pi/2 or 1.5707963267948966192313216916398 for 180 degrees

 

[pka]

Hi I have been trying to determine if there is any rhyme or reason to the ratios or arc to chords for various angles. For example if I knew the ratio for 180 degrees between the arc and the chord (which we do know as Pi/2) is there any formula devised that would give me the ratio for say 121 degrees ?
180 = 1.5707963268

121 = 1.2132099582

and that would work for all other angles.

Thanks in advance

db

Have look at this LINK

 

[Ahatmose]

There's a reason, but no rhyme!

That is, arcs and chords don't have any sort of proportionality; rather, the relationship has to be described in terms of a special function that ancient mathematicians and astronomers had to calculate in tables.

In modern terms, the relationship is [math]c = 2r\sin\left(\frac{\theta}{2}\right)[/math] where r is the radius, [imath]\theta[/imath] is the central angle of the arc, and c is the length of the chord. Equivalently, letting [imath]s=r\theta[/imath] be the length of the arc, [math]c = 2r\sin\left(\frac{s}{2r}\right)[/math]

Hi yes I have found that but is there any formula that allows you to calcualte the ratio between say 180 and 121 and then use this same forula and it would work on say 47 degres ?

 

[Ahatmose]

I am obviously not explaining myself very well. We know how to get the arc and we know how to calculate the chord for any given angle to get a ratio of we will call it 1 to x

If we do again for another angle we would get say 1 to y

Is there a mathematical formula that we could use to calculate y from x

 

[Ahatmose]

Here is a link to all the ratios for all angle from 1 to 180. what I am trying to determine is if there is a formula we could use to calculate these ratios that would be constant for all of the angles.

best
db

 

[Dr.Peterson]

Here is a link to all the ratios for all angle from 1 to 180. what I am trying to determine is if there is a formula we could use to calculate these ratios that would be constant for all of the angles.

best
db

Do you know how to find the ratio of two given formulas?

As I understand it, based on the table, you want the ratio s/c for a given angle [imath]\theta[/imath] (though your description sounded like you wanted something else). So divide the formula for s by the formula for c. I gave you both.