Arc Length to Radius Ratio: Why Equal?

[wajed]

(F) Thanks in advance (F)

 

[tiny-tim]

(F) Thanks in advance (F)

You're welcome!

Because that's the way the radian is defined …

one radian is the angle whose arc-length, S, equals the radius, R.

Since arc-length is proportional to angle, 2 radians has arc-length twice that: S = 2R, and so on.

 

[wajed]

but if I want to go further and ask such question: "how do you know that this equation holds and is 100% correct?", what field of mathematics should I study to be able to answer that question?

 

[quasar987]

None, because there is no further dept to the answer. At some point, some guy said "Wouldn't it be cool to define a new quantity, the "angle" as the ratio S/R?"

And that's all there is to it.

There is no a priori relation between what we call an "angle" and phenomena taking place in the physical world. That is to say, there is nothing to "test" the formula S/R against. It is not a theory that can to proved right or wrong; it is simply an abstraction of our mind.

 

[wajed]

None, because there is no further dept to the answer. At some point, some guy said "Wouldn't it be cool to define a new quantity, the "angle" as the ratio S/R?"

And that's all there is to it.

There is no a priori relation between what we call an "angle" and phenomena taking place in the physical world. That is to say, there is nothing to "test" the formula S/R against. It is not a theory that can to proved right or wrong; it is simply an abstraction of our mind.

So, I can simply say I want to define a new angle measurement and call it "wajed" and define it as the length of the radius over the distance between the two ends of the rays that form the angle, right?

 

[quasar987]

You mean "wajed"= R/S ?

Sure.

 

[wajed]

You mean "wajed"= R/S ?

Sure.

Yes, No (sorry) I mean "wajed"= R/S, where S is the distance between the blah blah blah (not the arc-length), but this still holds, so Thank you anyway :D
(Learned something new, today)

 

[quasar987]

I'm glad!

We go through school being taught that 1+1=2 and so on as if it these were irrevocable Grand Facts of the Universe, and so the moment one realizes that math is actually completely arbitrary and the work of man like you and me, a happy "ah-Ah!" moment is bound to result! That or utter insanity.

 

[tiny-tim]

are wajeds well-behaved?

but if I want to go further and ask such question: "how do you know that this equation holds and is 100% correct?", what field of mathematics should I study to be able to answer that question?

So, I can simply say I want to define a new angle measurement and call it "wajed" and define it as the length of the radius over the distance between the two ends of the rays that form the angle, right?

Hi wajed!

We can define anything we like, but some definitions are more useful than others.

We would prefer the "wajed" to be well-behaved (like its inventor? ), so we would want the sum of the wajeds of two angles to equal the wajed of the combined angle.

That works for the radian because rotations in one dimension form a one-parameter group (every rotation is a scalar multiple of every other rotation), and that parameter happens to be S/R.

 

[HallsofIvy]

The most important point to be made here is that relationship between angle and arc subtended is linear: one is simply a multiple of the other. That is a simplified version of what tiny-tim just said, "rotations in one dimension form a one-parameter group".

We know that a circle of radius R has circumference $2\pi R$ and that corresponds to a an angle, in radians, of $2\pi$. If the length of the arc subtended by angle $\theta$ is S, then we can set up the proportion
$$\frac{S}{\theta}= \frac{2\pi R`}{2\pi}$$
The "$2\pi$"s cancel and $S= R\theta$.

We can do the same thing with degrees: again the circumference of a circle is $2\pi$ but now the entire circle corresponds to 360 degrees so our proprotion is
$$\frac{S}{\theta}= \frac{2\pi R}{360}$$
and
$$S= \frac{2\pi}{360} R\theta$$
when $\theta$ is measured in degrees.