Deriving the Formula for S = theta * radius

[Miike012]

Homework Statement

S = theta * radius

I don't understand how they came up with this formula... can some one show me the proof how they derived this formula or can someone send me a link and I will just read it. Thank you.

Homework Equations

The Attempt at a Solution

 

[praharmitra]

Well, this formula is the DEFINITION of the unit 'radians'. The theta that you mentioned in this formula is measured in radians.

The measurement of an angle in 'radians' is defined as the length of the arc that it subtends divided by the radius.

 

[Kevin_Axion]

Just by definition, plug in theta with respect to it's radians and compare it to the cirumference formula (there is a a lot of similiarty)
$$C = 2\pi r$$
Now if you use the formula
$$S_{arc} = \theta_{rad} r$$
and plug in a degree $$\theta_{rad}$$ with respect to radians you end up with some result
$$N\pi r$$
where N is a rational number.

 

[Miike012]

Ok... but what is the mathamatical proof.

 

[praharmitra]

Ok... but what is the mathamatical proof.

There is no mathematical proof! That's how the quantities have been defined!

 

[Redbelly98]

Perhaps what is really wanted is a proof that arc length is proportional to the product of the angle and the radius.

 

[Miike012]

Perhaps what is really wanted is a proof that arc length is proportional to the product of the angle and the radius.

That would be nice..

 

[iRaid]

Here's the derivation:
http://en.wikipedia.org/wiki/Arc_length
Scroll down about half way..

 

[Redbelly98]

Here's the derivation:
http://en.wikipedia.org/wiki/Arc_length
Scroll down about half way..

I see no derivation of the s=rθ formula, or that s is proportional to r and θ.

 

[zketrouble]

Ok... but what is the mathamatical proof.

You can't have a mathematical proof for such a question. It's all part of the definition of the radian. One radian is the angle necessary within a circle of radius 1 to produce an arc that is also equal to 1, more formally 1rad = 180/pi. Further, radians are measured in terms of the length of the arc divided by the radius of the arc.

To ask for a mathematical "proof" is like proving that 1+1=2 instead of =zero. Why is it that that little + symbol doesn't decrease 1 from 1? You simply can't prove it, because humans have defined the symbols +, -, etc. at our discretion. We have likewise defined the definition of units of measure such as the radian. With other systems of measure, you can't prove that there are 24 hours in a day mathematically. Us humans have simply chosen to divide the day into 24 sections, and we gave these sections the name "hour."

 

[iRaid]

http://www.themathpage.com/atrig/arc-length.htm

There's some more stuff about this..

 

[Miike012]

thanks for all your help everyone