#3**+8 **

I'd like to take Alan's explanation and expand upon it a little.

**NOTE: THIS IS ONLY TRUE AS THE ANGLE TENDS TO 0**

$$The circumference of a circle is $2\pi r$\\

If the radius is 1 unit then the circumference is $2\pi$\\

So if the angle is measured in radians then the arc length will be equal to the angle that it is subtended from. This is for a unit circle.\\\\$$

Melody
Aug 9, 2014

#3**+8 **

Best Answer

I'd like to take Alan's explanation and expand upon it a little.

**NOTE: THIS IS ONLY TRUE AS THE ANGLE TENDS TO 0**

$$The circumference of a circle is $2\pi r$\\

If the radius is 1 unit then the circumference is $2\pi$\\

So if the angle is measured in radians then the arc length will be equal to the angle that it is subtended from. This is for a unit circle.\\\\$$

Melody
Aug 9, 2014

#4**+5 **

Here's the proof of this using the "Squeeze Theorem".....it's somewhat complicated, but if you have some patience, I think you will appreciate the geometry involved. The proof is the first one on the page.

http://tutorial.math.lamar.edu/Classes/CalcI/ProofTrigDeriv.aspx

Thanks to Paul's Online Math Notes for this one.....(I certainly couldn't have derived it!!!)

BTW....."Paul" is Dr. Paul Dawkins...he's a professor at Lamar University in Texas

CPhill
Aug 9, 2014