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sinΘ ≈Θ

 Aug 9, 2014

Best Answer 

 #3
avatar+109519 
+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

   $$The circumference of a circle is $2\pi r$\\
If the radius is 1 unit then the circumference is $2\pi$\\
So if the angle is measured in radians then the arc length will be equal to the angle that it is subtended from. This is for a unit circle.\\\\$$

 

 Aug 9, 2014
 #1
avatar+4473 
0

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 Aug 9, 2014
 #2
avatar+30085 
+5

The following images might help:

theta

theta2

So sin θ ≈ θ only when θ is small

 Aug 9, 2014
 #3
avatar+109519 
+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
avatar+111325 
+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

 

  

 Aug 9, 2014

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