About tangent function

  • #1
joshmccraney
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hey all

can anyone explain why, for small [itex] \alpha[/itex] we may allow [itex] \tan \alpha = \alpha[/itex] at an intuitive, geometrical perspective. i already understand the series explanation and higher order of tangent. im just trying for a picture.

thanks!
 

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  • #2
Simon Bridge
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Because it is a very good approximation.

To see why: what is the slope of the tangent at ##\alpha=0##?


More exactly - look at the definition of a tangent:

The length along the tangent to a circle radius R inside the some angle ##\alpha## is ##t=R\tan\alpha##
The arclength of a circle inside the same angle ##\alpha## is ##s=R\alpha##

When R>>s, then someone standing on the surface thinks the circle is actually flat.
i.e. it looks to be the same distance as the flat tangent measure. So ##t\approx s##
 
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  • #3
joshmccraney
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PF for the win!! thanks simon
 
  • #4
Simon Bridge
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Cool!

By the same token:
sinA = A
cosA = 1

When you realise that the trig functions are the names of lengths defined on a unit circle the whole thing makes a lot more sense.
 

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