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About tangent function

  1. Oct 13, 2013 #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!
     
  2. jcsd
  3. Oct 13, 2013 #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##
     
  4. Oct 13, 2013 #3

    joshmccraney

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    PF for the win!! thanks simon
     
  5. Oct 13, 2013 #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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