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Tangents and 18/pi

  1. Oct 11, 2013 #1
    In degrees, look at what the result is when you take tan(89), tan(89.9), tan(89.99) etc.

    The value as the number of 9's gets larger converges to 18/pi, with a shifted decimal point.

  2. jcsd
  3. Oct 11, 2013 #2

    Simon Bridge

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    Code (Text):
    octave:10> t
    t =


    octave:11> tan(t*pi/180)
    ans =

    18/pi is 5.72957795130823

    To 4dp that 5.6296 ... but notice that the calculation departs from the pattern at the e+10 stage?
    (That is when theta is 89.999999999 ... 9 "9"'s in a row.)

    Lets try it without rounding up:
    Code (Text):
    octave:12> format long

    octave:14> tan(t*pi/180)
    ans =

    What you are seeing is that the tangent function behave a bit like an exponential function for values close to 89.999999deg ... just like it behaves like y=x for values close to 0. What you are not seeing is any convergence to a pattern - far from it, continuing the calculation shows divergence as the argument approaches 90deg.
  4. Oct 12, 2013 #3


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    We have the identity


    it is better to work in radians so that


    when x is small we have (~ mean approximately)

    back in degrees

    tan(90-x)=1/tan(x)~180/(pi x)

    as you observed
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