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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.

    Why?
     
  2. jcsd
  3. Oct 11, 2013 #2

    Simon Bridge

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

       89.900
       89.990
       89.999
       90.000
       90.000
       90.000
       90.000
       90.000
       90.000
       90.000
       90.000
       90.000
       90.000
       90.000
       90.000

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

       5.7296e+02
       5.7296e+03
       5.7296e+04
       5.7296e+05
       5.7296e+06
       5.7296e+07
       5.7296e+08
       5.7296e+09
       5.7295e+10
       5.7296e+11
       5.7278e+12
       5.7535e+13
       5.4419e+14
       3.5301e+15
       1.6332e+16
    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 =

       5.72957213354303e+02
       5.72957789312165e+03
       5.72957795072129e+04
       5.72957795104345e+05
       5.72957794122192e+06
       5.72957798144568e+07
       5.72957787343207e+08
       5.72957898008453e+09
       5.72953173262481e+10
       5.72956950454798e+11
       5.72776101541460e+12
       5.75350505624601e+13
       5.44191874731457e+14
       3.53013952228678e+15
       1.63317787283838e+16
    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

    lurflurf

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    We have the identity

    tan(90-x)=1/tan(x)

    it is better to work in radians so that

    tan(pi/2-x)=1/tan(x)

    when x is small we have (~ mean approximately)
    tan(x)~x
    so
    tan(pi/2-x)=1/tan(x)~1/x

    back in degrees

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

    as you observed
     
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