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Simple Question: Converting arctan into pi

  1. Dec 11, 2007 #1
    Can anyone express arctan(11/2) in terms of pi. Is there an easy way to do this?
     
  2. jcsd
  3. Dec 12, 2007 #2

    Gib Z

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    Homework Helper

    I do not remember the proof, however I remember that arctan (1/2) is not expressible as a rational multiple of pi.

    Now, looking at 11 squared plus 2 squared, we get 125, which is a nice cube. Hence lets find z where [tex]z^3 = 2 + 11i[/tex].

    Expressing in Polar form, we get;
    [tex]z^3 = \sqrt{125}\exp ( i \arctan (11/2)[/tex].

    Since 125 is a nice cube, one value of z is [tex]\sqrt{5} \exp ( \frac{ i \arctan (11/2)}{3})[/tex]

    Some very hefty trig manipulations and converting back to Cartesian form gives 2+11i, cubing that we can easily verify the calculation. Essentially this implies that arctan (1/2) = 3 arctan (11/2).
     
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