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Derivation of a formula with trigonometric functions

  1. Mar 2, 2012 #1
    Hi everyone,

    1. The problem statement, all variables and given/known data

    My problem is just to derive a simple formula, which is

    http://www.texify.com/img/%5Cnormalsize%5C%21%28-1%29%5E%7Br%28r%2B1%29/2%7D%20%3D%5Csqrt%7B2%7D%20%5Cmbox%7Bcos%7D%20%5Cfrac%7B%5Cpi%7D%7B4%7D%282r%2B1%29.gif [Broken]

    Here r is a positive integer.
    3. The attempt at a solution

    I verified this formula by inserting r=4k ~ 4k+3 (k=0,1,2....), but I still have no idea how to derive it from the left hand side of the equation.


    Could anyone please help me out? Any help is appreciated.
     
    Last edited by a moderator: May 5, 2017
  2. jcsd
  3. Mar 2, 2012 #2

    HallsofIvy

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    I don't know what you mean by "r=4k ~ 4k+3" but the equation is clearly NOT true for n= 0, 1, 2, etc.
     
  4. Mar 2, 2012 #3
    http://www.wolframalpha.com/input/?i=%28-1%29^%28r%28r%2B1%29%2F2%29

    go to derivate and click show steps.
     
  5. Mar 2, 2012 #4
    Hello HallsofIvy,

    Thank you very much for pointing out my mistake. I typed the wrong formula. I have corrected it. Would you please check it out again?

    Thank you again.

     
  6. Mar 2, 2012 #5
    Hello the_epi,

    Thanks for your help. But I checked the website and check the Derivative part, I still do not understand how the Derivative related to the formula above. Could you please explain?

    Thanks a lot.


     
  7. Mar 2, 2012 #6

    HallsofIvy

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    For r a positive integer, 2r+ 1 is odd so, dropping multiples of [itex]2\pi[/itex], [itex]cos(\pi/4(2r+1)[/itex] is [itex]cos(\pi/4)= \sqrt{2}/2[/itex], [itex]cos(3\pi/4)= -\sqrt{2}/2[/itex], [itex]cos(5\pi/4)= \sqrt{2}/2[/itex], and [itex]cos(7\pi/4)= -\sqrt{2}/2[/itex]. So what does the left side give? I would look at r= 4n, 4n+1, 4n+2, and 4n+ 3 and compare to those values.
     
  8. Mar 2, 2012 #7
    Thank you very much HallsofIvy. I did the same thing to check this equation.

    But I do not know how to DERIVE it. Do you have any ideas? Thanks!!


     
  9. Mar 2, 2012 #8

    SammyS

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    This formula holds only for r being an integer. Right ?
     
  10. Mar 2, 2012 #9
    Yes!

     
  11. Mar 3, 2012 #10

    SammyS

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    Then [itex]\displaystyle\cos\left(\frac{\pi}{4}(2r+1)\right)=\cos\left(\frac{\pi}{2}r+\frac{\pi}{4}\right)\,.[/itex]
    Use the angle addition identity for the cosine.
     
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