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Stuck on Laplace Transform of Odd Trig Function

  1. Mar 30, 2013 #1
    Hey guys!

    I'm stuck on a Laplace transform. Following is the problematic function:

    [cos(t)]^3

    Seems simple, but I'm having issues doing the Laplace transform on odd trigonometric functions. When I use the half-angle formula, I get this, which I can't seem to solve:

    1/2cos(t) + 1/2cos(t)*cos(2t)

    How do I get this into a form on which I can perform a Laplace transform?
     
  2. jcsd
  3. Mar 30, 2013 #2

    SteamKing

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    You can use the cosine addition formula with A = 2t and B = t to obtain an identity which involves (cos t)^3
     
  4. Mar 30, 2013 #3

    HallsofIvy

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    [tex]cos(3t)= cos(2t)cos(t)- sin(2t)sin(t)= (cos^2(t)- sin^2(t))cos(t)- (2sin(t)cos(t))sin(t)[/tex]
    [tex]= cos^3(t)- sin^2(t)cos(t)- 2sin^2(t)cos(t)= cos^3(t)- 3(1- cos^2(t))cos(t)[/tex]
    [tex]= 4cos^3(t)- 3cos(t)[/tex]

    So [itex]cos^3(t)= cos(3t)/4+ 3cos(t)/4[/itex].
     
  5. Apr 2, 2013 #4
    Thanks both of you for your help.

    I eventually solved it by using a half-angle formula on [cos(t)]^2, and then using a trig product formula on the resulting expression. I got the same answer as HallsofIvy.

    Thanks again, guys!
     
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