1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Find a,b,c in sin^5(x) = asin(x) + bsin(3x) + csin(5x)

  1. Nov 1, 2009 #1
    1. The problem statement, all variables and given/known data
    Find A,B,C in sin^5(x) = Asin(x) + Bsin(3x) + Csin(5x).

    3. The attempt at a solution

    I get by Euler the double angle identities for [itex]sin(3x) and sin(5x)[/itex].
    They are

    [tex] sin(3x) = s(2x) c(x) + c(2x) s(x) [/tex]
    [tex] sin(5x) = s(3x) c(2x) + c(3x) s(2x) [/tex]


    I have the following expression now

    [tex]

    sin^5(x) = 1 - cos^5(x) - 5c^4(x)s(x) - 10c^3(x)s^2(x) - 10 c^2(x)s^3(x) - 5 c(x)s^4(x) [/tex]

    where the trigonometric terms are [itex] (1/2) s(2x) c^3(x), (1/4) (s(2x))^2 c(x), (1/4) (s2x)^2 s(x), (1/2) s(2x) s^3(x) [/itex], respectively.

    I get the common term [itex] [(1/2)s(2x) = c(x)s(x) [/itex] by the double angle identity of sine.

    So you have [itex] (-5/2) s(2x) (c^3(x) + c(x) + s(x) + s^(x) [/itex].

    By comparing the terms to the given equation, we get

    [tex] Bsin(3x) = B (s(2x) c(x) + c(2x) s(x) ) [/tex]
    [tex] Csin(5x) = C ( s(3x) c(2x) + c(3x) s(2x) ) [/tex]

    so we have

    [tex] B c(x) + C c(3x) = (-5/2) ( c^3(x) + c(x) + s(x) + s^3(x) ) [/tex]
    which implies that [itex] C = - \frac {-5} {2} [/itex]

    The other term is in the form

    [tex] B c(x) = (-5/2) ( c(x) + s(x) + s^3(x) ) [/tex]
    where where [itex] s(x) + s^3(x) = s(x) (1 + s^(x) = 2s(x) - c^2(x) s(x) [/itex] by Pythagoras.

    However, I do not see directly how to get the term [itex] c(x) [/tex] to the RHS of the equation.

    ---------------------------------------


    My first attempt seems to be useless, since the answer may be found by Fourier series too.
    However, I have little experience of them and cannot see to how use them here.


    How can you find A, B and C by Fourier series or by other method?
     
  2. jcsd
  3. Nov 10, 2009 #2

    tiny-tim

    User Avatar
    Science Advisor
    Homework Helper

    Hi soopo! :smile:

    How about expanding (eix - e-ix)5 ? :wink:
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Find a,b,c in sin^5(x) = asin(x) + bsin(3x) + csin(5x)
  1. Finding ∫x sin^3x dx (Replies: 4)

Loading...