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Simple Sinx Expansion

  1. Jan 10, 2013 #1
    1. The problem statement, all variables and given/known data

    Expand sinx about the point x= pi/4. Hint: Represent the function as sinx= sin(y+pi/4) and assume y to be small



    2. Relevant equations



    3. The attempt at a solution

    I thought the problem was simply asking to expand sinx with the McLauran expansions about the point pi/4 and get something like... (x-pi/4) - ((x-pi/4)^3)/3! + ((x-pi/4)^5)/5!.....so on and so forth.

    But the hint throws me off? What does that mean? Any help?

    Thanks
     
  2. jcsd
  3. Jan 10, 2013 #2

    haruspex

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    The hint is telling you to do precisely what you thought. It's just trying to make life simpler by substituting y for (x-pi/4) everywhere.
     
  4. Jan 10, 2013 #3

    jbunniii

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    Your expansion clearly cannot be correct: if you plug in x = pi/4, you get 0. But sin(pi/4) is not 0.

    What happens if you apply a trig identity to sin(y + pi/4)?
     
  5. Jan 10, 2013 #4

    lurflurf

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    [tex]\sin(x+h) \sim \sum^\infty_{k=0} \frac{h^k}{k!} \sin^{(k)}(x)=\sum^\infty_{k=0} \frac{h^k}{k!} \sin(x+k \, \pi/2)=\sum^\infty_{k=0} \frac{h^k}{k!} (\sin(x)\cos(k \, \pi/2)+\cos(x)\sin(k \, \pi/2))[/tex]

    [tex]\sin^{(k)}(x)[/tex]
    means the kth derivative of sine evaluated at x which we know to be
    [tex]\sin(x+k \, \pi/2)[/tex]
     
    Last edited: Jan 11, 2013
  6. Jan 11, 2013 #5

    jbunniii

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    I'm not sure what good that does. This isn't a Taylor series in x. I guess it's a Taylor series in h, but centered at h=0, which isn't what is asked for.

    I think the hint is intended to lead to the following:
    [tex]\sin(x) = \sin(y + \pi/4) = \sin(y) \cos(\pi/4) + \cos(y) \sin(\pi/4)[/tex]
    And we presumably know the Taylor series for [itex]\sin(y)[/itex] and [itex]\cos(y)[/itex].
     
  7. Jan 11, 2013 #6

    lurflurf

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    ^It does not matter what variables are used.
    In general h is not zero that is an uninteresting case.

    These all mean exactly the same thing.
    [tex]
    \sin(x+h) \sim \sum^\infty_{k=0}
    \frac{h^k}{k!} \sin^{(k)}(x)=\sum^\infty_{k=0}
    \frac{h^k}{k!} \sin(x+k \, \pi/2)=\sum^\infty_{k=0}
    \frac{h^k}{k!} (\sin(x)\cos(k \, \pi/2)+\cos(x)\sin(k \, \pi/2)) \\
    \sin(a+b) \sim \sum^\infty_{n=0}
    \frac{b^n}{n!} \sin^{(n)}(a)=\sum^\infty_{n=0}
    \frac{b^n}{n!} \sin(a+n \, \pi/2)=\sum^\infty_{n=0}
    \frac{b^n}{n!} (\sin(a)\cos(n \, \pi/2)+\cos(a)\sin(n \, \pi/2)) \\
    \sin(\mathrm{rock}+\mathrm{paper}) \sim \sum^\infty_{\mathrm{scissors}=0}
    \frac{\mathrm{paper}^\mathrm{scissors}}{\mathrm{scissors}!} \sin^{(\mathrm{scissors})}(\mathrm{rock})=
    \sum^\infty_{\mathrm{scissors}=0}
    \frac{\mathrm{paper}^\mathrm{scissors}}{\mathrm{scissors}!} \sin(\mathrm{rock}+\mathrm{scissors} \, \pi/2)=\sum^\infty_{\mathrm{scissors}=0}
    \frac{\mathrm{paper}^\mathrm{scissors}}{\mathrm{scissors}!} (\sin(\mathrm{rock})\cos(\mathrm{scissors} \, \pi/2)+\cos(\mathrm{rock})\sin(\mathrm{scissors} \, \pi/2)) [/tex]

    In the given exercise we can take
    x=rock+paper
    y=paper
    pi/4=rock
    k=scissors

    to give
    [tex]
    \sin(x)=\sin(\pi/4+y) \sim \sum^\infty_{k=0}
    \frac{y^k}{k!} \sin^{(k)}(\pi/4)=\sum^\infty_{k=0}
    \frac{y^k}{y!} \sin(\pi/4+k \, \pi/2)=\sum^\infty_{k=0}
    \frac{y^k}{k!} (\sin(\pi/4)\cos(k \, \pi/2)+\cos(\pi/4)\sin(k \, \pi/2))[/tex]
     
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