1. Limited time only! Sign up for a free 30min personal 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!

Sin^n(theta) as product of sin^1, cos^1 (m*theta)

  1. Nov 27, 2009 #1
    Hello,

    Somewhat urgent question, I would normally try and do this myself but I have a feeling it will take a while, and I sort of need to be working through this pretty quickly, so any help much appreciated. Plus I might end up wasting half a day trying formulas on this.

    I am expanding a function f(x + a sin(t)) as a Taylor series, so I get:

    f(x + a sin (t)) = f(x) + (a sin t)f'(x) + (a sin t)^2/2! *f''(x) + ...

    Now I am trying to decompose this into some sort of Fourier series (not exactly clear on final goal or method yet), so I need this in terms of:

    f(x + a sin (t)) = [ ... ] + [ ... ]*sin(t) + [ ... ]*sin(2t) + ... + [ ... ]*cos(t) + [ ... ]*cos(2t) + ...

    I am wondering if there is any simple way to do this. I have worked out the first one, eg:
    sin^2(t) = 1/2 - 1/2 *cos(t), and was wondering if there are any simple recursion formulas or anything out there to help me find an expression for sin^n(t) in terms of single powers of sines and cosines of multiples of t?

    Thanks for any help,
    Mike
     
  2. jcsd
  3. Nov 27, 2009 #2
    Ok, a little progress has been made:

    I am now working with cos^n (t) as that can be put into only cosine terms. I have also found the formula for this in terms of cos(t), cos(2t), etc. but it is not so helpful for finding what my bracket terms [ ... ] are, since there are many overlaps, as the expansion of f is a series of increasing powers of cos(t) each with a different weighting by a.

    For example, the coefficient of cos(t) would be: [a*f'(x) + (a^3)*f''(x)/2 + ... ]
     
  4. Nov 27, 2009 #3
    First of all note that the powers of trig function can be found from
    [tex]\cos^n(x)=\Re\left(e^{ix}\left(\frac{e^{ix}+e^{-ix}}{2}\right)^{n-1}\right)[/tex]
    and similarly for sine.
    I have to leave now, but I can think about the full problem later.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Sin^n(theta) as product of sin^1, cos^1 (m*theta)
  1. Sin/cos/tan of theta (Replies: 8)

Loading...