sum of sine functions

Discussion in 'Math' started by suzuki, May 7, 2012.

  1. suzuki

    Thread Starter Member

    Aug 10, 2011
    119
    0
    Hi,

    I have an equation that looks like

    \sin(x_1\beta)+sin(x_2\beta)+sin(x_3\beta)+...+sin(x8\beta) = Y

    If I know Y (its constant), is there a easy way to solve for \beta?

    I know for small values of \sin(\omega), that it can be approximated by \omega, so I did this for this first 2 terms where the value was < 0.3 rads. However, for the other terms, I don't think the small signal approximation applies anymore, and I can't think of a way to isolate \beta since x_3, x_4 etc are different.

    tia
     
  2. WBahn

    Moderator

    Mar 31, 2012
    17,716
    4,788
    I assume your last term is x_8\beta and not x8\beta.

    Do the various x_i values have any relationship to each other, or are they simply 8 arbitrary values?
     
  3. russ_hensel

    Well-Known Member

    Jan 11, 2009
    818
    47
    Constant with respect to what? If it is time then any values will do as time does not appear in any term on the left. If x sub n is the variable then no value will solve the equation unless another x is dependent on it, and you said they were all independent.
     
  4. suzuki

    Thread Starter Member

    Aug 10, 2011
    119
    0
    x_1 to x_8 are related by

    x_1 = \sin(\frac{\pi}{32})
    x_2 = \sin(\frac{3\pi}{32})
    x_3 = \sin(\frac{5\pi}{32})
    ...
    x_8 = \sin(\frac{15\pi}{32}

    The Y is a function of \beta.
     
  5. suzuki

    Thread Starter Member

    Aug 10, 2011
    119
    0
    x_1 to x_8 are related by

    x_1 = \sin(\frac{\pi}{32})
    x_2 = \sin(\frac{3\pi}{32})
    x_3 = \sin(\frac{5\pi}{32})
    ...
    x_8 = \sin(\frac{15\pi}{32})

    The Y is a function of \beta, but I know the result of Y(\beta).
     
  6. WBahn

    Moderator

    Mar 31, 2012
    17,716
    4,788
    The fact that all of the x_i values are specific constants makes the problem doable, whereas had they been variables (which 'x' usually is), there would have been no solution except for trivial cases, like Y=0.

    Having said that, I don't see much chance of obtaining an algebraic solution and, in fact, I would be surprised if ß is a function of Y at all (instead, I suspect it is a multivalued relation).

    So, what you have is the following:

    <br />
Y = \sum_{i=0}^{7} \sin \left( sin((2i+1)\frac{\pi}{32} ) \beta \right)<br />

    You could play some games with trig identities to see if you can break out sin(ß) and cos(ß), but I don't think you'll be able to do a whole lot with it.

    If you plot Y(ß), which is easily done using Excel or something similar, you will see that there is a LOT of seemingly random structure to it (of course, it is not random at all, but it makes divining any simply inverse relationship very difficult).
     
    suzuki likes this.
Loading...