# 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
18,085
4,917
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 Distinguished Member

Jan 11, 2009
820
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
18,085
4,917
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:

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

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.