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

 

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?

 

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.

 

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?

\(x_1\) to \(x_8\) are related by

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

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.

The Y is a function of \(\beta\).

 

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?

\(x_1\) to \(x_8\) are related by

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

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.

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

 

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).