Help with extracting alpha from: -alpha-asin(sin(alpha)(r/l)

[dbarger1225]

I am using the Law of Cosines to extract and angle I need from a shaft that is turning on an air engine by piston oscillation and I am having a brain fart on if there are any identities I am missing that can help me pull '$\alpha$' from the right side of my equation:

acos(($(h-pl)^{2}$-$r^{2}$-$l^{2}$)/(-2*r*l))-$\pi$=-$\alpha$-asin(sin($\alpha$)(r/l))

I would like to solve the entire equation for $\alpha$ and the other variables can be treated like constants.

Thanks for the help or attempts in advance.

 

[Simon Bridge]

well - you didn't close the brackets in the arcsine, so it's hard to tell - but asin(sin(x))=x

 

[dbarger1225]

I fixed the bracket issue, I apologize about that.

I am concerned with the (r/l) term inside of the equation. I understand that asin(sin(x))=x .. but I'm certain that asin(sin(x)(r/l)) --DNE-- (r/l)x

 

[Simon Bridge]

Oh I think I see, it wasn't clear to me if the r/l was multiplied with the alpha or the sine-alpha, or with the arcsine.
This what you mean?

$$\sin^{-1}( \frac{r}{l}\sin\alpha )$$

... afaik you can't extract alpha from this sort of equation - you need to be cleverer in the setup or use an approximation (or use a numerical method).

Equation is of form: x+sin^-1(A.sin(x))=b ... solve for x. argh. (assuming the RHS is all constants)

Note: the arcsin will be undefined for some alpha, depending on the value of rl - you want |sin(α)| < l/r (something...)

Where did you start from?