How to solve this expresion for the Beta factor? I understand this one has no closed form solution $$\sin(\beta-\theta)+\sin(\theta)e^{\beta/\omega\tau}=0$$ So besides to write it down in a software for numerical methods or symbolical algebra, how could it be at least aproximated the $\beta$ value?

I'll presume $\theta$ and $\omega \tau$ are real constants with $\omega \tau > 0$ and $\sin(\theta) \
e 0$. Note that the second term goes to $0$ as $\beta \to -\infty$, while the first oscillates between $-1$ and $1$ with period $2\pi$. So as long as $n$ is sufficiently large, there will be a solution near $\beta = \theta - n \pi$. If we write $\beta = \theta - n\pi + x$, the equation becomes $$ (-1)^n \sin(x) + \sin(\theta) e^{(\theta - n\pi + x)/(\omega \tau)} = 0$$ and a good approximation will be $$ x \approx \frac{-\omega \tau \sin(\theta)}{(-1)^n \omega \tau \exp((\pi n - \theta)/(\omega \tau))+\sin(\theta)}$$