How can I solve this trigonometric/lineal equation?

Question

$$9x-\dfrac{8}{3} \sin(3x)=s$$

I need to clear the $x$ in a closed expresion.

Este tipo de ecuaciones no se ve en ningún lado.

Answer 1

Hint

Because the lhs is an odd function, you can just considder the case where $s >0$. Using the bounds of the sine function the solution is such that $$\frac{3 s-8}{27}~\leq~ x~ \leq ~\frac{3 s+8}{27}$$ Compute the values of function $$f(x)=9x-\dfrac{8}{3} \sin(3x)-s$$ Compute the equation of the secant to have as an estimate $$x_0=\frac s 9+\frac 8{27}\times\frac{\cos \left(\frac{8}{9}\right) \sin \left(\frac{s}{3}\right)}{1-\sin \left(\frac{8}{9}\right) \cos \left(\frac{s}{3}\right)}$$

Trying for $s=123.456$, $x_0=13.6844$ while the solution, given by Newton method, is $x=13.6686$. Now, polish the root.

You could even be better using a third point $x=\frac s 9$. Using the three points, compute the parabola and select the solution which is in the range. For the worked case, this gives $x_0=13.6657$.