Can this equation be solved? x+sin⁡(x)=11π48

${\displaystyle f\left(x\right)=x+\sin x}$ is monotonically increasing, and without bounds, so there will be exactly one solution.
${\displaystyle y=\frac{11\pi }{48}}$ is (relatively) small, so a first guess is obtained by setting ${\displaystyle \sin x\approx x}$ to get ${\displaystyle x\approx \frac{11\pi }{96}}$. An improved value is obtained by including the next term of the sine series,
${\displaystyle 2x-\frac{1}{6}{x}^3\approx y\Rightarrow x\approx \frac{y}{2}+\frac{y^3}{96}}$
This gives the numerical value x=0.3638613210103829 which is already close to the (more) exact value 0.363965532996313