Marquis Neal

2022-01-25

Can this equation be solved? $x+\mathrm{sin}\left(x\right)=\frac{11\pi }{48}$

chaloideq1

Expert

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

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