Graph of the function intersects the axis X at f = 0
so we need to solve the equation:
$$2 \sin{\left(3 x \right)} + 1 = 0$$
Solve this equationThe points of intersection with the axis X:
Analytical solution$$x_{1} = - \frac{\pi}{18}$$
$$x_{2} = \frac{7 \pi}{18}$$
Numerical solution$$x_{1} = 70.3367688553715$$
$$x_{2} = -44.1568300754565$$
$$x_{3} = -61.6101225953998$$
$$x_{4} = -35.7792496658838$$
$$x_{5} = 646.2954220135$$
$$x_{6} = 91.9788515801012$$
$$x_{7} = 98.2620368872808$$
$$x_{8} = 68.2423737529783$$
$$x_{9} = 96.1676417848876$$
$$x_{10} = 85.6956662729216$$
$$x_{11} = 61.9591884457987$$
$$x_{12} = 72.4311639577647$$
$$x_{13} = -31.5904594610974$$
$$x_{14} = -19.7222205475359$$
$$x_{15} = 45.9021593274509$$
$$x_{16} = -28.0998009571087$$
$$x_{17} = -39.9680398706701$$
$$x_{18} = -26.0054058547155$$
$$x_{19} = -67.8933079025794$$
$$x_{20} = 41.7133691226645$$
$$x_{21} = 47.9965544298441$$
$$x_{22} = -54.6288055874225$$
$$x_{23} = 12.3918376891597$$
$$x_{24} = 39.6189740202713$$
$$x_{25} = 74.5255590601579$$
$$x_{26} = 54.2797397370236$$
$$x_{27} = 8.20304748437335$$
$$x_{28} = 56.3741348394168$$
$$x_{29} = -4.36332312998582$$
$$x_{30} = -57.4213323906134$$
$$x_{31} = 76.6199541625511$$
$$x_{32} = -86.0447321233204$$
$$x_{33} = -92.3279174305$$
$$x_{34} = -46.2512251778497$$
$$x_{35} = 64.0535835481919$$
$$x_{36} = -23.9110107523223$$
$$x_{37} = 1.91986217719376$$
$$x_{38} = 87.7900613753148$$
$$x_{39} = -98.6111027376796$$
$$x_{40} = -79.7615468161409$$
$$x_{41} = 22.165681500328$$
$$x_{42} = -37.873644768277$$
$$x_{43} = -42.0624349730633$$
$$x_{44} = -51.1381470834339$$
$$x_{45} = 52.1853446346305$$
$$x_{46} = -21.8166156499291$$
$$x_{47} = 43.8077642250577$$
$$x_{48} = -11.3446401379631$$
$$x_{49} = -16.929693744345$$
$$x_{50} = 10.2974425867665$$
$$x_{51} = -77.6671517137477$$
$$x_{52} = 20.0712863979348$$
$$x_{53} = -0.174532925199433$$
$$x_{54} = 50.0909495322373$$
$$x_{55} = 94.0732466824944$$
$$x_{56} = -48.3456202802429$$
$$x_{57} = 15.8824961931484$$
$$x_{58} = -72.0820981073658$$
$$x_{59} = -63.704517697793$$
$$x_{60} = -69.9877030049726$$
$$x_{61} = 24.2600766027212$$
$$x_{62} = -15.5334303427495$$
$$x_{63} = -65.7989128001862$$
$$x_{64} = -7.15584993317675$$
$$x_{65} = 89.884456477708$$
$$x_{66} = -81.8559419185341$$
$$x_{67} = 28.4488668075076$$
$$x_{68} = 17.9768912955416$$
$$x_{69} = 26.3544717051144$$
$$x_{70} = 4.01425727958696$$
$$x_{71} = 100.356431989674$$
$$x_{72} = 6.10865238198015$$
$$x_{73} = 33.3357887130917$$
$$x_{74} = -90.2335223281068$$
$$x_{75} = -75.5727566113545$$
$$x_{76} = 30.5432619099008$$
$$x_{77} = 32.637657012294$$
$$x_{78} = -83.9503370209273$$
$$x_{79} = 77.3180858633488$$
$$x_{80} = -88.1391272257137$$
$$x_{81} = -13.4390352403563$$
$$x_{82} = -2.26892802759263$$
$$x_{83} = -74.176493209759$$
$$x_{84} = 66.1479786505851$$
$$x_{85} = 83.6012711705284$$
$$x_{86} = -33.6848545634906$$
$$x_{87} = 59.8647933434055$$
$$x_{88} = -59.5157274930066$$
$$x_{89} = -17.6278254451427$$