El gráfico de la función cruce el eje X con f = 0
o sea hay que resolver la ecuación:
$$3 \sin{\left(x \right)} + 4 \cos{\left(2 x \right)} = 0$$
Resolvermos esta ecuaciónPuntos de cruce con el eje X:
Solución analítica$$x_{1} = - i \log{\left(- \frac{\sqrt{2} \sqrt{55 - 3 \sqrt{137}}}{16} + \frac{i \left(3 + \sqrt{137}\right)}{16} \right)}$$
$$x_{2} = - i \log{\left(\frac{\sqrt{2} \sqrt{55 - 3 \sqrt{137}}}{16} + \frac{i \left(3 + \sqrt{137}\right)}{16} \right)}$$
$$x_{3} = - i \log{\left(- \frac{\sqrt{2} \sqrt{3 \sqrt{137} + 55}}{16} + \frac{i \left(3 - \sqrt{137}\right)}{16} \right)}$$
$$x_{4} = - i \log{\left(\frac{\sqrt{2} \sqrt{3 \sqrt{137} + 55}}{16} + \frac{i \left(3 - \sqrt{137}\right)}{16} \right)}$$
Solución numérica$$x_{1} = 82.8470564882148$$
$$x_{2} = 55.973418750655$$
$$x_{3} = 99.9557159009121$$
$$x_{4} = -92.2718344489842$$
$$x_{5} = 26.2983887235986$$
$$x_{6} = -69.6902873929368$$
$$x_{7} = 47.6991388178082$$
$$x_{8} = -46.5486407898856$$
$$x_{9} = 52.2414276161463$$
$$x_{10} = -98.5550197561638$$
$$x_{11} = -55.3830202697361$$
$$x_{12} = -31.9911755498593$$
$$x_{13} = -90.5309379401427$$
$$x_{14} = -23.9670937338381$$
$$x_{15} = 33.3918716946075$$
$$x_{16} = 24.557492214757$$
$$x_{17} = 11.9911216003979$$
$$x_{18} = 20.015203416419$$
$$x_{19} = 18.2743069075774$$
$$x_{20} = -35.7231666843679$$
$$x_{21} = -61.6662055769156$$
$$x_{22} = 96.2237247664034$$
$$x_{23} = -33.9822701755264$$
$$x_{24} = 10.0000269747307$$
$$x_{25} = 93.6725305937325$$
$$x_{26} = 89.9405394592238$$
$$x_{27} = 30.8406775219366$$
$$x_{28} = 3.71684166755111$$
$$x_{29} = 97.9646212752449$$
$$x_{30} = -11.400723119479$$
$$x_{31} = 39.6750570017871$$
$$x_{32} = 60.2655094321674$$
$$x_{33} = 68.5397893650141$$
$$x_{34} = 63.9975005666761$$
$$x_{35} = -75.9734727001164$$
$$x_{36} = 74.8229746721937$$
$$x_{37} = -48.2895372987271$$
$$x_{38} = -77.9645673257835$$
$$x_{39} = -79.705463834625$$
$$x_{40} = -38.2743608570388$$
$$x_{41} = -65.3981967114243$$
$$x_{42} = 66.548694739347$$
$$x_{43} = 70.2806858738557$$
$$x_{44} = -17.6839084266585$$
$$x_{45} = -29.4399813771884$$
$$x_{46} = -10.5904254556496$$
$$x_{47} = -27.6990848683468$$
$$x_{48} = -2.56634363962847$$
$$x_{49} = -84.2477526329631$$
$$x_{50} = 16.2832122819103$$
$$x_{51} = -99.3653174199932$$
$$x_{52} = 45.9582423089667$$
$$x_{53} = -4.30724014847001$$
$$x_{54} = -57.1239167785776$$
$$x_{55} = 81.1061599793733$$
$$x_{56} = -42.0063519915475$$
$$x_{57} = -54.5727226059067$$
$$x_{58} = -25.7079902426797$$
$$x_{59} = 62.2566040578345$$
$$x_{60} = -19.4248049355001$$
$$x_{61} = -63.4071020857572$$
$$x_{62} = 91.6814359680653$$
$$x_{63} = 85.3982506608857$$
$$x_{64} = 22.5663975890899$$
$$x_{65} = 41.4159535106286$$
$$x_{66} = 53.9823241249878$$
$$x_{67} = -74.2325761912748$$
$$x_{68} = 8.25913046588916$$
$$x_{69} = 77.3741688448646$$
$$x_{70} = 83.6573541520442$$
$$x_{71} = 76.5638711810353$$
$$x_{72} = -40.265455482706$$
$$x_{73} = 1.97594515870957$$
$$x_{74} = -21.4158995611672$$
$$x_{75} = -85.9886491418046$$
$$x_{76} = 38.8647593379577$$
$$x_{77} = -71.6813820186039$$
$$x_{78} = 57.7143152594965$$
$$x_{79} = -13.1416196283205$$
$$x_{80} = -82.2566580072959$$
$$x_{81} = -67.9493908840952$$
$$x_{82} = 49.6902334434754$$
$$x_{83} = 32.5815740307781$$
$$x_{84} = 5.70793629321827$$