Graph of the function intersects the axis X at f = 0
so we need to solve the equation:
$$- \sin{\left(x \right)} + 2 \cos{\left(2 x \right)} = 0$$
Solve this equationThe points of intersection with the axis X:
Analytical solution$$x_{1} = - i \log{\left(- \frac{\sqrt{2} \sqrt{15 - \sqrt{33}}}{8} - \frac{i \left(1 + \sqrt{33}\right)}{8} \right)}$$
$$x_{2} = - i \log{\left(- \frac{\sqrt{2} \sqrt{\sqrt{33} + 15}}{8} - \frac{i \left(1 - \sqrt{33}\right)}{8} \right)}$$
$$x_{3} = - i \log{\left(\frac{\sqrt{2} \sqrt{\sqrt{33} + 15}}{8} - \frac{i \left(1 - \sqrt{33}\right)}{8} \right)}$$
$$x_{4} = - i \log{\left(\frac{\sqrt{2} \sqrt{15 - \sqrt{33}}}{8} - \frac{\sqrt{33} i}{8} - \frac{i}{8} \right)}$$
Numerical solution$$x_{1} = 0.634866871133571$$
$$x_{2} = -16.3428301390825$$
$$x_{3} = 76.0330905572886$$
$$x_{4} = 71.6217641614317$$
$$x_{5} = -57.5516347184825$$
$$x_{6} = -51.2684494113029$$
$$x_{7} = -18.2146890504052$$
$$x_{8} = 99.5279979610071$$
$$x_{9} = 60.6932273720723$$
$$x_{10} = 16.7109302218152$$
$$x_{11} = 46.4890229327133$$
$$x_{12} = -2.13862569972354$$
$$x_{13} = -47.7587566749805$$
$$x_{14} = 41.8436714505336$$
$$x_{15} = 40.2058376255337$$
$$x_{16} = 30.4129595820317$$
$$x_{17} = 38715.9848958867$$
$$x_{18} = 52.7722082398929$$
$$x_{19} = -8561.47484790332$$
$$x_{20} = 10.4277449146356$$
$$x_{21} = -5.64831843604602$$
$$x_{22} = 90.4713200829704$$
$$x_{23} = 19.4844227926723$$
$$x_{24} = 33.9226523183542$$
$$x_{25} = 24.1297742748521$$
$$x_{26} = -3.77645952472336$$
$$x_{27} = 77.9049494686113$$
$$x_{28} = -33.5545522356215$$
$$x_{29} = -55.9138008934827$$
$$x_{30} = -1.00296695386625$$
$$x_{31} = -93.6129127365602$$
$$x_{32} = 63.4667199429294$$
$$x_{33} = -49.6306155863031$$
$$x_{34} = -99.8960980437398$$
$$x_{35} = 55.54570081075$$
$$x_{36} = -41.4755713678009$$
$$x_{37} = -10.059644831903$$
$$x_{38} = -79.1746832108784$$
$$x_{39} = 48.1268567577131$$
$$x_{40} = 74.3952567322888$$
$$x_{41} = -11.9315037432256$$
$$x_{42} = -63.8348200256621$$
$$x_{43} = 17.8465889676725$$
$$x_{44} = -39.8377375428011$$
$$x_{45} = -98.0242391324172$$
$$x_{46} = -116.873795053956$$
$$x_{47} = 98.3923392151498$$
$$x_{48} = -46.1209228499806$$
$$x_{49} = 29.2773008361744$$
$$x_{50} = -32.4188934897642$$
$$x_{51} = 82.3162758644682$$
$$x_{52} = 121.887246618868$$
$$x_{53} = 4.14455960745605$$
$$x_{54} = 8.78991108963581$$
$$x_{55} = 54.4100420648927$$
$$x_{56} = -13.5693375682254$$
$$x_{57} = 61.8288861179296$$
$$x_{58} = 96.75450539015$$
$$x_{59} = -43.3474302791235$$
$$x_{60} = 2.50672578245622$$
$$x_{61} = -90.1032200002378$$
$$x_{62} = 85.8259686007907$$
$$x_{63} = -70.1180053328417$$
$$x_{64} = -87.3297274293806$$
$$x_{65} = -24.4978743575848$$
$$x_{66} = -26.1357081825846$$
$$x_{67} = 25.7676080998519$$
$$x_{68} = -71.253664078699$$
$$x_{69} = 92.1091539079703$$
$$x_{70} = 69.749905250109$$
$$x_{71} = -91.7410538252376$$
$$x_{72} = -366.56337351614$$
$$x_{73} = -27.2713669284419$$
$$x_{74} = 11.5634036604929$$
$$x_{75} = 38.3339787142111$$
$$x_{76} = 88.5994611716478$$
$$x_{77} = -77.5368493858786$$
$$x_{78} = 27.6394670111746$$
$$x_{79} = -76.4011906400213$$
$$x_{80} = 44.6171640213907$$
$$x_{81} = 32.0507934070315$$
$$x_{82} = -54.0419419821601$$
$$x_{83} = -68.4801715078419$$
$$x_{84} = -85.457868518058$$
$$x_{85} = -62.1969862006623$$
$$x_{86} = -19.852522875405$$
$$x_{87} = -60.3251272893396$$
$$x_{88} = -35.1923860606213$$
$$x_{89} = 68.1120714251092$$
$$x_{90} = -83.8200346930582$$
$$x_{91} = 84.1881347757908$$