Se da la desigualdad:
$$\left(\left(- 2 \sin{\left(x \right)} + 3 \sin{\left(2 x \right)}\right) - 3 \cos{\left(x \right)}\right) + 4 > 0$$
Para resolver esta desigualdad primero hay que resolver la ecuación correspondiente:
$$\left(\left(- 2 \sin{\left(x \right)} + 3 \sin{\left(2 x \right)}\right) - 3 \cos{\left(x \right)}\right) + 4 = 0$$
Resolvemos:
$$x_{1} = -57.2326136979463$$
$$x_{2} = 187.811613282058$$
$$x_{3} = 68.8001327840634$$
$$x_{4} = -1873.07316747285$$
$$x_{5} = -88.6485402338442$$
$$x_{6} = 68.4310924456455$$
$$x_{7} = -38.0140174379896$$
$$x_{8} = -94.5626852026059$$
$$x_{9} = 24.4487952953884$$
$$x_{10} = -19.5335018548687$$
$$x_{11} = -94.9317255410238$$
$$x_{12} = 5.96827971226751$$
$$x_{13} = 74.7142777528251$$
$$x_{14} = 100.216059319961$$
$$x_{15} = -69.7989843123054$$
$$x_{16} = 87.6496887056021$$
$$x_{17} = 30.731980602568$$
$$x_{18} = 93.5638336743638$$
$$x_{19} = -44.6662430835871$$
$$x_{20} = -163.677723581581$$
$$x_{21} = 5.59923937384961$$
$$x_{22} = -50.9494283907667$$
$$x_{23} = 80.9974630600046$$
$$x_{24} = 43.667391555345$$
$$x_{25} = -6.59809090209166$$
$$x_{26} = 37.3842062481654$$
$$x_{27} = -50.5803880523488$$
$$x_{28} = 18.1656099882088$$
$$x_{29} = -75.7131292810671$$
$$x_{30} = -81.9963145882467$$
$$x_{31} = 62.1479071384659$$
$$x_{32} = 81.3665033984226$$
$$x_{33} = 62.5169474768838$$
$$x_{34} = -31.73083213081$$
$$x_{35} = 49.9505768625246$$
$$x_{36} = 43.2983512169271$$
$$x_{37} = 872.678811764633$$
$$x_{38} = -151.111352967222$$
$$x_{39} = 87.2806483671842$$
$$x_{40} = 11.8824246810292$$
$$x_{41} = -220.226391346198$$
$$x_{42} = -1703.427164179$$
$$x_{43} = 93.9328740127817$$
$$x_{44} = 12.2514650194471$$
$$x_{45} = -25.8166871620483$$
$$x_{46} = -6.96713124050957$$
$$x_{47} = -107.498096155383$$
$$x_{48} = -12.8812762092712$$
$$x_{49} = -13.2503165476892$$
$$x_{50} = 56.2337621697042$$
$$x_{51} = -0.683945933329981$$
$$x_{52} = 37.0151659097475$$
$$x_{53} = -63.5157990051258$$
$$x_{54} = -88.2794998954263$$
$$x_{55} = -201.376835424659$$
$$x_{56} = -44.2972027451692$$
$$x_{57} = 18.5346503266267$$
$$x_{58} = -0.314905594912077$$
$$x_{59} = -578.367953855434$$
$$x_{1} = -57.2326136979463$$
$$x_{2} = 187.811613282058$$
$$x_{3} = 68.8001327840634$$
$$x_{4} = -1873.07316747285$$
$$x_{5} = -88.6485402338442$$
$$x_{6} = 68.4310924456455$$
$$x_{7} = -38.0140174379896$$
$$x_{8} = -94.5626852026059$$
$$x_{9} = 24.4487952953884$$
$$x_{10} = -19.5335018548687$$
$$x_{11} = -94.9317255410238$$
$$x_{12} = 5.96827971226751$$
$$x_{13} = 74.7142777528251$$
$$x_{14} = 100.216059319961$$
$$x_{15} = -69.7989843123054$$
$$x_{16} = 87.6496887056021$$
$$x_{17} = 30.731980602568$$
$$x_{18} = 93.5638336743638$$
$$x_{19} = -44.6662430835871$$
$$x_{20} = -163.677723581581$$
$$x_{21} = 5.59923937384961$$
$$x_{22} = -50.9494283907667$$
$$x_{23} = 80.9974630600046$$
$$x_{24} = 43.667391555345$$
$$x_{25} = -6.59809090209166$$
$$x_{26} = 37.3842062481654$$
$$x_{27} = -50.5803880523488$$
$$x_{28} = 18.1656099882088$$
$$x_{29} = -75.7131292810671$$
$$x_{30} = -81.9963145882467$$
$$x_{31} = 62.1479071384659$$
$$x_{32} = 81.3665033984226$$
$$x_{33} = 62.5169474768838$$
$$x_{34} = -31.73083213081$$
$$x_{35} = 49.9505768625246$$
$$x_{36} = 43.2983512169271$$
$$x_{37} = 872.678811764633$$
$$x_{38} = -151.111352967222$$
$$x_{39} = 87.2806483671842$$
$$x_{40} = 11.8824246810292$$
$$x_{41} = -220.226391346198$$
$$x_{42} = -1703.427164179$$
$$x_{43} = 93.9328740127817$$
$$x_{44} = 12.2514650194471$$
$$x_{45} = -25.8166871620483$$
$$x_{46} = -6.96713124050957$$
$$x_{47} = -107.498096155383$$
$$x_{48} = -12.8812762092712$$
$$x_{49} = -13.2503165476892$$
$$x_{50} = 56.2337621697042$$
$$x_{51} = -0.683945933329981$$
$$x_{52} = 37.0151659097475$$
$$x_{53} = -63.5157990051258$$
$$x_{54} = -88.2794998954263$$
$$x_{55} = -201.376835424659$$
$$x_{56} = -44.2972027451692$$
$$x_{57} = 18.5346503266267$$
$$x_{58} = -0.314905594912077$$
$$x_{59} = -578.367953855434$$
Las raíces dadas
$$x_{4} = -1873.07316747285$$
$$x_{42} = -1703.427164179$$
$$x_{59} = -578.367953855434$$
$$x_{41} = -220.226391346198$$
$$x_{55} = -201.376835424659$$
$$x_{20} = -163.677723581581$$
$$x_{38} = -151.111352967222$$
$$x_{47} = -107.498096155383$$
$$x_{11} = -94.9317255410238$$
$$x_{8} = -94.5626852026059$$
$$x_{5} = -88.6485402338442$$
$$x_{54} = -88.2794998954263$$
$$x_{30} = -81.9963145882467$$
$$x_{29} = -75.7131292810671$$
$$x_{15} = -69.7989843123054$$
$$x_{53} = -63.5157990051258$$
$$x_{1} = -57.2326136979463$$
$$x_{22} = -50.9494283907667$$
$$x_{27} = -50.5803880523488$$
$$x_{19} = -44.6662430835871$$
$$x_{56} = -44.2972027451692$$
$$x_{7} = -38.0140174379896$$
$$x_{34} = -31.73083213081$$
$$x_{45} = -25.8166871620483$$
$$x_{10} = -19.5335018548687$$
$$x_{49} = -13.2503165476892$$
$$x_{48} = -12.8812762092712$$
$$x_{46} = -6.96713124050957$$
$$x_{25} = -6.59809090209166$$
$$x_{51} = -0.683945933329981$$
$$x_{58} = -0.314905594912077$$
$$x_{21} = 5.59923937384961$$
$$x_{12} = 5.96827971226751$$
$$x_{40} = 11.8824246810292$$
$$x_{44} = 12.2514650194471$$
$$x_{28} = 18.1656099882088$$
$$x_{57} = 18.5346503266267$$
$$x_{9} = 24.4487952953884$$
$$x_{17} = 30.731980602568$$
$$x_{52} = 37.0151659097475$$
$$x_{26} = 37.3842062481654$$
$$x_{36} = 43.2983512169271$$
$$x_{24} = 43.667391555345$$
$$x_{35} = 49.9505768625246$$
$$x_{50} = 56.2337621697042$$
$$x_{31} = 62.1479071384659$$
$$x_{33} = 62.5169474768838$$
$$x_{6} = 68.4310924456455$$
$$x_{3} = 68.8001327840634$$
$$x_{13} = 74.7142777528251$$
$$x_{23} = 80.9974630600046$$
$$x_{32} = 81.3665033984226$$
$$x_{39} = 87.2806483671842$$
$$x_{16} = 87.6496887056021$$
$$x_{18} = 93.5638336743638$$
$$x_{43} = 93.9328740127817$$
$$x_{14} = 100.216059319961$$
$$x_{2} = 187.811613282058$$
$$x_{37} = 872.678811764633$$
son puntos de cambio del signo de desigualdad en las soluciones.
Primero definámonos con el signo hasta el punto extremo izquierdo:
$$x_{0} < x_{4}$$
Consideremos, por ejemplo, el punto
$$x_{0} = x_{4} - \frac{1}{10}$$
=
$$-1873.07316747285 + - \frac{1}{10}$$
=
$$-1873.17316747285$$
lo sustituimos en la expresión
$$\left(\left(- 2 \sin{\left(x \right)} + 3 \sin{\left(2 x \right)}\right) - 3 \cos{\left(x \right)}\right) + 4 > 0$$
$$\left(- 3 \cos{\left(-1873.17316747285 \right)} + \left(3 \sin{\left(\left(-1873.17316747285\right) 2 \right)} - 2 \sin{\left(-1873.17316747285 \right)}\right)\right) + 4 > 0$$
0.287772211433395 > 0
significa que una de las soluciones de nuestra ecuación será con:
$$x < -1873.07316747285$$
_____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____
\ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \
-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------
x4 x42 x59 x41 x55 x20 x38 x47 x11 x8 x5 x54 x30 x29 x15 x53 x1 x22 x27 x19 x56 x7 x34 x45 x10 x49 x48 x46 x25 x51 x58 x21 x12 x40 x44 x28 x57 x9 x17 x52 x26 x36 x24 x35 x50 x31 x33 x6 x3 x13 x23 x32 x39 x16 x18 x43 x14 x2 x37
Recibiremos otras soluciones de la desigualdad pasando al polo siguiente etc.
etc.
Respuesta:
$$x < -1873.07316747285$$
$$x > -1703.427164179 \wedge x < -578.367953855434$$
$$x > -220.226391346198 \wedge x < -201.376835424659$$
$$x > -163.677723581581 \wedge x < -151.111352967222$$
$$x > -107.498096155383 \wedge x < -94.9317255410238$$
$$x > -94.5626852026059 \wedge x < -88.6485402338442$$
$$x > -88.2794998954263 \wedge x < -81.9963145882467$$
$$x > -75.7131292810671 \wedge x < -69.7989843123054$$
$$x > -63.5157990051258 \wedge x < -57.2326136979463$$
$$x > -50.9494283907667 \wedge x < -50.5803880523488$$
$$x > -44.6662430835871 \wedge x < -44.2972027451692$$
$$x > -38.0140174379896 \wedge x < -31.73083213081$$
$$x > -25.8166871620483 \wedge x < -19.5335018548687$$
$$x > -13.2503165476892 \wedge x < -12.8812762092712$$
$$x > -6.96713124050957 \wedge x < -6.59809090209166$$
$$x > -0.683945933329981 \wedge x < -0.314905594912077$$
$$x > 5.59923937384961 \wedge x < 5.96827971226751$$
$$x > 11.8824246810292 \wedge x < 12.2514650194471$$
$$x > 18.1656099882088 \wedge x < 18.5346503266267$$
$$x > 24.4487952953884 \wedge x < 30.731980602568$$
$$x > 37.0151659097475 \wedge x < 37.3842062481654$$
$$x > 43.2983512169271 \wedge x < 43.667391555345$$
$$x > 49.9505768625246 \wedge x < 56.2337621697042$$
$$x > 62.1479071384659 \wedge x < 62.5169474768838$$
$$x > 68.4310924456455 \wedge x < 68.8001327840634$$
$$x > 74.7142777528251 \wedge x < 80.9974630600046$$
$$x > 81.3665033984226 \wedge x < 87.2806483671842$$
$$x > 87.6496887056021 \wedge x < 93.5638336743638$$
$$x > 93.9328740127817 \wedge x < 100.216059319961$$
$$x > 187.811613282058 \wedge x < 872.678811764633$$