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3*sin(2*x)-2*sin(x)-3*cos(x)+4>0 desigualdades

En la desigualdad la incógnita

Solución

Ha introducido [src]
3*sin(2*x) - 2*sin(x) - 3*cos(x) + 4 > 0
$$\left(\left(- 2 \sin{\left(x \right)} + 3 \sin{\left(2 x \right)}\right) - 3 \cos{\left(x \right)}\right) + 4 > 0$$
-2*sin(x) + 3*sin(2*x) - 3*cos(x) + 4 > 0
Solución detallada
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:
$$x < -1873.07316747285$$
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       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:
$$x < -1873.07316747285$$
$$x > -1703.427164179 \wedge x < -578.367953855434$$
$$x > -220.226391346198 \wedge x < -201.376835424659$$
$$x > -163.677723581581 \wedge x < -151.111352967222$$
$$x > -107.498096155383 \wedge x < -94.9317255410238$$
$$x > -94.5626852026059 \wedge x < -88.6485402338442$$
$$x > -88.2794998954263 \wedge x < -81.9963145882467$$
$$x > -75.7131292810671 \wedge x < -69.7989843123054$$
$$x > -63.5157990051258 \wedge x < -57.2326136979463$$
$$x > -50.9494283907667 \wedge x < -50.5803880523488$$
$$x > -44.6662430835871 \wedge x < -44.2972027451692$$
$$x > -38.0140174379896 \wedge x < -31.73083213081$$
$$x > -25.8166871620483 \wedge x < -19.5335018548687$$
$$x > -13.2503165476892 \wedge x < -12.8812762092712$$
$$x > -6.96713124050957 \wedge x < -6.59809090209166$$
$$x > -0.683945933329981 \wedge x < -0.314905594912077$$
$$x > 5.59923937384961 \wedge x < 5.96827971226751$$
$$x > 11.8824246810292 \wedge x < 12.2514650194471$$
$$x > 18.1656099882088 \wedge x < 18.5346503266267$$
$$x > 24.4487952953884 \wedge x < 30.731980602568$$
$$x > 37.0151659097475 \wedge x < 37.3842062481654$$
$$x > 43.2983512169271 \wedge x < 43.667391555345$$
$$x > 49.9505768625246 \wedge x < 56.2337621697042$$
$$x > 62.1479071384659 \wedge x < 62.5169474768838$$
$$x > 68.4310924456455 \wedge x < 68.8001327840634$$
$$x > 74.7142777528251 \wedge x < 80.9974630600046$$
$$x > 81.3665033984226 \wedge x < 87.2806483671842$$
$$x > 87.6496887056021 \wedge x < 93.5638336743638$$
$$x > 93.9328740127817 \wedge x < 100.216059319961$$
$$x > 187.811613282058 \wedge x < 872.678811764633$$
Solución de la desigualdad en el gráfico