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sin^2(2x)+2sin(x)>0 desigualdades

En la desigualdad la incógnita

Solución

Ha introducido [src]
   2                    
sin (2*x) + 2*sin(x) > 0
$$2 \sin{\left(x \right)} + \sin^{2}{\left(2 x \right)} > 0$$
2*sin(x) + sin(2*x)^2 > 0
Solución detallada
Se da la desigualdad:
$$2 \sin{\left(x \right)} + \sin^{2}{\left(2 x \right)} > 0$$
Para resolver esta desigualdad primero hay que resolver la ecuación correspondiente:
$$2 \sin{\left(x \right)} + \sin^{2}{\left(2 x \right)} = 0$$
Resolvemos:
$$x_{1} = 53.4070751110265$$
$$x_{2} = -97.3893722612836$$
$$x_{3} = 37.6991118430775$$
$$x_{4} = 97.3893722612836$$
$$x_{5} = 78.5398163397448$$
$$x_{6} = -380.132711084365$$
$$x_{7} = -59.6902604182061$$
$$x_{8} = -65.9734457253857$$
$$x_{9} = 0$$
$$x_{10} = -31.4159265358979$$
$$x_{11} = -50.2654824574367$$
$$x_{12} = -21.9911485751286$$
$$x_{13} = 6.28318530717959$$
$$x_{14} = -34.5575191894877$$
$$x_{15} = -94.2477796076938$$
$$x_{16} = -15.707963267949$$
$$x_{17} = 21.9911485751286$$
$$x_{18} = -301.59289474462$$
$$x_{19} = 50.2654824574367$$
$$x_{20} = 81.6814089933346$$
$$x_{21} = 29018.8913412089$$
$$x_{22} = -40.8407044966673$$
$$x_{23} = 9.42477796076938$$
$$x_{24} = -87.9645943005142$$
$$x_{25} = 34.5575191894877$$
$$x_{26} = 65.9734457253857$$
$$x_{27} = -62.8318530717959$$
$$x_{28} = -18.8495559215388$$
$$x_{29} = -28.2743338823081$$
$$x_{30} = -56.5486677646163$$
$$x_{31} = -53.4070751110265$$
$$x_{32} = -37.6991118430775$$
$$x_{33} = 103.672557568463$$
$$x_{34} = -100.530964914873$$
$$x_{35} = -9.42477796076938$$
$$x_{36} = -91.106186954104$$
$$x_{37} = 87.9645943005142$$
$$x_{38} = 59.6902604182061$$
$$x_{39} = -6.28318530717959$$
$$x_{40} = 25.1327412287183$$
$$x_{41} = 56102.5616078065$$
$$x_{42} = 999.026463841554$$
$$x_{43} = -7772.30022498115$$
$$x_{44} = 28.2743338823081$$
$$x_{45} = 135.088484104361$$
$$x_{46} = 56.5486677646163$$
$$x_{47} = -43.9822971502571$$
$$x_{48} = -3.14159265358979$$
$$x_{49} = 31.4159265358979$$
$$x_{50} = 94.2477796076938$$
$$x_{51} = 109.955742875643$$
$$x_{52} = -12.5663706143592$$
$$x_{53} = 75.398223686155$$
$$x_{54} = -72.2566310325652$$
$$x_{55} = -84.8230016469244$$
$$x_{56} = 84.8230016469244$$
$$x_{57} = 72.2566310325652$$
$$x_{58} = -81.6814089933346$$
$$x_{59} = 43.9822971502571$$
$$x_{60} = -78.5398163397448$$
$$x_{61} = 3.14159265358979$$
$$x_{1} = 53.4070751110265$$
$$x_{2} = -97.3893722612836$$
$$x_{3} = 37.6991118430775$$
$$x_{4} = 97.3893722612836$$
$$x_{5} = 78.5398163397448$$
$$x_{6} = -380.132711084365$$
$$x_{7} = -59.6902604182061$$
$$x_{8} = -65.9734457253857$$
$$x_{9} = 0$$
$$x_{10} = -31.4159265358979$$
$$x_{11} = -50.2654824574367$$
$$x_{12} = -21.9911485751286$$
$$x_{13} = 6.28318530717959$$
$$x_{14} = -34.5575191894877$$
$$x_{15} = -94.2477796076938$$
$$x_{16} = -15.707963267949$$
$$x_{17} = 21.9911485751286$$
$$x_{18} = -301.59289474462$$
$$x_{19} = 50.2654824574367$$
$$x_{20} = 81.6814089933346$$
$$x_{21} = 29018.8913412089$$
$$x_{22} = -40.8407044966673$$
$$x_{23} = 9.42477796076938$$
$$x_{24} = -87.9645943005142$$
$$x_{25} = 34.5575191894877$$
$$x_{26} = 65.9734457253857$$
$$x_{27} = -62.8318530717959$$
$$x_{28} = -18.8495559215388$$
$$x_{29} = -28.2743338823081$$
$$x_{30} = -56.5486677646163$$
$$x_{31} = -53.4070751110265$$
$$x_{32} = -37.6991118430775$$
$$x_{33} = 103.672557568463$$
$$x_{34} = -100.530964914873$$
$$x_{35} = -9.42477796076938$$
$$x_{36} = -91.106186954104$$
$$x_{37} = 87.9645943005142$$
$$x_{38} = 59.6902604182061$$
$$x_{39} = -6.28318530717959$$
$$x_{40} = 25.1327412287183$$
$$x_{41} = 56102.5616078065$$
$$x_{42} = 999.026463841554$$
$$x_{43} = -7772.30022498115$$
$$x_{44} = 28.2743338823081$$
$$x_{45} = 135.088484104361$$
$$x_{46} = 56.5486677646163$$
$$x_{47} = -43.9822971502571$$
$$x_{48} = -3.14159265358979$$
$$x_{49} = 31.4159265358979$$
$$x_{50} = 94.2477796076938$$
$$x_{51} = 109.955742875643$$
$$x_{52} = -12.5663706143592$$
$$x_{53} = 75.398223686155$$
$$x_{54} = -72.2566310325652$$
$$x_{55} = -84.8230016469244$$
$$x_{56} = 84.8230016469244$$
$$x_{57} = 72.2566310325652$$
$$x_{58} = -81.6814089933346$$
$$x_{59} = 43.9822971502571$$
$$x_{60} = -78.5398163397448$$
$$x_{61} = 3.14159265358979$$
Las raíces dadas
$$x_{43} = -7772.30022498115$$
$$x_{6} = -380.132711084365$$
$$x_{18} = -301.59289474462$$
$$x_{34} = -100.530964914873$$
$$x_{2} = -97.3893722612836$$
$$x_{15} = -94.2477796076938$$
$$x_{36} = -91.106186954104$$
$$x_{24} = -87.9645943005142$$
$$x_{55} = -84.8230016469244$$
$$x_{58} = -81.6814089933346$$
$$x_{60} = -78.5398163397448$$
$$x_{54} = -72.2566310325652$$
$$x_{8} = -65.9734457253857$$
$$x_{27} = -62.8318530717959$$
$$x_{7} = -59.6902604182061$$
$$x_{30} = -56.5486677646163$$
$$x_{31} = -53.4070751110265$$
$$x_{11} = -50.2654824574367$$
$$x_{47} = -43.9822971502571$$
$$x_{22} = -40.8407044966673$$
$$x_{32} = -37.6991118430775$$
$$x_{14} = -34.5575191894877$$
$$x_{10} = -31.4159265358979$$
$$x_{29} = -28.2743338823081$$
$$x_{12} = -21.9911485751286$$
$$x_{28} = -18.8495559215388$$
$$x_{16} = -15.707963267949$$
$$x_{52} = -12.5663706143592$$
$$x_{35} = -9.42477796076938$$
$$x_{39} = -6.28318530717959$$
$$x_{48} = -3.14159265358979$$
$$x_{9} = 0$$
$$x_{61} = 3.14159265358979$$
$$x_{13} = 6.28318530717959$$
$$x_{23} = 9.42477796076938$$
$$x_{17} = 21.9911485751286$$
$$x_{40} = 25.1327412287183$$
$$x_{44} = 28.2743338823081$$
$$x_{49} = 31.4159265358979$$
$$x_{25} = 34.5575191894877$$
$$x_{3} = 37.6991118430775$$
$$x_{59} = 43.9822971502571$$
$$x_{19} = 50.2654824574367$$
$$x_{1} = 53.4070751110265$$
$$x_{46} = 56.5486677646163$$
$$x_{38} = 59.6902604182061$$
$$x_{26} = 65.9734457253857$$
$$x_{57} = 72.2566310325652$$
$$x_{53} = 75.398223686155$$
$$x_{5} = 78.5398163397448$$
$$x_{20} = 81.6814089933346$$
$$x_{56} = 84.8230016469244$$
$$x_{37} = 87.9645943005142$$
$$x_{50} = 94.2477796076938$$
$$x_{4} = 97.3893722612836$$
$$x_{33} = 103.672557568463$$
$$x_{51} = 109.955742875643$$
$$x_{45} = 135.088484104361$$
$$x_{42} = 999.026463841554$$
$$x_{21} = 29018.8913412089$$
$$x_{41} = 56102.5616078065$$
son puntos de cambio del signo de desigualdad en las soluciones.
Primero definámonos con el signo hasta el punto extremo izquierdo:
$$x_{0} < x_{43}$$
Consideremos, por ejemplo, el punto
$$x_{0} = x_{43} - \frac{1}{10}$$
=
$$-7772.30022498115 + - \frac{1}{10}$$
=
$$-7772.40022498115$$
lo sustituimos en la expresión
$$2 \sin{\left(x \right)} + \sin^{2}{\left(2 x \right)} > 0$$
$$2 \sin{\left(-7772.40022498115 \right)} + \sin^{2}{\left(\left(-7772.40022498115\right) 2 \right)} > 0$$
-0.160197330296007 > 0

Entonces
$$x < -7772.30022498115$$
no se cumple
significa que una de las soluciones de nuestra ecuación será con:
$$x > -7772.30022498115 \wedge x < -380.132711084365$$
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        /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /
-------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-------ο-------ο-------
       x43      x6      x18      x34      x2      x15      x36      x24      x55      x58      x60      x54      x8      x27      x7      x30      x31      x11      x47      x22      x32      x14      x10      x29      x12      x28      x16      x52      x35      x39      x48      x9      x61      x13      x23      x17      x40      x44      x49      x25      x3      x59      x19      x1      x46      x38      x26      x57      x53      x5      x20      x56      x37      x50      x4      x33      x51      x45      x42      x21      x41

Recibiremos otras soluciones de la desigualdad pasando al polo siguiente etc.
etc.
Respuesta:
$$x > -7772.30022498115 \wedge x < -380.132711084365$$
$$x > -301.59289474462 \wedge x < -100.530964914873$$
$$x > -97.3893722612836 \wedge x < -94.2477796076938$$
$$x > -91.106186954104 \wedge x < -87.9645943005142$$
$$x > -84.8230016469244 \wedge x < -81.6814089933346$$
$$x > -78.5398163397448 \wedge x < -72.2566310325652$$
$$x > -65.9734457253857 \wedge x < -62.8318530717959$$
$$x > -59.6902604182061 \wedge x < -56.5486677646163$$
$$x > -53.4070751110265 \wedge x < -50.2654824574367$$
$$x > -43.9822971502571 \wedge x < -40.8407044966673$$
$$x > -37.6991118430775 \wedge x < -34.5575191894877$$
$$x > -31.4159265358979 \wedge x < -28.2743338823081$$
$$x > -21.9911485751286 \wedge x < -18.8495559215388$$
$$x > -15.707963267949 \wedge x < -12.5663706143592$$
$$x > -9.42477796076938 \wedge x < -6.28318530717959$$
$$x > -3.14159265358979 \wedge x < 0$$
$$x > 3.14159265358979 \wedge x < 6.28318530717959$$
$$x > 9.42477796076938 \wedge x < 21.9911485751286$$
$$x > 25.1327412287183 \wedge x < 28.2743338823081$$
$$x > 31.4159265358979 \wedge x < 34.5575191894877$$
$$x > 37.6991118430775 \wedge x < 43.9822971502571$$
$$x > 50.2654824574367 \wedge x < 53.4070751110265$$
$$x > 56.5486677646163 \wedge x < 59.6902604182061$$
$$x > 65.9734457253857 \wedge x < 72.2566310325652$$
$$x > 75.398223686155 \wedge x < 78.5398163397448$$
$$x > 81.6814089933346 \wedge x < 84.8230016469244$$
$$x > 87.9645943005142 \wedge x < 94.2477796076938$$
$$x > 97.3893722612836 \wedge x < 103.672557568463$$
$$x > 109.955742875643 \wedge x < 135.088484104361$$
$$x > 999.026463841554 \wedge x < 29018.8913412089$$
$$x > 56102.5616078065$$
Solución de la desigualdad en el gráfico
Respuesta rápida 2 [src]
(0, pi)
$$x\ in\ \left(0, \pi\right)$$
x in Interval.open(0, pi)
Respuesta rápida [src]
And(0 < x, x < pi)
$$0 < x \wedge x < \pi$$
(0 < x)∧(x < pi)