|sin(x)|<=0 desigualdades

Se da la desigualdad:
$$\left|{\sin{\left(x \right)}}\right| \leq 0$$
Para resolver esta desigualdad primero hay que resolver la ecuación correspondiente:
$$\left|{\sin{\left(x \right)}}\right| = 0$$
Resolvemos:
Tenemos la ecuación
$$\left|{\sin{\left(x \right)}}\right| = 0$$
cambiamos
$$\left|{\sin{\left(x \right)}}\right| - 1 = 0$$
$$\left|{\sin{\left(x \right)}}\right| - 1 = 0$$
Sustituimos
$$w = \left|{\sin{\left(x \right)}}\right|$$
Transportamos los términos libres (sin w)
del miembro izquierdo al derecho, obtenemos:
$$w = 1$$
Obtenemos la respuesta: w = 1
hacemos cambio inverso
$$\left|{\sin{\left(x \right)}}\right| = w$$
sustituimos w:
$$x_{1} = 72.2566310325652$$
$$x_{2} = -59.6902604182061$$
$$x_{3} = 81.6814089933346$$
$$x_{4} = -43.9822971502571$$
$$x_{5} = -100.530964914873$$
$$x_{6} = -427.256600888212$$
$$x_{7} = 28.2743338823081$$
$$x_{8} = 65.9734457253857$$
$$x_{9} = -31.4159265358979$$
$$x_{10} = -9.42477796076938$$
$$x_{11} = 40.8407044966673$$
$$x_{12} = 56.5486677646163$$
$$x_{13} = -56.5486677646163$$
$$x_{14} = 12.5663706143592$$
$$x_{15} = 43.9822971502571$$
$$x_{16} = 100.530964914873$$
$$x_{17} = 650.309679293087$$
$$x_{18} = -3.14159265358979$$
$$x_{19} = -15.707963267949$$
$$x_{20} = 59.6902604182061$$
$$x_{21} = 6.28318530717959$$
$$x_{22} = 9.42477796076938$$
$$x_{23} = -53.4070751110265$$
$$x_{24} = -47.1238898038469$$
$$x_{25} = -87.9645943005142$$
$$x_{26} = 21.9911485751286$$
$$x_{27} = 87.9645943005142$$
$$x_{28} = 18.8495559215388$$
$$x_{29} = -25.1327412287183$$
$$x_{30} = -72.2566310325652$$
$$x_{31} = 37.6991118430775$$
$$x_{32} = 0$$
$$x_{33} = 50.2654824574367$$
$$x_{34} = -6.28318530717959$$
$$x_{35} = -65.9734457253857$$
$$x_{36} = -21.9911485751286$$
$$x_{37} = 75.398223686155$$
$$x_{38} = 84.8230016469244$$
$$x_{39} = 34.5575191894877$$
$$x_{40} = 53.4070751110265$$
$$x_{41} = 15.707963267949$$
$$x_{42} = -28.2743338823081$$
$$x_{43} = -285.884931476671$$
$$x_{44} = -91.106186954104$$
$$x_{45} = 97.3893722612836$$
$$x_{46} = -69.1150383789755$$
$$x_{47} = 94.2477796076938$$
$$x_{48} = -50.2654824574367$$
$$x_{49} = -37.6991118430775$$
$$x_{50} = -81.6814089933346$$
$$x_{51} = -3760.48640634698$$
$$x_{52} = 62.8318530717959$$
$$x_{53} = 78.5398163397448$$
$$x_{54} = 31.4159265358979$$
$$x_{55} = -78.5398163397448$$
$$x_{56} = -97.3893722612836$$
$$x_{57} = -75.398223686155$$
$$x_{58} = -12.5663706143592$$
$$x_{59} = -94.2477796076938$$
$$x_{60} = -34.5575191894877$$
$$x_{1} = 72.2566310325652$$
$$x_{2} = -59.6902604182061$$
$$x_{3} = 81.6814089933346$$
$$x_{4} = -43.9822971502571$$
$$x_{5} = -100.530964914873$$
$$x_{6} = -427.256600888212$$
$$x_{7} = 28.2743338823081$$
$$x_{8} = 65.9734457253857$$
$$x_{9} = -31.4159265358979$$
$$x_{10} = -9.42477796076938$$
$$x_{11} = 40.8407044966673$$
$$x_{12} = 56.5486677646163$$
$$x_{13} = -56.5486677646163$$
$$x_{14} = 12.5663706143592$$
$$x_{15} = 43.9822971502571$$
$$x_{16} = 100.530964914873$$
$$x_{17} = 650.309679293087$$
$$x_{18} = -3.14159265358979$$
$$x_{19} = -15.707963267949$$
$$x_{20} = 59.6902604182061$$
$$x_{21} = 6.28318530717959$$
$$x_{22} = 9.42477796076938$$
$$x_{23} = -53.4070751110265$$
$$x_{24} = -47.1238898038469$$
$$x_{25} = -87.9645943005142$$
$$x_{26} = 21.9911485751286$$
$$x_{27} = 87.9645943005142$$
$$x_{28} = 18.8495559215388$$
$$x_{29} = -25.1327412287183$$
$$x_{30} = -72.2566310325652$$
$$x_{31} = 37.6991118430775$$
$$x_{32} = 0$$
$$x_{33} = 50.2654824574367$$
$$x_{34} = -6.28318530717959$$
$$x_{35} = -65.9734457253857$$
$$x_{36} = -21.9911485751286$$
$$x_{37} = 75.398223686155$$
$$x_{38} = 84.8230016469244$$
$$x_{39} = 34.5575191894877$$
$$x_{40} = 53.4070751110265$$
$$x_{41} = 15.707963267949$$
$$x_{42} = -28.2743338823081$$
$$x_{43} = -285.884931476671$$
$$x_{44} = -91.106186954104$$
$$x_{45} = 97.3893722612836$$
$$x_{46} = -69.1150383789755$$
$$x_{47} = 94.2477796076938$$
$$x_{48} = -50.2654824574367$$
$$x_{49} = -37.6991118430775$$
$$x_{50} = -81.6814089933346$$
$$x_{51} = -3760.48640634698$$
$$x_{52} = 62.8318530717959$$
$$x_{53} = 78.5398163397448$$
$$x_{54} = 31.4159265358979$$
$$x_{55} = -78.5398163397448$$
$$x_{56} = -97.3893722612836$$
$$x_{57} = -75.398223686155$$
$$x_{58} = -12.5663706143592$$
$$x_{59} = -94.2477796076938$$
$$x_{60} = -34.5575191894877$$
Las raíces dadas
$$x_{51} = -3760.48640634698$$
$$x_{6} = -427.256600888212$$
$$x_{43} = -285.884931476671$$
$$x_{5} = -100.530964914873$$
$$x_{56} = -97.3893722612836$$
$$x_{59} = -94.2477796076938$$
$$x_{44} = -91.106186954104$$
$$x_{25} = -87.9645943005142$$
$$x_{50} = -81.6814089933346$$
$$x_{55} = -78.5398163397448$$
$$x_{57} = -75.398223686155$$
$$x_{30} = -72.2566310325652$$
$$x_{46} = -69.1150383789755$$
$$x_{35} = -65.9734457253857$$
$$x_{2} = -59.6902604182061$$
$$x_{13} = -56.5486677646163$$
$$x_{23} = -53.4070751110265$$
$$x_{48} = -50.2654824574367$$
$$x_{24} = -47.1238898038469$$
$$x_{4} = -43.9822971502571$$
$$x_{49} = -37.6991118430775$$
$$x_{60} = -34.5575191894877$$
$$x_{9} = -31.4159265358979$$
$$x_{42} = -28.2743338823081$$
$$x_{29} = -25.1327412287183$$
$$x_{36} = -21.9911485751286$$
$$x_{19} = -15.707963267949$$
$$x_{58} = -12.5663706143592$$
$$x_{10} = -9.42477796076938$$
$$x_{34} = -6.28318530717959$$
$$x_{18} = -3.14159265358979$$
$$x_{32} = 0$$
$$x_{21} = 6.28318530717959$$
$$x_{22} = 9.42477796076938$$
$$x_{14} = 12.5663706143592$$
$$x_{41} = 15.707963267949$$
$$x_{28} = 18.8495559215388$$
$$x_{26} = 21.9911485751286$$
$$x_{7} = 28.2743338823081$$
$$x_{54} = 31.4159265358979$$
$$x_{39} = 34.5575191894877$$
$$x_{31} = 37.6991118430775$$
$$x_{11} = 40.8407044966673$$
$$x_{15} = 43.9822971502571$$
$$x_{33} = 50.2654824574367$$
$$x_{40} = 53.4070751110265$$
$$x_{12} = 56.5486677646163$$
$$x_{20} = 59.6902604182061$$
$$x_{52} = 62.8318530717959$$
$$x_{8} = 65.9734457253857$$
$$x_{1} = 72.2566310325652$$
$$x_{37} = 75.398223686155$$
$$x_{53} = 78.5398163397448$$
$$x_{3} = 81.6814089933346$$
$$x_{38} = 84.8230016469244$$
$$x_{27} = 87.9645943005142$$
$$x_{47} = 94.2477796076938$$
$$x_{45} = 97.3893722612836$$
$$x_{16} = 100.530964914873$$
$$x_{17} = 650.309679293087$$
son puntos de cambio del signo de desigualdad en las soluciones.
Primero definámonos con el signo hasta el punto extremo izquierdo:
$$x_{0} \leq x_{51}$$
Consideremos, por ejemplo, el punto
$$x_{0} = x_{51} - \frac{1}{10}$$
=
$$-3760.48640634698 + - \frac{1}{10}$$
=
$$-3760.58640634698$$
lo sustituimos en la expresión
$$\left|{\sin{\left(x \right)}}\right| \leq 0$$
$$\left|{\sin{\left(-3760.58640634698 \right)}}\right| \leq 0$$

0.0998334166469064 <= 0

pero

0.0998334166469064 >= 0

Entonces
$$x \leq -3760.48640634698$$
no se cumple
significa que una de las soluciones de nuestra ecuación será con:
$$x \geq -3760.48640634698 \wedge x \leq -427.256600888212$$

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       x51      x6      x43      x5      x56      x59      x44      x25      x50      x55      x57      x30      x46      x35      x2      x13      x23      x48      x24      x4      x49      x60      x9      x42      x29      x36      x19      x58      x10      x34      x18      x32      x21      x22      x14      x41      x28      x26      x7      x54      x39      x31      x11      x15      x33      x40      x12      x20      x52      x8      x1      x37      x53      x3      x38      x27      x47      x45      x16      x17

Recibiremos otras soluciones de la desigualdad pasando al polo siguiente etc.
etc.
Respuesta:
$$x \geq -3760.48640634698 \wedge x \leq -427.256600888212$$
$$x \geq -285.884931476671 \wedge x \leq -100.530964914873$$
$$x \geq -97.3893722612836 \wedge x \leq -94.2477796076938$$
$$x \geq -91.106186954104 \wedge x \leq -87.9645943005142$$
$$x \geq -81.6814089933346 \wedge x \leq -78.5398163397448$$
$$x \geq -75.398223686155 \wedge x \leq -72.2566310325652$$
$$x \geq -69.1150383789755 \wedge x \leq -65.9734457253857$$
$$x \geq -59.6902604182061 \wedge x \leq -56.5486677646163$$
$$x \geq -53.4070751110265 \wedge x \leq -50.2654824574367$$
$$x \geq -47.1238898038469 \wedge x \leq -43.9822971502571$$
$$x \geq -37.6991118430775 \wedge x \leq -34.5575191894877$$
$$x \geq -31.4159265358979 \wedge x \leq -28.2743338823081$$
$$x \geq -25.1327412287183 \wedge x \leq -21.9911485751286$$
$$x \geq -15.707963267949 \wedge x \leq -12.5663706143592$$
$$x \geq -9.42477796076938 \wedge x \leq -6.28318530717959$$
$$x \geq -3.14159265358979 \wedge x \leq 0$$
$$x \geq 6.28318530717959 \wedge x \leq 9.42477796076938$$
$$x \geq 12.5663706143592 \wedge x \leq 15.707963267949$$
$$x \geq 18.8495559215388 \wedge x \leq 21.9911485751286$$
$$x \geq 28.2743338823081 \wedge x \leq 31.4159265358979$$
$$x \geq 34.5575191894877 \wedge x \leq 37.6991118430775$$
$$x \geq 40.8407044966673 \wedge x \leq 43.9822971502571$$
$$x \geq 50.2654824574367 \wedge x \leq 53.4070751110265$$
$$x \geq 56.5486677646163 \wedge x \leq 59.6902604182061$$
$$x \geq 62.8318530717959 \wedge x \leq 65.9734457253857$$
$$x \geq 72.2566310325652 \wedge x \leq 75.398223686155$$
$$x \geq 78.5398163397448 \wedge x \leq 81.6814089933346$$
$$x \geq 84.8230016469244 \wedge x \leq 87.9645943005142$$
$$x \geq 94.2477796076938 \wedge x \leq 97.3893722612836$$
$$x \geq 100.530964914873 \wedge x \leq 650.309679293087$$