cost<=1 desigualdades

Se da la desigualdad:
$$\cos{\left(t \right)} \leq 1$$
Para resolver esta desigualdad primero hay que resolver la ecuación correspondiente:
$$\cos{\left(t \right)} = 1$$
Resolvemos:
Tenemos la ecuación
$$\cos{\left(t \right)} = 1$$
cambiamos
$$\cos{\left(t \right)} - 1 = 0$$
$$\cos{\left(t \right)} - 1 = 0$$
Sustituimos
$$w = \cos{\left(t \right)}$$
Transportamos los términos libres (sin w)
del miembro izquierdo al derecho, obtenemos:
$$w = 1$$
Obtenemos la respuesta: w = 1
hacemos cambio inverso
$$\cos{\left(t \right)} = w$$
sustituimos w:
$$x_{1} = 94.2477796093523$$
$$x_{2} = -81.6814084945807$$
$$x_{3} = -100.530964626003$$
$$x_{4} = -50.2654829667315$$
$$x_{5} = -37.6991121287155$$
$$x_{6} = 62.8318535568358$$
$$x_{7} = -25.1327415297174$$
$$x_{8} = 0$$
$$x_{9} = 18.8495564031971$$
$$x_{10} = 6.28318579821791$$
$$x_{11} = 43.9822966661001$$
$$x_{12} = -4.79511606158654 \cdot 10^{-7}$$
$$x_{13} = -43.9822976246252$$
$$x_{14} = 75.3982232188727$$
$$x_{15} = 100.530965157364$$
$$x_{16} = -75.3982231720141$$
$$x_{17} = -12.5663710889626$$
$$x_{18} = 69.115038794053$$
$$x_{19} = -81.6814092565354$$
$$x_{20} = 81.6814091897036$$
$$x_{21} = -75.3982231045728$$
$$x_{22} = -56.5486682426592$$
$$x_{23} = -6.28318555849548$$
$$x_{24} = 87.9645938121814$$
$$x_{25} = -56.5486674685864$$
$$x_{26} = 43.9822974733639$$
$$x_{27} = 50.2654829439723$$
$$x_{28} = 56.5486680806249$$
$$x_{29} = -87.9645947692094$$
$$x_{30} = -43.9822967932182$$
$$x_{31} = 56.5486682809363$$
$$x_{32} = 50.2654821322586$$
$$x_{33} = 18.8495556275525$$
$$x_{34} = -6.2831851275477$$
$$x_{35} = -81.6814090382277$$
$$x_{36} = -87.9645943586158$$
$$x_{37} = 94.2477800892631$$
$$x_{38} = 56.5486668532011$$
$$x_{39} = -50.2654822863493$$
$$x_{40} = 25.1327416384075$$
$$x_{41} = -37.6991113479743$$
$$x_{42} = -31.4159260208155$$
$$x_{43} = 100.530964759815$$
$$x_{44} = -69.1150379045123$$
$$x_{45} = 81.6814085860518$$
$$x_{46} = 87.9645946044253$$
$$x_{47} = -6.2831858160515$$
$$x_{48} = -18.8495555173448$$
$$x_{49} = 6.28318528420851$$
$$x_{50} = 12.5663704426592$$
$$x_{51} = 12.5663711301703$$
$$x_{52} = -18.8495552124105$$
$$x_{53} = -94.2477797298079$$
$$x_{54} = -81.6814075578313$$
$$x_{55} = -31.4159267157965$$
$$x_{56} = 6.28318626747926$$
$$x_{57} = -43.9822971745392$$
$$x_{58} = 43.9822971694647$$
$$x_{59} = 37.6991115173992$$
$$x_{60} = 75.3982240031607$$
$$x_{61} = 81.6814084860076$$
$$x_{62} = -87.964593928489$$
$$x_{63} = 62.8318527849002$$
$$x_{64} = 3.40772025642919 \cdot 10^{-7}$$
$$x_{65} = 31.4159260648825$$
$$x_{66} = -94.2477801171671$$
$$x_{67} = -12.5663703112531$$
$$x_{68} = -94.2477794452815$$
$$x_{69} = -18.8495563230046$$
$$x_{70} = -75.3982238741744$$
$$x_{71} = 25.1327408328211$$
$$x_{72} = -37.6991118772631$$
$$x_{73} = 69.1150379887504$$
$$x_{74} = -25.1327407505866$$
$$x_{75} = -31.4159260507536$$
$$x_{76} = 56.5486676011951$$
$$x_{77} = -69.1150386869085$$
$$x_{78} = 12.5663710110881$$
$$x_{79} = 37.6991120311338$$
$$x_{80} = -62.831852673202$$
$$x_{81} = 94.2477792651059$$
$$x_{82} = 50.2654824463392$$
$$x_{83} = 31.4159268459961$$
$$x_{84} = -50.265482641087$$
$$x_{85} = -62.8318534787248$$
$$x_{86} = 87.964594335905$$
$$x_{87} = 6.28318500093652$$
$$x_{88} = 37.6991113348642$$
$$x_{1} = 94.2477796093523$$
$$x_{2} = -81.6814084945807$$
$$x_{3} = -100.530964626003$$
$$x_{4} = -50.2654829667315$$
$$x_{5} = -37.6991121287155$$
$$x_{6} = 62.8318535568358$$
$$x_{7} = -25.1327415297174$$
$$x_{8} = 0$$
$$x_{9} = 18.8495564031971$$
$$x_{10} = 6.28318579821791$$
$$x_{11} = 43.9822966661001$$
$$x_{12} = -4.79511606158654 \cdot 10^{-7}$$
$$x_{13} = -43.9822976246252$$
$$x_{14} = 75.3982232188727$$
$$x_{15} = 100.530965157364$$
$$x_{16} = -75.3982231720141$$
$$x_{17} = -12.5663710889626$$
$$x_{18} = 69.115038794053$$
$$x_{19} = -81.6814092565354$$
$$x_{20} = 81.6814091897036$$
$$x_{21} = -75.3982231045728$$
$$x_{22} = -56.5486682426592$$
$$x_{23} = -6.28318555849548$$
$$x_{24} = 87.9645938121814$$
$$x_{25} = -56.5486674685864$$
$$x_{26} = 43.9822974733639$$
$$x_{27} = 50.2654829439723$$
$$x_{28} = 56.5486680806249$$
$$x_{29} = -87.9645947692094$$
$$x_{30} = -43.9822967932182$$
$$x_{31} = 56.5486682809363$$
$$x_{32} = 50.2654821322586$$
$$x_{33} = 18.8495556275525$$
$$x_{34} = -6.2831851275477$$
$$x_{35} = -81.6814090382277$$
$$x_{36} = -87.9645943586158$$
$$x_{37} = 94.2477800892631$$
$$x_{38} = 56.5486668532011$$
$$x_{39} = -50.2654822863493$$
$$x_{40} = 25.1327416384075$$
$$x_{41} = -37.6991113479743$$
$$x_{42} = -31.4159260208155$$
$$x_{43} = 100.530964759815$$
$$x_{44} = -69.1150379045123$$
$$x_{45} = 81.6814085860518$$
$$x_{46} = 87.9645946044253$$
$$x_{47} = -6.2831858160515$$
$$x_{48} = -18.8495555173448$$
$$x_{49} = 6.28318528420851$$
$$x_{50} = 12.5663704426592$$
$$x_{51} = 12.5663711301703$$
$$x_{52} = -18.8495552124105$$
$$x_{53} = -94.2477797298079$$
$$x_{54} = -81.6814075578313$$
$$x_{55} = -31.4159267157965$$
$$x_{56} = 6.28318626747926$$
$$x_{57} = -43.9822971745392$$
$$x_{58} = 43.9822971694647$$
$$x_{59} = 37.6991115173992$$
$$x_{60} = 75.3982240031607$$
$$x_{61} = 81.6814084860076$$
$$x_{62} = -87.964593928489$$
$$x_{63} = 62.8318527849002$$
$$x_{64} = 3.40772025642919 \cdot 10^{-7}$$
$$x_{65} = 31.4159260648825$$
$$x_{66} = -94.2477801171671$$
$$x_{67} = -12.5663703112531$$
$$x_{68} = -94.2477794452815$$
$$x_{69} = -18.8495563230046$$
$$x_{70} = -75.3982238741744$$
$$x_{71} = 25.1327408328211$$
$$x_{72} = -37.6991118772631$$
$$x_{73} = 69.1150379887504$$
$$x_{74} = -25.1327407505866$$
$$x_{75} = -31.4159260507536$$
$$x_{76} = 56.5486676011951$$
$$x_{77} = -69.1150386869085$$
$$x_{78} = 12.5663710110881$$
$$x_{79} = 37.6991120311338$$
$$x_{80} = -62.831852673202$$
$$x_{81} = 94.2477792651059$$
$$x_{82} = 50.2654824463392$$
$$x_{83} = 31.4159268459961$$
$$x_{84} = -50.265482641087$$
$$x_{85} = -62.8318534787248$$
$$x_{86} = 87.964594335905$$
$$x_{87} = 6.28318500093652$$
$$x_{88} = 37.6991113348642$$
Las raíces dadas
$$x_{3} = -100.530964626003$$
$$x_{66} = -94.2477801171671$$
$$x_{53} = -94.2477797298079$$
$$x_{68} = -94.2477794452815$$
$$x_{29} = -87.9645947692094$$
$$x_{36} = -87.9645943586158$$
$$x_{62} = -87.964593928489$$
$$x_{19} = -81.6814092565354$$
$$x_{35} = -81.6814090382277$$
$$x_{2} = -81.6814084945807$$
$$x_{54} = -81.6814075578313$$
$$x_{70} = -75.3982238741744$$
$$x_{16} = -75.3982231720141$$
$$x_{21} = -75.3982231045728$$
$$x_{77} = -69.1150386869085$$
$$x_{44} = -69.1150379045123$$
$$x_{85} = -62.8318534787248$$
$$x_{80} = -62.831852673202$$
$$x_{22} = -56.5486682426592$$
$$x_{25} = -56.5486674685864$$
$$x_{4} = -50.2654829667315$$
$$x_{84} = -50.265482641087$$
$$x_{39} = -50.2654822863493$$
$$x_{13} = -43.9822976246252$$
$$x_{57} = -43.9822971745392$$
$$x_{30} = -43.9822967932182$$
$$x_{5} = -37.6991121287155$$
$$x_{72} = -37.6991118772631$$
$$x_{41} = -37.6991113479743$$
$$x_{55} = -31.4159267157965$$
$$x_{75} = -31.4159260507536$$
$$x_{42} = -31.4159260208155$$
$$x_{7} = -25.1327415297174$$
$$x_{74} = -25.1327407505866$$
$$x_{69} = -18.8495563230046$$
$$x_{48} = -18.8495555173448$$
$$x_{52} = -18.8495552124105$$
$$x_{17} = -12.5663710889626$$
$$x_{67} = -12.5663703112531$$
$$x_{47} = -6.2831858160515$$
$$x_{23} = -6.28318555849548$$
$$x_{34} = -6.2831851275477$$
$$x_{12} = -4.79511606158654 \cdot 10^{-7}$$
$$x_{8} = 0$$
$$x_{64} = 3.40772025642919 \cdot 10^{-7}$$
$$x_{87} = 6.28318500093652$$
$$x_{49} = 6.28318528420851$$
$$x_{10} = 6.28318579821791$$
$$x_{56} = 6.28318626747926$$
$$x_{50} = 12.5663704426592$$
$$x_{78} = 12.5663710110881$$
$$x_{51} = 12.5663711301703$$
$$x_{33} = 18.8495556275525$$
$$x_{9} = 18.8495564031971$$
$$x_{71} = 25.1327408328211$$
$$x_{40} = 25.1327416384075$$
$$x_{65} = 31.4159260648825$$
$$x_{83} = 31.4159268459961$$
$$x_{88} = 37.6991113348642$$
$$x_{59} = 37.6991115173992$$
$$x_{79} = 37.6991120311338$$
$$x_{11} = 43.9822966661001$$
$$x_{58} = 43.9822971694647$$
$$x_{26} = 43.9822974733639$$
$$x_{32} = 50.2654821322586$$
$$x_{82} = 50.2654824463392$$
$$x_{27} = 50.2654829439723$$
$$x_{38} = 56.5486668532011$$
$$x_{76} = 56.5486676011951$$
$$x_{28} = 56.5486680806249$$
$$x_{31} = 56.5486682809363$$
$$x_{63} = 62.8318527849002$$
$$x_{6} = 62.8318535568358$$
$$x_{73} = 69.1150379887504$$
$$x_{18} = 69.115038794053$$
$$x_{14} = 75.3982232188727$$
$$x_{60} = 75.3982240031607$$
$$x_{61} = 81.6814084860076$$
$$x_{45} = 81.6814085860518$$
$$x_{20} = 81.6814091897036$$
$$x_{24} = 87.9645938121814$$
$$x_{86} = 87.964594335905$$
$$x_{46} = 87.9645946044253$$
$$x_{81} = 94.2477792651059$$
$$x_{1} = 94.2477796093523$$
$$x_{37} = 94.2477800892631$$
$$x_{43} = 100.530964759815$$
$$x_{15} = 100.530965157364$$
son puntos de cambio del signo de desigualdad en las soluciones.
Primero definámonos con el signo hasta el punto extremo izquierdo:
$$x_{0} \leq x_{3}$$
Consideremos, por ejemplo, el punto
$$x_{0} = x_{3} - \frac{1}{10}$$
=
$$-100.530964626003 + - \frac{1}{10}$$
=
$$-100.630964626003$$
lo sustituimos en la expresión
$$\cos{\left(t \right)} \leq 1$$
$$\cos{\left(t \right)} \leq 1$$

cos(t) <= 1

Entonces
$$x \leq -100.530964626003$$
no se cumple
significa que una de las soluciones de nuestra ecuación será con:
$$x \geq -100.530964626003 \wedge x \leq -94.2477801171671$$

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       x3      x66      x53      x68      x29      x36      x62      x19      x35      x2      x54      x70      x16      x21      x77      x44      x85      x80      x22      x25      x4      x84      x39      x13      x57      x30      x5      x72      x41      x55      x75      x42      x7      x74      x69      x48      x52      x17      x67      x47      x23      x34      x12      x8      x64      x87      x49      x10      x56      x50      x78      x51      x33      x9      x71      x40      x65      x83      x88      x59      x79      x11      x58      x26      x32      x82      x27      x38      x76      x28      x31      x63      x6      x73      x18      x14      x60      x61      x45      x20      x24      x86      x46      x81      x1      x37      x43      x15

Recibiremos otras soluciones de la desigualdad pasando al polo siguiente etc.
etc.
Respuesta:
$$x \geq -100.530964626003 \wedge x \leq -94.2477801171671$$
$$x \geq -94.2477797298079 \wedge x \leq -94.2477794452815$$
$$x \geq -87.9645947692094 \wedge x \leq -87.9645943586158$$
$$x \geq -87.964593928489 \wedge x \leq -81.6814092565354$$
$$x \geq -81.6814090382277 \wedge x \leq -81.6814084945807$$
$$x \geq -81.6814075578313 \wedge x \leq -75.3982238741744$$
$$x \geq -75.3982231720141 \wedge x \leq -75.3982231045728$$
$$x \geq -69.1150386869085 \wedge x \leq -69.1150379045123$$
$$x \geq -62.8318534787248 \wedge x \leq -62.831852673202$$
$$x \geq -56.5486682426592 \wedge x \leq -56.5486674685864$$
$$x \geq -50.2654829667315 \wedge x \leq -50.265482641087$$
$$x \geq -50.2654822863493 \wedge x \leq -43.9822976246252$$
$$x \geq -43.9822971745392 \wedge x \leq -43.9822967932182$$
$$x \geq -37.6991121287155 \wedge x \leq -37.6991118772631$$
$$x \geq -37.6991113479743 \wedge x \leq -31.4159267157965$$
$$x \geq -31.4159260507536 \wedge x \leq -31.4159260208155$$
$$x \geq -25.1327415297174 \wedge x \leq -25.1327407505866$$
$$x \geq -18.8495563230046 \wedge x \leq -18.8495555173448$$
$$x \geq -18.8495552124105 \wedge x \leq -12.5663710889626$$
$$x \geq -12.5663703112531 \wedge x \leq -6.2831858160515$$
$$x \geq -6.28318555849548 \wedge x \leq -6.2831851275477$$
$$x \geq -4.79511606158654 \cdot 10^{-7} \wedge x \leq 0$$
$$x \geq 3.40772025642919 \cdot 10^{-7} \wedge x \leq 6.28318500093652$$
$$x \geq 6.28318528420851 \wedge x \leq 6.28318579821791$$
$$x \geq 6.28318626747926 \wedge x \leq 12.5663704426592$$
$$x \geq 12.5663710110881 \wedge x \leq 12.5663711301703$$
$$x \geq 18.8495556275525 \wedge x \leq 18.8495564031971$$
$$x \geq 25.1327408328211 \wedge x \leq 25.1327416384075$$
$$x \geq 31.4159260648825 \wedge x \leq 31.4159268459961$$
$$x \geq 37.6991113348642 \wedge x \leq 37.6991115173992$$
$$x \geq 37.6991120311338 \wedge x \leq 43.9822966661001$$
$$x \geq 43.9822971694647 \wedge x \leq 43.9822974733639$$
$$x \geq 50.2654821322586 \wedge x \leq 50.2654824463392$$
$$x \geq 50.2654829439723 \wedge x \leq 56.5486668532011$$
$$x \geq 56.5486676011951 \wedge x \leq 56.5486680806249$$
$$x \geq 56.5486682809363 \wedge x \leq 62.8318527849002$$
$$x \geq 62.8318535568358 \wedge x \leq 69.1150379887504$$
$$x \geq 69.115038794053 \wedge x \leq 75.3982232188727$$
$$x \geq 75.3982240031607 \wedge x \leq 81.6814084860076$$
$$x \geq 81.6814085860518 \wedge x \leq 81.6814091897036$$
$$x \geq 87.9645938121814 \wedge x \leq 87.964594335905$$
$$x \geq 87.9645946044253 \wedge x \leq 94.2477792651059$$
$$x \geq 94.2477796093523 \wedge x \leq 94.2477800892631$$
$$x \geq 100.530964759815 \wedge x \leq 100.530965157364$$