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cos(t)>=-sqrt(2)/2 desigualdades

En la desigualdad la incógnita

Solución

Ha introducido [src]
             ___ 
          -\/ 2  
cos(t) >= -------
             2   
$$\cos{\left(t \right)} \geq \frac{\left(-1\right) \sqrt{2}}{2}$$
cos(t) >= (-sqrt(2))/2
Solución detallada
Se da la desigualdad:
$$\cos{\left(t \right)} \geq \frac{\left(-1\right) \sqrt{2}}{2}$$
Para resolver esta desigualdad primero hay que resolver la ecuación correspondiente:
$$\cos{\left(t \right)} = \frac{\left(-1\right) \sqrt{2}}{2}$$
Resolvemos:
$$x_{1} = -27.4889357189107$$
$$x_{2} = 3.92699081698724$$
$$x_{3} = -40.0553063332699$$
$$x_{4} = -22.776546738526$$
$$x_{5} = -90.3207887907066$$
$$x_{6} = 47.9092879672443$$
$$x_{7} = 65.1880475619882$$
$$x_{8} = -21.2057504117311$$
$$x_{9} = -79.3252145031423$$
$$x_{10} = 40.0553063332699$$
$$x_{11} = -29.0597320457056$$
$$x_{12} = -77.7544181763474$$
$$x_{13} = -52.621676947629$$
$$x_{14} = 84.037603483527$$
$$x_{15} = -71.4712328691678$$
$$x_{16} = 85.6083998103219$$
$$x_{17} = 54.1924732744239$$
$$x_{18} = -10.2101761241668$$
$$x_{19} = -8.63937979737193$$
$$x_{20} = -33.7721210260903$$
$$x_{21} = 21.2057504117311$$
$$x_{22} = 35.3429173528852$$
$$x_{23} = -58.9048622548086$$
$$x_{24} = 104.457955731861$$
$$x_{25} = -41.6261026600648$$
$$x_{26} = 2.35619449019234$$
$$x_{27} = -73.0420291959627$$
$$x_{28} = -14.9225651045515$$
$$x_{29} = -66.7588438887831$$
$$x_{30} = 58.9048622548086$$
$$x_{31} = 79.3252145031423$$
$$x_{32} = -16.4933614313464$$
$$x_{33} = 14.9225651045515$$
$$x_{34} = -65.1880475619882$$
$$x_{35} = 10.2101761241668$$
$$x_{36} = -54.1924732744239$$
$$x_{37} = 71.4712328691678$$
$$x_{38} = 96.6039740978861$$
$$x_{39} = 60.4756585816035$$
$$x_{40} = -47.9092879672443$$
$$x_{41} = 98.174770424681$$
$$x_{42} = 8.63937979737193$$
$$x_{43} = 77.7544181763474$$
$$x_{44} = -335.36501577071$$
$$x_{45} = -60.4756585816035$$
$$x_{46} = 39349.2333843756$$
$$x_{47} = 73.0420291959627$$
$$x_{48} = -85.6083998103219$$
$$x_{49} = -35.3429173528852$$
$$x_{50} = 255.254403104171$$
$$x_{51} = -91.8915851175014$$
$$x_{52} = -2.35619449019234$$
$$x_{53} = 90.3207887907066$$
$$x_{54} = 27.4889357189107$$
$$x_{55} = -96.6039740978861$$
$$x_{56} = -98.174770424681$$
$$x_{57} = 33.7721210260903$$
$$x_{58} = -46.3384916404494$$
$$x_{59} = 22.776546738526$$
$$x_{60} = 66.7588438887831$$
$$x_{61} = 2584.745355741$$
$$x_{62} = 46.3384916404494$$
$$x_{63} = 41.6261026600648$$
$$x_{64} = 16.4933614313464$$
$$x_{65} = -3.92699081698724$$
$$x_{66} = -84.037603483527$$
$$x_{67} = 29.0597320457056$$
$$x_{68} = 52.621676947629$$
$$x_{69} = 91.8915851175014$$
$$x_{1} = -27.4889357189107$$
$$x_{2} = 3.92699081698724$$
$$x_{3} = -40.0553063332699$$
$$x_{4} = -22.776546738526$$
$$x_{5} = -90.3207887907066$$
$$x_{6} = 47.9092879672443$$
$$x_{7} = 65.1880475619882$$
$$x_{8} = -21.2057504117311$$
$$x_{9} = -79.3252145031423$$
$$x_{10} = 40.0553063332699$$
$$x_{11} = -29.0597320457056$$
$$x_{12} = -77.7544181763474$$
$$x_{13} = -52.621676947629$$
$$x_{14} = 84.037603483527$$
$$x_{15} = -71.4712328691678$$
$$x_{16} = 85.6083998103219$$
$$x_{17} = 54.1924732744239$$
$$x_{18} = -10.2101761241668$$
$$x_{19} = -8.63937979737193$$
$$x_{20} = -33.7721210260903$$
$$x_{21} = 21.2057504117311$$
$$x_{22} = 35.3429173528852$$
$$x_{23} = -58.9048622548086$$
$$x_{24} = 104.457955731861$$
$$x_{25} = -41.6261026600648$$
$$x_{26} = 2.35619449019234$$
$$x_{27} = -73.0420291959627$$
$$x_{28} = -14.9225651045515$$
$$x_{29} = -66.7588438887831$$
$$x_{30} = 58.9048622548086$$
$$x_{31} = 79.3252145031423$$
$$x_{32} = -16.4933614313464$$
$$x_{33} = 14.9225651045515$$
$$x_{34} = -65.1880475619882$$
$$x_{35} = 10.2101761241668$$
$$x_{36} = -54.1924732744239$$
$$x_{37} = 71.4712328691678$$
$$x_{38} = 96.6039740978861$$
$$x_{39} = 60.4756585816035$$
$$x_{40} = -47.9092879672443$$
$$x_{41} = 98.174770424681$$
$$x_{42} = 8.63937979737193$$
$$x_{43} = 77.7544181763474$$
$$x_{44} = -335.36501577071$$
$$x_{45} = -60.4756585816035$$
$$x_{46} = 39349.2333843756$$
$$x_{47} = 73.0420291959627$$
$$x_{48} = -85.6083998103219$$
$$x_{49} = -35.3429173528852$$
$$x_{50} = 255.254403104171$$
$$x_{51} = -91.8915851175014$$
$$x_{52} = -2.35619449019234$$
$$x_{53} = 90.3207887907066$$
$$x_{54} = 27.4889357189107$$
$$x_{55} = -96.6039740978861$$
$$x_{56} = -98.174770424681$$
$$x_{57} = 33.7721210260903$$
$$x_{58} = -46.3384916404494$$
$$x_{59} = 22.776546738526$$
$$x_{60} = 66.7588438887831$$
$$x_{61} = 2584.745355741$$
$$x_{62} = 46.3384916404494$$
$$x_{63} = 41.6261026600648$$
$$x_{64} = 16.4933614313464$$
$$x_{65} = -3.92699081698724$$
$$x_{66} = -84.037603483527$$
$$x_{67} = 29.0597320457056$$
$$x_{68} = 52.621676947629$$
$$x_{69} = 91.8915851175014$$
Las raíces dadas
$$x_{44} = -335.36501577071$$
$$x_{56} = -98.174770424681$$
$$x_{55} = -96.6039740978861$$
$$x_{51} = -91.8915851175014$$
$$x_{5} = -90.3207887907066$$
$$x_{48} = -85.6083998103219$$
$$x_{66} = -84.037603483527$$
$$x_{9} = -79.3252145031423$$
$$x_{12} = -77.7544181763474$$
$$x_{27} = -73.0420291959627$$
$$x_{15} = -71.4712328691678$$
$$x_{29} = -66.7588438887831$$
$$x_{34} = -65.1880475619882$$
$$x_{45} = -60.4756585816035$$
$$x_{23} = -58.9048622548086$$
$$x_{36} = -54.1924732744239$$
$$x_{13} = -52.621676947629$$
$$x_{40} = -47.9092879672443$$
$$x_{58} = -46.3384916404494$$
$$x_{25} = -41.6261026600648$$
$$x_{3} = -40.0553063332699$$
$$x_{49} = -35.3429173528852$$
$$x_{20} = -33.7721210260903$$
$$x_{11} = -29.0597320457056$$
$$x_{1} = -27.4889357189107$$
$$x_{4} = -22.776546738526$$
$$x_{8} = -21.2057504117311$$
$$x_{32} = -16.4933614313464$$
$$x_{28} = -14.9225651045515$$
$$x_{18} = -10.2101761241668$$
$$x_{19} = -8.63937979737193$$
$$x_{65} = -3.92699081698724$$
$$x_{52} = -2.35619449019234$$
$$x_{26} = 2.35619449019234$$
$$x_{2} = 3.92699081698724$$
$$x_{42} = 8.63937979737193$$
$$x_{35} = 10.2101761241668$$
$$x_{33} = 14.9225651045515$$
$$x_{64} = 16.4933614313464$$
$$x_{21} = 21.2057504117311$$
$$x_{59} = 22.776546738526$$
$$x_{54} = 27.4889357189107$$
$$x_{67} = 29.0597320457056$$
$$x_{57} = 33.7721210260903$$
$$x_{22} = 35.3429173528852$$
$$x_{10} = 40.0553063332699$$
$$x_{63} = 41.6261026600648$$
$$x_{62} = 46.3384916404494$$
$$x_{6} = 47.9092879672443$$
$$x_{68} = 52.621676947629$$
$$x_{17} = 54.1924732744239$$
$$x_{30} = 58.9048622548086$$
$$x_{39} = 60.4756585816035$$
$$x_{7} = 65.1880475619882$$
$$x_{60} = 66.7588438887831$$
$$x_{37} = 71.4712328691678$$
$$x_{47} = 73.0420291959627$$
$$x_{43} = 77.7544181763474$$
$$x_{31} = 79.3252145031423$$
$$x_{14} = 84.037603483527$$
$$x_{16} = 85.6083998103219$$
$$x_{53} = 90.3207887907066$$
$$x_{69} = 91.8915851175014$$
$$x_{38} = 96.6039740978861$$
$$x_{41} = 98.174770424681$$
$$x_{24} = 104.457955731861$$
$$x_{50} = 255.254403104171$$
$$x_{61} = 2584.745355741$$
$$x_{46} = 39349.2333843756$$
son puntos de cambio del signo de desigualdad en las soluciones.
Primero definámonos con el signo hasta el punto extremo izquierdo:
$$x_{0} \leq x_{44}$$
Consideremos, por ejemplo, el punto
$$x_{0} = x_{44} - \frac{1}{10}$$
=
$$-335.36501577071 + - \frac{1}{10}$$
=
$$-335.46501577071$$
lo sustituimos en la expresión
$$\cos{\left(t \right)} \geq \frac{\left(-1\right) \sqrt{2}}{2}$$
$$\cos{\left(t \right)} \geq \frac{\left(-1\right) \sqrt{2}}{2}$$
             ___ 
          -\/ 2  
cos(t) >= -------
             2   
          

Entonces
$$x \leq -335.36501577071$$
no se cumple
significa que una de las soluciones de nuestra ecuación será con:
$$x \geq -335.36501577071 \wedge x \leq -98.174770424681$$
         _____           _____           _____           _____           _____           _____           _____           _____           _____           _____           _____           _____           _____           _____           _____           _____           _____           _____           _____           _____           _____           _____           _____           _____           _____           _____           _____           _____           _____           _____           _____           _____           _____           _____           _____  
        /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /
-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------
       x44      x56      x55      x51      x5      x48      x66      x9      x12      x27      x15      x29      x34      x45      x23      x36      x13      x40      x58      x25      x3      x49      x20      x11      x1      x4      x8      x32      x28      x18      x19      x65      x52      x26      x2      x42      x35      x33      x64      x21      x59      x54      x67      x57      x22      x10      x63      x62      x6      x68      x17      x30      x39      x7      x60      x37      x47      x43      x31      x14      x16      x53      x69      x38      x41      x24      x50      x61      x46

Recibiremos otras soluciones de la desigualdad pasando al polo siguiente etc.
etc.
Respuesta:
$$x \geq -335.36501577071 \wedge x \leq -98.174770424681$$
$$x \geq -96.6039740978861 \wedge x \leq -91.8915851175014$$
$$x \geq -90.3207887907066 \wedge x \leq -85.6083998103219$$
$$x \geq -84.037603483527 \wedge x \leq -79.3252145031423$$
$$x \geq -77.7544181763474 \wedge x \leq -73.0420291959627$$
$$x \geq -71.4712328691678 \wedge x \leq -66.7588438887831$$
$$x \geq -65.1880475619882 \wedge x \leq -60.4756585816035$$
$$x \geq -58.9048622548086 \wedge x \leq -54.1924732744239$$
$$x \geq -52.621676947629 \wedge x \leq -47.9092879672443$$
$$x \geq -46.3384916404494 \wedge x \leq -41.6261026600648$$
$$x \geq -40.0553063332699 \wedge x \leq -35.3429173528852$$
$$x \geq -33.7721210260903 \wedge x \leq -29.0597320457056$$
$$x \geq -27.4889357189107 \wedge x \leq -22.776546738526$$
$$x \geq -21.2057504117311 \wedge x \leq -16.4933614313464$$
$$x \geq -14.9225651045515 \wedge x \leq -10.2101761241668$$
$$x \geq -8.63937979737193 \wedge x \leq -3.92699081698724$$
$$x \geq -2.35619449019234 \wedge x \leq 2.35619449019234$$
$$x \geq 3.92699081698724 \wedge x \leq 8.63937979737193$$
$$x \geq 10.2101761241668 \wedge x \leq 14.9225651045515$$
$$x \geq 16.4933614313464 \wedge x \leq 21.2057504117311$$
$$x \geq 22.776546738526 \wedge x \leq 27.4889357189107$$
$$x \geq 29.0597320457056 \wedge x \leq 33.7721210260903$$
$$x \geq 35.3429173528852 \wedge x \leq 40.0553063332699$$
$$x \geq 41.6261026600648 \wedge x \leq 46.3384916404494$$
$$x \geq 47.9092879672443 \wedge x \leq 52.621676947629$$
$$x \geq 54.1924732744239 \wedge x \leq 58.9048622548086$$
$$x \geq 60.4756585816035 \wedge x \leq 65.1880475619882$$
$$x \geq 66.7588438887831 \wedge x \leq 71.4712328691678$$
$$x \geq 73.0420291959627 \wedge x \leq 77.7544181763474$$
$$x \geq 79.3252145031423 \wedge x \leq 84.037603483527$$
$$x \geq 85.6083998103219 \wedge x \leq 90.3207887907066$$
$$x \geq 91.8915851175014 \wedge x \leq 96.6039740978861$$
$$x \geq 98.174770424681 \wedge x \leq 104.457955731861$$
$$x \geq 255.254403104171 \wedge x \leq 2584.745355741$$
$$x \geq 39349.2333843756$$
Respuesta rápida [src]
  /   /             3*pi\     /5*pi                \\
Or|And|0 <= x, x <= ----|, And|---- <= x, x <= 2*pi||
  \   \              4  /     \ 4                  //
$$\left(0 \leq x \wedge x \leq \frac{3 \pi}{4}\right) \vee \left(\frac{5 \pi}{4} \leq x \wedge x \leq 2 \pi\right)$$
((0 <= x)∧(x <= 3*pi/4))∨((5*pi/4 <= x)∧(x <= 2*pi))
Respuesta rápida 2 [src]
    3*pi     5*pi       
[0, ----] U [----, 2*pi]
     4        4         
$$x\ in\ \left[0, \frac{3 \pi}{4}\right] \cup \left[\frac{5 \pi}{4}, 2 \pi\right]$$
x in Union(Interval(0, 3*pi/4), Interval(5*pi/4, 2*pi))