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sint<-√2/2 desigualdades

En la desigualdad la incógnita

Solución

Ha introducido [src]
            ___ 
         -\/ 2  
sin(t) < -------
            2   
$$\sin{\left(t \right)} < \frac{\left(-1\right) \sqrt{2}}{2}$$
sin(t) < (-sqrt(2))/2
Solución detallada
Se da la desigualdad:
$$\sin{\left(t \right)} < \frac{\left(-1\right) \sqrt{2}}{2}$$
Para resolver esta desigualdad primero hay que resolver la ecuación correspondiente:
$$\sin{\left(t \right)} = \frac{\left(-1\right) \sqrt{2}}{2}$$
Resolvemos:
$$x_{1} = -27.4889357189107$$
$$x_{2} = 3.92699081698724$$
$$x_{3} = -40.0553063332699$$
$$x_{4} = 47.9092879672443$$
$$x_{5} = 24.3473430653209$$
$$x_{6} = -21.2057504117311$$
$$x_{7} = -77.7544181763474$$
$$x_{8} = 93.4623814442964$$
$$x_{9} = -128.019900633784$$
$$x_{10} = -52.621676947629$$
$$x_{11} = -63.6172512351933$$
$$x_{12} = -71.4712328691678$$
$$x_{13} = 29.0597320457056$$
$$x_{14} = 5.49778714378214$$
$$x_{15} = 85.6083998103219$$
$$x_{16} = 36.9137136796801$$
$$x_{17} = 54.1924732744239$$
$$x_{18} = -13.3517687777566$$
$$x_{19} = 80.8960108299372$$
$$x_{20} = -8.63937979737193$$
$$x_{21} = 68.329640215578$$
$$x_{22} = -33.7721210260903$$
$$x_{23} = -57.3340659280137$$
$$x_{24} = 91.8915851175014$$
$$x_{25} = 35.3429173528852$$
$$x_{26} = 99.7455667514759$$
$$x_{27} = -371.493331286993$$
$$x_{28} = -2865.91789823729$$
$$x_{29} = -215482.625507887$$
$$x_{30} = -58.9048622548086$$
$$x_{31} = -101.316363078271$$
$$x_{32} = 55.7632696012188$$
$$x_{33} = -38.484510006475$$
$$x_{34} = -14.9225651045515$$
$$x_{35} = -44.7676953136546$$
$$x_{36} = 79.3252145031423$$
$$x_{37} = -76.1836218495525$$
$$x_{38} = 30.6305283725005$$
$$x_{39} = -95.0331777710912$$
$$x_{40} = 74.6128255227576$$
$$x_{41} = 10.2101761241668$$
$$x_{42} = 87.1791961371168$$
$$x_{43} = -19.6349540849362$$
$$x_{44} = 60.4756585816035$$
$$x_{45} = -69.9004365423729$$
$$x_{46} = 98.174770424681$$
$$x_{47} = -84.037603483527$$
$$x_{48} = 11.7809724509617$$
$$x_{49} = 73.0420291959627$$
$$x_{50} = -82.4668071567321$$
$$x_{51} = 49.4800842940392$$
$$x_{52} = -32.2013246992954$$
$$x_{53} = -2.35619449019234$$
$$x_{54} = -88.7499924639117$$
$$x_{55} = -25.9181393921158$$
$$x_{56} = 43.1968989868597$$
$$x_{57} = 62.0464549083984$$
$$x_{58} = -96.6039740978861$$
$$x_{59} = -46.3384916404494$$
$$x_{60} = -704.502152567511$$
$$x_{61} = 66.7588438887831$$
$$x_{62} = 22.776546738526$$
$$x_{63} = 41.6261026600648$$
$$x_{64} = -830.165858711103$$
$$x_{65} = 16.4933614313464$$
$$x_{66} = -7.06858347057703$$
$$x_{67} = 18.0641577581413$$
$$x_{68} = -51.0508806208341$$
$$x_{69} = 652.665873783279$$
$$x_{70} = -0.785398163397448$$
$$x_{71} = -65.1880475619882$$
$$x_{72} = -90.3207887907066$$
$$x_{1} = -27.4889357189107$$
$$x_{2} = 3.92699081698724$$
$$x_{3} = -40.0553063332699$$
$$x_{4} = 47.9092879672443$$
$$x_{5} = 24.3473430653209$$
$$x_{6} = -21.2057504117311$$
$$x_{7} = -77.7544181763474$$
$$x_{8} = 93.4623814442964$$
$$x_{9} = -128.019900633784$$
$$x_{10} = -52.621676947629$$
$$x_{11} = -63.6172512351933$$
$$x_{12} = -71.4712328691678$$
$$x_{13} = 29.0597320457056$$
$$x_{14} = 5.49778714378214$$
$$x_{15} = 85.6083998103219$$
$$x_{16} = 36.9137136796801$$
$$x_{17} = 54.1924732744239$$
$$x_{18} = -13.3517687777566$$
$$x_{19} = 80.8960108299372$$
$$x_{20} = -8.63937979737193$$
$$x_{21} = 68.329640215578$$
$$x_{22} = -33.7721210260903$$
$$x_{23} = -57.3340659280137$$
$$x_{24} = 91.8915851175014$$
$$x_{25} = 35.3429173528852$$
$$x_{26} = 99.7455667514759$$
$$x_{27} = -371.493331286993$$
$$x_{28} = -2865.91789823729$$
$$x_{29} = -215482.625507887$$
$$x_{30} = -58.9048622548086$$
$$x_{31} = -101.316363078271$$
$$x_{32} = 55.7632696012188$$
$$x_{33} = -38.484510006475$$
$$x_{34} = -14.9225651045515$$
$$x_{35} = -44.7676953136546$$
$$x_{36} = 79.3252145031423$$
$$x_{37} = -76.1836218495525$$
$$x_{38} = 30.6305283725005$$
$$x_{39} = -95.0331777710912$$
$$x_{40} = 74.6128255227576$$
$$x_{41} = 10.2101761241668$$
$$x_{42} = 87.1791961371168$$
$$x_{43} = -19.6349540849362$$
$$x_{44} = 60.4756585816035$$
$$x_{45} = -69.9004365423729$$
$$x_{46} = 98.174770424681$$
$$x_{47} = -84.037603483527$$
$$x_{48} = 11.7809724509617$$
$$x_{49} = 73.0420291959627$$
$$x_{50} = -82.4668071567321$$
$$x_{51} = 49.4800842940392$$
$$x_{52} = -32.2013246992954$$
$$x_{53} = -2.35619449019234$$
$$x_{54} = -88.7499924639117$$
$$x_{55} = -25.9181393921158$$
$$x_{56} = 43.1968989868597$$
$$x_{57} = 62.0464549083984$$
$$x_{58} = -96.6039740978861$$
$$x_{59} = -46.3384916404494$$
$$x_{60} = -704.502152567511$$
$$x_{61} = 66.7588438887831$$
$$x_{62} = 22.776546738526$$
$$x_{63} = 41.6261026600648$$
$$x_{64} = -830.165858711103$$
$$x_{65} = 16.4933614313464$$
$$x_{66} = -7.06858347057703$$
$$x_{67} = 18.0641577581413$$
$$x_{68} = -51.0508806208341$$
$$x_{69} = 652.665873783279$$
$$x_{70} = -0.785398163397448$$
$$x_{71} = -65.1880475619882$$
$$x_{72} = -90.3207887907066$$
Las raíces dadas
$$x_{29} = -215482.625507887$$
$$x_{28} = -2865.91789823729$$
$$x_{64} = -830.165858711103$$
$$x_{60} = -704.502152567511$$
$$x_{27} = -371.493331286993$$
$$x_{9} = -128.019900633784$$
$$x_{31} = -101.316363078271$$
$$x_{58} = -96.6039740978861$$
$$x_{39} = -95.0331777710912$$
$$x_{72} = -90.3207887907066$$
$$x_{54} = -88.7499924639117$$
$$x_{47} = -84.037603483527$$
$$x_{50} = -82.4668071567321$$
$$x_{7} = -77.7544181763474$$
$$x_{37} = -76.1836218495525$$
$$x_{12} = -71.4712328691678$$
$$x_{45} = -69.9004365423729$$
$$x_{71} = -65.1880475619882$$
$$x_{11} = -63.6172512351933$$
$$x_{30} = -58.9048622548086$$
$$x_{23} = -57.3340659280137$$
$$x_{10} = -52.621676947629$$
$$x_{68} = -51.0508806208341$$
$$x_{59} = -46.3384916404494$$
$$x_{35} = -44.7676953136546$$
$$x_{3} = -40.0553063332699$$
$$x_{33} = -38.484510006475$$
$$x_{22} = -33.7721210260903$$
$$x_{52} = -32.2013246992954$$
$$x_{1} = -27.4889357189107$$
$$x_{55} = -25.9181393921158$$
$$x_{6} = -21.2057504117311$$
$$x_{43} = -19.6349540849362$$
$$x_{34} = -14.9225651045515$$
$$x_{18} = -13.3517687777566$$
$$x_{20} = -8.63937979737193$$
$$x_{66} = -7.06858347057703$$
$$x_{53} = -2.35619449019234$$
$$x_{70} = -0.785398163397448$$
$$x_{2} = 3.92699081698724$$
$$x_{14} = 5.49778714378214$$
$$x_{41} = 10.2101761241668$$
$$x_{48} = 11.7809724509617$$
$$x_{65} = 16.4933614313464$$
$$x_{67} = 18.0641577581413$$
$$x_{62} = 22.776546738526$$
$$x_{5} = 24.3473430653209$$
$$x_{13} = 29.0597320457056$$
$$x_{38} = 30.6305283725005$$
$$x_{25} = 35.3429173528852$$
$$x_{16} = 36.9137136796801$$
$$x_{63} = 41.6261026600648$$
$$x_{56} = 43.1968989868597$$
$$x_{4} = 47.9092879672443$$
$$x_{51} = 49.4800842940392$$
$$x_{17} = 54.1924732744239$$
$$x_{32} = 55.7632696012188$$
$$x_{44} = 60.4756585816035$$
$$x_{57} = 62.0464549083984$$
$$x_{61} = 66.7588438887831$$
$$x_{21} = 68.329640215578$$
$$x_{49} = 73.0420291959627$$
$$x_{40} = 74.6128255227576$$
$$x_{36} = 79.3252145031423$$
$$x_{19} = 80.8960108299372$$
$$x_{15} = 85.6083998103219$$
$$x_{42} = 87.1791961371168$$
$$x_{24} = 91.8915851175014$$
$$x_{8} = 93.4623814442964$$
$$x_{46} = 98.174770424681$$
$$x_{26} = 99.7455667514759$$
$$x_{69} = 652.665873783279$$
son puntos de cambio del signo de desigualdad en las soluciones.
Primero definámonos con el signo hasta el punto extremo izquierdo:
$$x_{0} < x_{29}$$
Consideremos, por ejemplo, el punto
$$x_{0} = x_{29} - \frac{1}{10}$$
=
$$-215482.625507887 + - \frac{1}{10}$$
=
$$-215482.725507887$$
lo sustituimos en la expresión
$$\sin{\left(t \right)} < \frac{\left(-1\right) \sqrt{2}}{2}$$
$$\sin{\left(t \right)} < \frac{\left(-1\right) \sqrt{2}}{2}$$
            ___ 
         -\/ 2  
sin(t) < -------
            2   
         

Entonces
$$x < -215482.625507887$$
no se cumple
significa que una de las soluciones de nuestra ecuación será con:
$$x > -215482.625507887 \wedge x < -2865.91789823729$$
         _____           _____           _____           _____           _____           _____           _____           _____           _____           _____           _____           _____           _____           _____           _____           _____           _____           _____           _____           _____           _____           _____           _____           _____           _____           _____           _____           _____           _____           _____           _____           _____           _____           _____           _____           _____  
        /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \  
-------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-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------
       x29      x28      x64      x60      x27      x9      x31      x58      x39      x72      x54      x47      x50      x7      x37      x12      x45      x71      x11      x30      x23      x10      x68      x59      x35      x3      x33      x22      x52      x1      x55      x6      x43      x34      x18      x20      x66      x53      x70      x2      x14      x41      x48      x65      x67      x62      x5      x13      x38      x25      x16      x63      x56      x4      x51      x17      x32      x44      x57      x61      x21      x49      x40      x36      x19      x15      x42      x24      x8      x46      x26      x69

Recibiremos otras soluciones de la desigualdad pasando al polo siguiente etc.
etc.
Respuesta:
$$x > -215482.625507887 \wedge x < -2865.91789823729$$
$$x > -830.165858711103 \wedge x < -704.502152567511$$
$$x > -371.493331286993 \wedge x < -128.019900633784$$
$$x > -101.316363078271 \wedge x < -96.6039740978861$$
$$x > -95.0331777710912 \wedge x < -90.3207887907066$$
$$x > -88.7499924639117 \wedge x < -84.037603483527$$
$$x > -82.4668071567321 \wedge x < -77.7544181763474$$
$$x > -76.1836218495525 \wedge x < -71.4712328691678$$
$$x > -69.9004365423729 \wedge x < -65.1880475619882$$
$$x > -63.6172512351933 \wedge x < -58.9048622548086$$
$$x > -57.3340659280137 \wedge x < -52.621676947629$$
$$x > -51.0508806208341 \wedge x < -46.3384916404494$$
$$x > -44.7676953136546 \wedge x < -40.0553063332699$$
$$x > -38.484510006475 \wedge x < -33.7721210260903$$
$$x > -32.2013246992954 \wedge x < -27.4889357189107$$
$$x > -25.9181393921158 \wedge x < -21.2057504117311$$
$$x > -19.6349540849362 \wedge x < -14.9225651045515$$
$$x > -13.3517687777566 \wedge x < -8.63937979737193$$
$$x > -7.06858347057703 \wedge x < -2.35619449019234$$
$$x > -0.785398163397448 \wedge x < 3.92699081698724$$
$$x > 5.49778714378214 \wedge x < 10.2101761241668$$
$$x > 11.7809724509617 \wedge x < 16.4933614313464$$
$$x > 18.0641577581413 \wedge x < 22.776546738526$$
$$x > 24.3473430653209 \wedge x < 29.0597320457056$$
$$x > 30.6305283725005 \wedge x < 35.3429173528852$$
$$x > 36.9137136796801 \wedge x < 41.6261026600648$$
$$x > 43.1968989868597 \wedge x < 47.9092879672443$$
$$x > 49.4800842940392 \wedge x < 54.1924732744239$$
$$x > 55.7632696012188 \wedge x < 60.4756585816035$$
$$x > 62.0464549083984 \wedge x < 66.7588438887831$$
$$x > 68.329640215578 \wedge x < 73.0420291959627$$
$$x > 74.6128255227576 \wedge x < 79.3252145031423$$
$$x > 80.8960108299372 \wedge x < 85.6083998103219$$
$$x > 87.1791961371168 \wedge x < 91.8915851175014$$
$$x > 93.4623814442964 \wedge x < 98.174770424681$$
$$x > 99.7455667514759 \wedge x < 652.665873783279$$
Respuesta rápida [src]
   /5*pi          7*pi\
And|---- < x, x < ----|
   \ 4             4  /
$$\frac{5 \pi}{4} < x \wedge x < \frac{7 \pi}{4}$$
(5*pi/4 < x)∧(x < 7*pi/4)
Respuesta rápida 2 [src]
 5*pi  7*pi 
(----, ----)
  4     4   
$$x\ in\ \left(\frac{5 \pi}{4}, \frac{7 \pi}{4}\right)$$
x in Interval.open(5*pi/4, 7*pi/4)