Sr Examen

sin(t)<-2:3 desigualdades

En la desigualdad la incógnita

Solución

Ha introducido [src]
sin(t) < -2/3
$$\sin{\left(t \right)} < - \frac{2}{3}$$
sin(t) < -2/3
Solución detallada
Se da la desigualdad:
$$\sin{\left(t \right)} < - \frac{2}{3}$$
Para resolver esta desigualdad primero hay que resolver la ecuación correspondiente:
$$\sin{\left(t \right)} = - \frac{2}{3}$$
Resolvemos:
Tenemos la ecuación
$$\sin{\left(t \right)} = - \frac{2}{3}$$
cambiamos
$$\sin{\left(t \right)} + \frac{2}{3} = 0$$
$$\sin{\left(t \right)} + \frac{2}{3} = 0$$
Sustituimos
$$w = \sin{\left(t \right)}$$
Transportamos los términos libres (sin w)
del miembro izquierdo al derecho, obtenemos:
$$w = - \frac{2}{3}$$
Obtenemos la respuesta: w = -2/3
hacemos cambio inverso
$$\sin{\left(t \right)} = w$$
sustituimos w:
$$x_{1} = 412.27836527649$$
$$x_{2} = 10.1545056169963$$
$$x_{3} = 93.5180519514668$$
$$x_{4} = 1216.52608459548$$
$$x_{5} = -32.1456541921249$$
$$x_{6} = -65.2437180691587$$
$$x_{7} = 30.686198879671$$
$$x_{8} = -107.54387787828$$
$$x_{9} = -302.322622400847$$
$$x_{10} = -77.8100886835179$$
$$x_{11} = -8.69505030454241$$
$$x_{12} = -38.4288394993045$$
$$x_{13} = 5.55345765095262$$
$$x_{14} = 16.4376909241759$$
$$x_{15} = -88.6943219567412$$
$$x_{16} = 18.1198282653118$$
$$x_{17} = -50.9952101136637$$
$$x_{18} = -57.2783954208432$$
$$x_{19} = 24.4030135724914$$
$$x_{20} = 55.8189401083893$$
$$x_{21} = -14.978235611722$$
$$x_{22} = 85.5527293031514$$
$$x_{23} = 72.9863586887922$$
$$x_{24} = -69.8447660352024$$
$$x_{25} = 11.8366429581322$$
$$x_{26} = 74.6684960299281$$
$$x_{27} = -33.8277915332608$$
$$x_{28} = -101.2606925711$$
$$x_{29} = -7.01291296340655$$
$$x_{30} = -0.729727656226966$$
$$x_{31} = -63.5615807280228$$
$$x_{32} = -84.0932739906975$$
$$x_{33} = 68.3853107227485$$
$$x_{34} = 60.419988074433$$
$$x_{35} = 99.8012372586464$$
$$x_{36} = -13.2960982705861$$
$$x_{37} = -58.9605327619791$$
$$x_{38} = -21.2614209189016$$
$$x_{39} = 29.0040615385351$$
$$x_{40} = 87.2348666442873$$
$$x_{41} = -2.41186499736283$$
$$x_{42} = 62.1021254155689$$
$$x_{43} = -27.5446062260812$$
$$x_{44} = -44.7120248064841$$
$$x_{45} = -94.9775072639208$$
$$x_{46} = 54.1368027672534$$
$$x_{47} = -25.8624688849453$$
$$x_{48} = 98.1190999175106$$
$$x_{49} = -836.393373511112$$
$$x_{50} = 3.87132030981676$$
$$x_{51} = -46.3941621476199$$
$$x_{52} = 22.7208762313555$$
$$x_{53} = 36.9693841868506$$
$$x_{54} = -90.376459297877$$
$$x_{55} = 175.199460944801$$
$$x_{56} = -82.4111366495616$$
$$x_{57} = 91.835914610331$$
$$x_{58} = -1755.42056570047$$
$$x_{59} = -19.5792835777657$$
$$x_{60} = 79.2695439959718$$
$$x_{61} = 49.5357548012097$$
$$x_{62} = 66.7031733816126$$
$$x_{63} = 80.9516813371077$$
$$x_{64} = -40.1109768404403$$
$$x_{65} = 47.8536174600739$$
$$x_{66} = -96.6596446050566$$
$$x_{67} = 43.2525694940301$$
$$x_{68} = -1979.9330994178$$
$$x_{69} = -71.5269033763383$$
$$x_{70} = 41.5704321528943$$
$$x_{71} = -52.6773474547995$$
$$x_{72} = 35.2872468457147$$
$$x_{73} = -76.127951342382$$
$$x_{1} = 412.27836527649$$
$$x_{2} = 10.1545056169963$$
$$x_{3} = 93.5180519514668$$
$$x_{4} = 1216.52608459548$$
$$x_{5} = -32.1456541921249$$
$$x_{6} = -65.2437180691587$$
$$x_{7} = 30.686198879671$$
$$x_{8} = -107.54387787828$$
$$x_{9} = -302.322622400847$$
$$x_{10} = -77.8100886835179$$
$$x_{11} = -8.69505030454241$$
$$x_{12} = -38.4288394993045$$
$$x_{13} = 5.55345765095262$$
$$x_{14} = 16.4376909241759$$
$$x_{15} = -88.6943219567412$$
$$x_{16} = 18.1198282653118$$
$$x_{17} = -50.9952101136637$$
$$x_{18} = -57.2783954208432$$
$$x_{19} = 24.4030135724914$$
$$x_{20} = 55.8189401083893$$
$$x_{21} = -14.978235611722$$
$$x_{22} = 85.5527293031514$$
$$x_{23} = 72.9863586887922$$
$$x_{24} = -69.8447660352024$$
$$x_{25} = 11.8366429581322$$
$$x_{26} = 74.6684960299281$$
$$x_{27} = -33.8277915332608$$
$$x_{28} = -101.2606925711$$
$$x_{29} = -7.01291296340655$$
$$x_{30} = -0.729727656226966$$
$$x_{31} = -63.5615807280228$$
$$x_{32} = -84.0932739906975$$
$$x_{33} = 68.3853107227485$$
$$x_{34} = 60.419988074433$$
$$x_{35} = 99.8012372586464$$
$$x_{36} = -13.2960982705861$$
$$x_{37} = -58.9605327619791$$
$$x_{38} = -21.2614209189016$$
$$x_{39} = 29.0040615385351$$
$$x_{40} = 87.2348666442873$$
$$x_{41} = -2.41186499736283$$
$$x_{42} = 62.1021254155689$$
$$x_{43} = -27.5446062260812$$
$$x_{44} = -44.7120248064841$$
$$x_{45} = -94.9775072639208$$
$$x_{46} = 54.1368027672534$$
$$x_{47} = -25.8624688849453$$
$$x_{48} = 98.1190999175106$$
$$x_{49} = -836.393373511112$$
$$x_{50} = 3.87132030981676$$
$$x_{51} = -46.3941621476199$$
$$x_{52} = 22.7208762313555$$
$$x_{53} = 36.9693841868506$$
$$x_{54} = -90.376459297877$$
$$x_{55} = 175.199460944801$$
$$x_{56} = -82.4111366495616$$
$$x_{57} = 91.835914610331$$
$$x_{58} = -1755.42056570047$$
$$x_{59} = -19.5792835777657$$
$$x_{60} = 79.2695439959718$$
$$x_{61} = 49.5357548012097$$
$$x_{62} = 66.7031733816126$$
$$x_{63} = 80.9516813371077$$
$$x_{64} = -40.1109768404403$$
$$x_{65} = 47.8536174600739$$
$$x_{66} = -96.6596446050566$$
$$x_{67} = 43.2525694940301$$
$$x_{68} = -1979.9330994178$$
$$x_{69} = -71.5269033763383$$
$$x_{70} = 41.5704321528943$$
$$x_{71} = -52.6773474547995$$
$$x_{72} = 35.2872468457147$$
$$x_{73} = -76.127951342382$$
Las raíces dadas
$$x_{68} = -1979.9330994178$$
$$x_{58} = -1755.42056570047$$
$$x_{49} = -836.393373511112$$
$$x_{9} = -302.322622400847$$
$$x_{8} = -107.54387787828$$
$$x_{28} = -101.2606925711$$
$$x_{66} = -96.6596446050566$$
$$x_{45} = -94.9775072639208$$
$$x_{54} = -90.376459297877$$
$$x_{15} = -88.6943219567412$$
$$x_{32} = -84.0932739906975$$
$$x_{56} = -82.4111366495616$$
$$x_{10} = -77.8100886835179$$
$$x_{73} = -76.127951342382$$
$$x_{69} = -71.5269033763383$$
$$x_{24} = -69.8447660352024$$
$$x_{6} = -65.2437180691587$$
$$x_{31} = -63.5615807280228$$
$$x_{37} = -58.9605327619791$$
$$x_{18} = -57.2783954208432$$
$$x_{71} = -52.6773474547995$$
$$x_{17} = -50.9952101136637$$
$$x_{51} = -46.3941621476199$$
$$x_{44} = -44.7120248064841$$
$$x_{64} = -40.1109768404403$$
$$x_{12} = -38.4288394993045$$
$$x_{27} = -33.8277915332608$$
$$x_{5} = -32.1456541921249$$
$$x_{43} = -27.5446062260812$$
$$x_{47} = -25.8624688849453$$
$$x_{38} = -21.2614209189016$$
$$x_{59} = -19.5792835777657$$
$$x_{21} = -14.978235611722$$
$$x_{36} = -13.2960982705861$$
$$x_{11} = -8.69505030454241$$
$$x_{29} = -7.01291296340655$$
$$x_{41} = -2.41186499736283$$
$$x_{30} = -0.729727656226966$$
$$x_{50} = 3.87132030981676$$
$$x_{13} = 5.55345765095262$$
$$x_{2} = 10.1545056169963$$
$$x_{25} = 11.8366429581322$$
$$x_{14} = 16.4376909241759$$
$$x_{16} = 18.1198282653118$$
$$x_{52} = 22.7208762313555$$
$$x_{19} = 24.4030135724914$$
$$x_{39} = 29.0040615385351$$
$$x_{7} = 30.686198879671$$
$$x_{72} = 35.2872468457147$$
$$x_{53} = 36.9693841868506$$
$$x_{70} = 41.5704321528943$$
$$x_{67} = 43.2525694940301$$
$$x_{65} = 47.8536174600739$$
$$x_{61} = 49.5357548012097$$
$$x_{46} = 54.1368027672534$$
$$x_{20} = 55.8189401083893$$
$$x_{34} = 60.419988074433$$
$$x_{42} = 62.1021254155689$$
$$x_{62} = 66.7031733816126$$
$$x_{33} = 68.3853107227485$$
$$x_{23} = 72.9863586887922$$
$$x_{26} = 74.6684960299281$$
$$x_{60} = 79.2695439959718$$
$$x_{63} = 80.9516813371077$$
$$x_{22} = 85.5527293031514$$
$$x_{40} = 87.2348666442873$$
$$x_{57} = 91.835914610331$$
$$x_{3} = 93.5180519514668$$
$$x_{48} = 98.1190999175106$$
$$x_{35} = 99.8012372586464$$
$$x_{55} = 175.199460944801$$
$$x_{1} = 412.27836527649$$
$$x_{4} = 1216.52608459548$$
son puntos de cambio del signo de desigualdad en las soluciones.
Primero definámonos con el signo hasta el punto extremo izquierdo:
$$x_{0} < x_{68}$$
Consideremos, por ejemplo, el punto
$$x_{0} = x_{68} - \frac{1}{10}$$
=
$$-1979.9330994178 + - \frac{1}{10}$$
=
$$-1980.0330994178$$
lo sustituimos en la expresión
$$\sin{\left(t \right)} < - \frac{2}{3}$$
$$\sin{\left(t \right)} < - \frac{2}{3}$$
sin(t) < -2/3

Entonces
$$x < -1979.9330994178$$
no se cumple
significa que una de las soluciones de nuestra ecuación será con:
$$x > -1979.9330994178 \wedge x < -1755.42056570047$$
         _____           _____           _____           _____           _____           _____           _____           _____           _____           _____           _____           _____           _____           _____           _____           _____           _____           _____           _____           _____           _____           _____           _____           _____           _____           _____           _____           _____           _____           _____           _____           _____           _____           _____           _____           _____           _____  
        /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /
-------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-------ο-------
       x68      x58      x49      x9      x8      x28      x66      x45      x54      x15      x32      x56      x10      x73      x69      x24      x6      x31      x37      x18      x71      x17      x51      x44      x64      x12      x27      x5      x43      x47      x38      x59      x21      x36      x11      x29      x41      x30      x50      x13      x2      x25      x14      x16      x52      x19      x39      x7      x72      x53      x70      x67      x65      x61      x46      x20      x34      x42      x62      x33      x23      x26      x60      x63      x22      x40      x57      x3      x48      x35      x55      x1      x4

Recibiremos otras soluciones de la desigualdad pasando al polo siguiente etc.
etc.
Respuesta:
$$x > -1979.9330994178 \wedge x < -1755.42056570047$$
$$x > -836.393373511112 \wedge x < -302.322622400847$$
$$x > -107.54387787828 \wedge x < -101.2606925711$$
$$x > -96.6596446050566 \wedge x < -94.9775072639208$$
$$x > -90.376459297877 \wedge x < -88.6943219567412$$
$$x > -84.0932739906975 \wedge x < -82.4111366495616$$
$$x > -77.8100886835179 \wedge x < -76.127951342382$$
$$x > -71.5269033763383 \wedge x < -69.8447660352024$$
$$x > -65.2437180691587 \wedge x < -63.5615807280228$$
$$x > -58.9605327619791 \wedge x < -57.2783954208432$$
$$x > -52.6773474547995 \wedge x < -50.9952101136637$$
$$x > -46.3941621476199 \wedge x < -44.7120248064841$$
$$x > -40.1109768404403 \wedge x < -38.4288394993045$$
$$x > -33.8277915332608 \wedge x < -32.1456541921249$$
$$x > -27.5446062260812 \wedge x < -25.8624688849453$$
$$x > -21.2614209189016 \wedge x < -19.5792835777657$$
$$x > -14.978235611722 \wedge x < -13.2960982705861$$
$$x > -8.69505030454241 \wedge x < -7.01291296340655$$
$$x > -2.41186499736283 \wedge x < -0.729727656226966$$
$$x > 3.87132030981676 \wedge x < 5.55345765095262$$
$$x > 10.1545056169963 \wedge x < 11.8366429581322$$
$$x > 16.4376909241759 \wedge x < 18.1198282653118$$
$$x > 22.7208762313555 \wedge x < 24.4030135724914$$
$$x > 29.0040615385351 \wedge x < 30.686198879671$$
$$x > 35.2872468457147 \wedge x < 36.9693841868506$$
$$x > 41.5704321528943 \wedge x < 43.2525694940301$$
$$x > 47.8536174600739 \wedge x < 49.5357548012097$$
$$x > 54.1368027672534 \wedge x < 55.8189401083893$$
$$x > 60.419988074433 \wedge x < 62.1021254155689$$
$$x > 66.7031733816126 \wedge x < 68.3853107227485$$
$$x > 72.9863586887922 \wedge x < 74.6684960299281$$
$$x > 79.2695439959718 \wedge x < 80.9516813371077$$
$$x > 85.5527293031514 \wedge x < 87.2348666442873$$
$$x > 91.835914610331 \wedge x < 93.5180519514668$$
$$x > 98.1190999175106 \wedge x < 99.8012372586464$$
$$x > 175.199460944801 \wedge x < 412.27836527649$$
$$x > 1216.52608459548$$
Respuesta rápida [src]
   /          /    ___\                  /    ___\    \
   |          |2*\/ 5 |                  |2*\/ 5 |    |
And|t < - atan|-------| + 2*pi, pi + atan|-------| < t|
   \          \   5   /                  \   5   /    /
$$t < - \operatorname{atan}{\left(\frac{2 \sqrt{5}}{5} \right)} + 2 \pi \wedge \operatorname{atan}{\left(\frac{2 \sqrt{5}}{5} \right)} + \pi < t$$
(pi + atan(2*sqrt(5)/5) < t)∧(t < -atan(2*sqrt(5)/5) + 2*pi)
Respuesta rápida 2 [src]
          /    ___\        /    ___\        
          |2*\/ 5 |        |2*\/ 5 |        
(pi + atan|-------|, - atan|-------| + 2*pi)
          \   5   /        \   5   /        
$$x\ in\ \left(\operatorname{atan}{\left(\frac{2 \sqrt{5}}{5} \right)} + \pi, - \operatorname{atan}{\left(\frac{2 \sqrt{5}}{5} \right)} + 2 \pi\right)$$
x in Interval.open(atan(2*sqrt(5)/5) + pi, -atan(2*sqrt(5)/5) + 2*pi)