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sint<1/2 desigualdades

En la desigualdad la incógnita

Solución

Ha introducido [src]
sin(t) < 1/2
$$\sin{\left(t \right)} < \frac{1}{2}$$
sin(t) < 1/2
Solución detallada
Se da la desigualdad:
$$\sin{\left(t \right)} < \frac{1}{2}$$
Para resolver esta desigualdad primero hay que resolver la ecuación correspondiente:
$$\sin{\left(t \right)} = \frac{1}{2}$$
Resolvemos:
Tenemos la ecuación
$$\sin{\left(t \right)} = \frac{1}{2}$$
cambiamos
$$\sin{\left(t \right)} - \frac{1}{2} = 0$$
$$\sin{\left(t \right)} - \frac{1}{2} = 0$$
Sustituimos
$$w = \sin{\left(t \right)}$$
Transportamos los términos libres (sin w)
del miembro izquierdo al derecho, obtenemos:
$$w = \frac{1}{2}$$
Obtenemos la respuesta: w = 1/2
hacemos cambio inverso
$$\sin{\left(t \right)} = w$$
sustituimos w:
$$x_{1} = 84.2994028713261$$
$$x_{2} = -93.7241808320955$$
$$x_{3} = 65.4498469497874$$
$$x_{4} = -627.79493194236$$
$$x_{5} = 63.3554518473942$$
$$x_{6} = -5.75958653158129$$
$$x_{7} = -22.5147473507269$$
$$x_{8} = 57.0722665402146$$
$$x_{9} = 38.2227106186758$$
$$x_{10} = -4454.25478401473$$
$$x_{11} = -60.2138591938044$$
$$x_{12} = -62.3082542961976$$
$$x_{13} = 101.054563690472$$
$$x_{14} = -41.3643032722656$$
$$x_{15} = -9.94837673636768$$
$$x_{16} = 8.90117918517108$$
$$x_{17} = -47.6474885794452$$
$$x_{18} = -28.7979326579064$$
$$x_{19} = -37.1755130674792$$
$$x_{20} = 88.4881930761125$$
$$x_{21} = -100.007366139275$$
$$x_{22} = 31.9395253114962$$
$$x_{23} = 46.6002910282486$$
$$x_{24} = 2.61799387799149$$
$$x_{25} = 50.789081233035$$
$$x_{26} = -74.8746249105567$$
$$x_{27} = 52.8834763354282$$
$$x_{28} = -18.3259571459405$$
$$x_{29} = 94.7713783832921$$
$$x_{30} = 40.317105721069$$
$$x_{31} = 17438.4572213013$$
$$x_{32} = 44.5058959258554$$
$$x_{33} = -68.5914396033772$$
$$x_{34} = -85.3466004225227$$
$$x_{35} = -81.1578102177363$$
$$x_{36} = 13.0899693899575$$
$$x_{37} = -24.60914245312$$
$$x_{38} = 69.6386371545737$$
$$x_{39} = -16.2315620435473$$
$$x_{40} = -87.4409955249159$$
$$x_{41} = -53.9306738866248$$
$$x_{42} = -97.9129710368819$$
$$x_{43} = 82.2050077689329$$
$$x_{44} = 25.6563400043166$$
$$x_{45} = 59.1666616426078$$
$$x_{46} = -66.497044500984$$
$$x_{47} = 19.3731546971371$$
$$x_{48} = 15.1843644923507$$
$$x_{49} = 96.8657734856853$$
$$x_{50} = -43.4586983746588$$
$$x_{51} = 78.0162175641465$$
$$x_{52} = 90.5825881785057$$
$$x_{53} = -3.66519142918809$$
$$x_{54} = -49.7418836818384$$
$$x_{55} = 138.753675533549$$
$$x_{56} = -79.0634151153431$$
$$x_{57} = -56.025068989018$$
$$x_{58} = 71.733032256967$$
$$x_{59} = 21.4675497995303$$
$$x_{60} = 75.9218224617533$$
$$x_{61} = 0.523598775598299$$
$$x_{62} = 34.0339204138894$$
$$x_{63} = -12.0427718387609$$
$$x_{64} = 134.564885328763$$
$$x_{65} = -91.6297857297023$$
$$x_{66} = -30.8923277602996$$
$$x_{67} = -2650.98060085419$$
$$x_{68} = -72.7802298081635$$
$$x_{69} = 27.7507351067098$$
$$x_{70} = -35.081117965086$$
$$x_{71} = 6.80678408277789$$
$$x_{1} = 84.2994028713261$$
$$x_{2} = -93.7241808320955$$
$$x_{3} = 65.4498469497874$$
$$x_{4} = -627.79493194236$$
$$x_{5} = 63.3554518473942$$
$$x_{6} = -5.75958653158129$$
$$x_{7} = -22.5147473507269$$
$$x_{8} = 57.0722665402146$$
$$x_{9} = 38.2227106186758$$
$$x_{10} = -4454.25478401473$$
$$x_{11} = -60.2138591938044$$
$$x_{12} = -62.3082542961976$$
$$x_{13} = 101.054563690472$$
$$x_{14} = -41.3643032722656$$
$$x_{15} = -9.94837673636768$$
$$x_{16} = 8.90117918517108$$
$$x_{17} = -47.6474885794452$$
$$x_{18} = -28.7979326579064$$
$$x_{19} = -37.1755130674792$$
$$x_{20} = 88.4881930761125$$
$$x_{21} = -100.007366139275$$
$$x_{22} = 31.9395253114962$$
$$x_{23} = 46.6002910282486$$
$$x_{24} = 2.61799387799149$$
$$x_{25} = 50.789081233035$$
$$x_{26} = -74.8746249105567$$
$$x_{27} = 52.8834763354282$$
$$x_{28} = -18.3259571459405$$
$$x_{29} = 94.7713783832921$$
$$x_{30} = 40.317105721069$$
$$x_{31} = 17438.4572213013$$
$$x_{32} = 44.5058959258554$$
$$x_{33} = -68.5914396033772$$
$$x_{34} = -85.3466004225227$$
$$x_{35} = -81.1578102177363$$
$$x_{36} = 13.0899693899575$$
$$x_{37} = -24.60914245312$$
$$x_{38} = 69.6386371545737$$
$$x_{39} = -16.2315620435473$$
$$x_{40} = -87.4409955249159$$
$$x_{41} = -53.9306738866248$$
$$x_{42} = -97.9129710368819$$
$$x_{43} = 82.2050077689329$$
$$x_{44} = 25.6563400043166$$
$$x_{45} = 59.1666616426078$$
$$x_{46} = -66.497044500984$$
$$x_{47} = 19.3731546971371$$
$$x_{48} = 15.1843644923507$$
$$x_{49} = 96.8657734856853$$
$$x_{50} = -43.4586983746588$$
$$x_{51} = 78.0162175641465$$
$$x_{52} = 90.5825881785057$$
$$x_{53} = -3.66519142918809$$
$$x_{54} = -49.7418836818384$$
$$x_{55} = 138.753675533549$$
$$x_{56} = -79.0634151153431$$
$$x_{57} = -56.025068989018$$
$$x_{58} = 71.733032256967$$
$$x_{59} = 21.4675497995303$$
$$x_{60} = 75.9218224617533$$
$$x_{61} = 0.523598775598299$$
$$x_{62} = 34.0339204138894$$
$$x_{63} = -12.0427718387609$$
$$x_{64} = 134.564885328763$$
$$x_{65} = -91.6297857297023$$
$$x_{66} = -30.8923277602996$$
$$x_{67} = -2650.98060085419$$
$$x_{68} = -72.7802298081635$$
$$x_{69} = 27.7507351067098$$
$$x_{70} = -35.081117965086$$
$$x_{71} = 6.80678408277789$$
Las raíces dadas
$$x_{10} = -4454.25478401473$$
$$x_{67} = -2650.98060085419$$
$$x_{4} = -627.79493194236$$
$$x_{21} = -100.007366139275$$
$$x_{42} = -97.9129710368819$$
$$x_{2} = -93.7241808320955$$
$$x_{65} = -91.6297857297023$$
$$x_{40} = -87.4409955249159$$
$$x_{34} = -85.3466004225227$$
$$x_{35} = -81.1578102177363$$
$$x_{56} = -79.0634151153431$$
$$x_{26} = -74.8746249105567$$
$$x_{68} = -72.7802298081635$$
$$x_{33} = -68.5914396033772$$
$$x_{46} = -66.497044500984$$
$$x_{12} = -62.3082542961976$$
$$x_{11} = -60.2138591938044$$
$$x_{57} = -56.025068989018$$
$$x_{41} = -53.9306738866248$$
$$x_{54} = -49.7418836818384$$
$$x_{17} = -47.6474885794452$$
$$x_{50} = -43.4586983746588$$
$$x_{14} = -41.3643032722656$$
$$x_{19} = -37.1755130674792$$
$$x_{70} = -35.081117965086$$
$$x_{66} = -30.8923277602996$$
$$x_{18} = -28.7979326579064$$
$$x_{37} = -24.60914245312$$
$$x_{7} = -22.5147473507269$$
$$x_{28} = -18.3259571459405$$
$$x_{39} = -16.2315620435473$$
$$x_{63} = -12.0427718387609$$
$$x_{15} = -9.94837673636768$$
$$x_{6} = -5.75958653158129$$
$$x_{53} = -3.66519142918809$$
$$x_{61} = 0.523598775598299$$
$$x_{24} = 2.61799387799149$$
$$x_{71} = 6.80678408277789$$
$$x_{16} = 8.90117918517108$$
$$x_{36} = 13.0899693899575$$
$$x_{48} = 15.1843644923507$$
$$x_{47} = 19.3731546971371$$
$$x_{59} = 21.4675497995303$$
$$x_{44} = 25.6563400043166$$
$$x_{69} = 27.7507351067098$$
$$x_{22} = 31.9395253114962$$
$$x_{62} = 34.0339204138894$$
$$x_{9} = 38.2227106186758$$
$$x_{30} = 40.317105721069$$
$$x_{32} = 44.5058959258554$$
$$x_{23} = 46.6002910282486$$
$$x_{25} = 50.789081233035$$
$$x_{27} = 52.8834763354282$$
$$x_{8} = 57.0722665402146$$
$$x_{45} = 59.1666616426078$$
$$x_{5} = 63.3554518473942$$
$$x_{3} = 65.4498469497874$$
$$x_{38} = 69.6386371545737$$
$$x_{58} = 71.733032256967$$
$$x_{60} = 75.9218224617533$$
$$x_{51} = 78.0162175641465$$
$$x_{43} = 82.2050077689329$$
$$x_{1} = 84.2994028713261$$
$$x_{20} = 88.4881930761125$$
$$x_{52} = 90.5825881785057$$
$$x_{29} = 94.7713783832921$$
$$x_{49} = 96.8657734856853$$
$$x_{13} = 101.054563690472$$
$$x_{64} = 134.564885328763$$
$$x_{55} = 138.753675533549$$
$$x_{31} = 17438.4572213013$$
son puntos de cambio del signo de desigualdad en las soluciones.
Primero definámonos con el signo hasta el punto extremo izquierdo:
$$x_{0} < x_{10}$$
Consideremos, por ejemplo, el punto
$$x_{0} = x_{10} - \frac{1}{10}$$
=
$$-4454.25478401473 + - \frac{1}{10}$$
=
$$-4454.35478401473$$
lo sustituimos en la expresión
$$\sin{\left(t \right)} < \frac{1}{2}$$
$$\sin{\left(t \right)} < \frac{1}{2}$$
sin(t) < 1/2

Entonces
$$x < -4454.25478401473$$
no se cumple
significa que una de las soluciones de nuestra ecuación será con:
$$x > -4454.25478401473 \wedge x < -2650.98060085419$$
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-------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-------
       x10      x67      x4      x21      x42      x2      x65      x40      x34      x35      x56      x26      x68      x33      x46      x12      x11      x57      x41      x54      x17      x50      x14      x19      x70      x66      x18      x37      x7      x28      x39      x63      x15      x6      x53      x61      x24      x71      x16      x36      x48      x47      x59      x44      x69      x22      x62      x9      x30      x32      x23      x25      x27      x8      x45      x5      x3      x38      x58      x60      x51      x43      x1      x20      x52      x29      x49      x13      x64      x55      x31

Recibiremos otras soluciones de la desigualdad pasando al polo siguiente etc.
etc.
Respuesta:
$$x > -4454.25478401473 \wedge x < -2650.98060085419$$
$$x > -627.79493194236 \wedge x < -100.007366139275$$
$$x > -97.9129710368819 \wedge x < -93.7241808320955$$
$$x > -91.6297857297023 \wedge x < -87.4409955249159$$
$$x > -85.3466004225227 \wedge x < -81.1578102177363$$
$$x > -79.0634151153431 \wedge x < -74.8746249105567$$
$$x > -72.7802298081635 \wedge x < -68.5914396033772$$
$$x > -66.497044500984 \wedge x < -62.3082542961976$$
$$x > -60.2138591938044 \wedge x < -56.025068989018$$
$$x > -53.9306738866248 \wedge x < -49.7418836818384$$
$$x > -47.6474885794452 \wedge x < -43.4586983746588$$
$$x > -41.3643032722656 \wedge x < -37.1755130674792$$
$$x > -35.081117965086 \wedge x < -30.8923277602996$$
$$x > -28.7979326579064 \wedge x < -24.60914245312$$
$$x > -22.5147473507269 \wedge x < -18.3259571459405$$
$$x > -16.2315620435473 \wedge x < -12.0427718387609$$
$$x > -9.94837673636768 \wedge x < -5.75958653158129$$
$$x > -3.66519142918809 \wedge x < 0.523598775598299$$
$$x > 2.61799387799149 \wedge x < 6.80678408277789$$
$$x > 8.90117918517108 \wedge x < 13.0899693899575$$
$$x > 15.1843644923507 \wedge x < 19.3731546971371$$
$$x > 21.4675497995303 \wedge x < 25.6563400043166$$
$$x > 27.7507351067098 \wedge x < 31.9395253114962$$
$$x > 34.0339204138894 \wedge x < 38.2227106186758$$
$$x > 40.317105721069 \wedge x < 44.5058959258554$$
$$x > 46.6002910282486 \wedge x < 50.789081233035$$
$$x > 52.8834763354282 \wedge x < 57.0722665402146$$
$$x > 59.1666616426078 \wedge x < 63.3554518473942$$
$$x > 65.4498469497874 \wedge x < 69.6386371545737$$
$$x > 71.733032256967 \wedge x < 75.9218224617533$$
$$x > 78.0162175641465 \wedge x < 82.2050077689329$$
$$x > 84.2994028713261 \wedge x < 88.4881930761125$$
$$x > 90.5825881785057 \wedge x < 94.7713783832921$$
$$x > 96.8657734856853 \wedge x < 101.054563690472$$
$$x > 134.564885328763 \wedge x < 138.753675533549$$
$$x > 17438.4572213013$$
Respuesta rápida [src]
  /   /            pi\     /           5*pi    \\
Or|And|0 <= t, t < --|, And|t <= 2*pi, ---- < t||
  \   \            6 /     \            6      //
$$\left(0 \leq t \wedge t < \frac{\pi}{6}\right) \vee \left(t \leq 2 \pi \wedge \frac{5 \pi}{6} < t\right)$$
((0 <= t)∧(t < pi/6))∨((t <= 2*pi)∧(5*pi/6 < t))
Respuesta rápida 2 [src]
    pi     5*pi       
[0, --) U (----, 2*pi]
    6       6         
$$x\ in\ \left[0, \frac{\pi}{6}\right) \cup \left(\frac{5 \pi}{6}, 2 \pi\right]$$
x in Union(Interval.Ropen(0, pi/6), Interval.Lopen(5*pi/6, 2*pi))