Sr Examen
Otras calculadoras
- Sistemas de desigualdades paso a paso
- Ecuaciones básicas paso a paso
- Sistemas de ecuaciones paso a paso
- Solución de desigualdades trigonométricas en línea
- Resolución de desigualdades exponenciales en línea
- Resolución de desigualdades con un módulo en línea
-
¿Cómo usar?
- Desigualdades:
-
4+12x>7+13x
-
x-4x^2/x-1>0
-
x^2-17x+72>0
-
(x-9)*(x-1)>0
- Integral de d{x}:
- |sin(x)|
- Derivada de:
-
1/2
- Suma de la serie:
-
1/2
- Límite de la función:
-
1/2
-
Expresiones idénticas
- |sin(x)|< uno / dos
- módulo de seno de (x)| menos 1 dividir por 2
- módulo de seno de (x)| menos uno dividir por dos
- |sinx|<1/2
- |sin(x)|<1 dividir por 2
-
Expresiones semejantes
- |sinx|+|cosx|>1
- |sinx|*cosx>1/4
- (x-2)*(2x-1)/(2x^2+7x+3)>0
- |sinx|>|cosx|
- |sinx|>=√3/2
- |sinx|<1/2
-
Expresiones con funciones
- Seno sin
- sin(x)>=-√3/2
- sin(x)<=-√3/2
- sin(x)>2
- sinx/2cosx/2>1/2
- sin(x)<=1
- Módulo |
- |2-x|-|6+2x|<3
- |x^3-3x+1|<=x^3+x^2-1
- |x+2|-|x-3|>=2x-1
- |x^3+2x^2-2|>=2+3x
- |x-3|<1
|sin(x)|<1/2 desigualdades
Solución
Ha introducido
[src]
|sin(x)| < 1/2
$$\left|{\sin{\left(x \right)}}\right| < \frac{1}{2}$$
Abs(sin(x)) < 1/2
Solución detallada
Se da la desigualdad:
$$\left|{\sin{\left(x \right)}}\right| < \frac{1}{2}$$
Para resolver esta desigualdad primero hay que resolver la ecuación correspondiente:
$$\left|{\sin{\left(x \right)}}\right| = \frac{1}{2}$$
Resolvemos:
Tenemos la ecuación
$$\left|{\sin{\left(x \right)}}\right| = \frac{1}{2}$$
cambiamos
$$\left|{\sin{\left(x \right)}}\right| - \frac{1}{2} = 0$$
$$\left|{\sin{\left(x \right)}}\right| - \frac{1}{2} = 0$$
Sustituimos
$$w = \left|{\sin{\left(x \right)}}\right|$$
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
$$\left|{\sin{\left(x \right)}}\right| = w$$
sustituimos w:
$$x_{1} = -27.7507351067098$$
$$x_{2} = -47.6474885794452$$
$$x_{3} = 27.7507351067098$$
$$x_{4} = 91.6297857297023$$
$$x_{5} = -31.9395253114962$$
$$x_{6} = 34.0339204138894$$
$$x_{7} = -41.3643032722656$$
$$x_{8} = -5.75958653158129$$
$$x_{9} = 31.9395253114962$$
$$x_{10} = 96.8657734856853$$
$$x_{11} = 40.317105721069$$
$$x_{12} = 44.5058959258554$$
$$x_{13} = -46.6002910282486$$
$$x_{14} = 16.2315620435473$$
$$x_{15} = -18.3259571459405$$
$$x_{16} = -100.007366139275$$
$$x_{17} = 5.75958653158129$$
$$x_{18} = -57.0722665402146$$
$$x_{19} = -90.5825881785057$$
$$x_{20} = -78.0162175641465$$
$$x_{21} = -91.6297857297023$$
$$x_{22} = -60.2138591938044$$
$$x_{23} = -137.706477982353$$
$$x_{24} = -3.66519142918809$$
$$x_{25} = 46.6002910282486$$
$$x_{26} = -34.0339204138894$$
$$x_{27} = 71.733032256967$$
$$x_{28} = -69.6386371545737$$
$$x_{29} = -97.9129710368819$$
$$x_{30} = -12.0427718387609$$
$$x_{31} = 60.2138591938044$$
$$x_{32} = -19.3731546971371$$
$$x_{33} = 69.6386371545737$$
$$x_{34} = 75.9218224617533$$
$$x_{35} = 78.0162175641465$$
$$x_{36} = -38.2227106186758$$
$$x_{37} = -75.9218224617533$$
$$x_{38} = -25.6563400043166$$
$$x_{39} = 3.66519142918809$$
$$x_{40} = -87.4409955249159$$
$$x_{41} = 53.9306738866248$$
$$x_{42} = 38.2227106186758$$
$$x_{43} = -21.4675497995303$$
$$x_{44} = -71.733032256967$$
$$x_{45} = 2.61799387799149$$
$$x_{46} = 12.0427718387609$$
$$x_{47} = 24.60914245312$$
$$x_{48} = -382.750704962356$$
$$x_{49} = 93.7241808320955$$
$$x_{50} = -63.3554518473942$$
$$x_{51} = -9.94837673636768$$
$$x_{52} = -24.60914245312$$
$$x_{53} = -16.2315620435473$$
$$x_{54} = 9.94837673636768$$
$$x_{55} = -62.3082542961976$$
$$x_{56} = 0.523598775598299$$
$$x_{57} = 49.7418836818384$$
$$x_{58} = 18.3259571459405$$
$$x_{59} = 62.3082542961976$$
$$x_{60} = -56.025068989018$$
$$x_{61} = -49.7418836818384$$
$$x_{62} = 47.6474885794452$$
$$x_{63} = 97.9129710368819$$
$$x_{64} = 84.2994028713261$$
$$x_{65} = 56.025068989018$$
$$x_{66} = 90.5825881785057$$
$$x_{67} = 66.497044500984$$
$$x_{68} = -82.2050077689329$$
$$x_{69} = -40.317105721069$$
$$x_{70} = -93.7241808320955$$
$$x_{71} = 68.5914396033772$$
$$x_{72} = -53.9306738866248$$
$$x_{73} = 25.6563400043166$$
$$x_{74} = -85.3466004225227$$
$$x_{75} = 100.007366139275$$
$$x_{76} = 82.2050077689329$$
$$x_{77} = -2.61799387799149$$
$$x_{78} = -43.4586983746588$$
$$x_{79} = 22.5147473507269$$
$$x_{80} = -84.2994028713261$$
$$x_{81} = -65.4498469497874$$
$$x_{82} = -68.5914396033772$$
$$x_{83} = 88.4881930761125$$
$$x_{84} = -66.497044500984$$
$$x_{1} = -27.7507351067098$$
$$x_{2} = -47.6474885794452$$
$$x_{3} = 27.7507351067098$$
$$x_{4} = 91.6297857297023$$
$$x_{5} = -31.9395253114962$$
$$x_{6} = 34.0339204138894$$
$$x_{7} = -41.3643032722656$$
$$x_{8} = -5.75958653158129$$
$$x_{9} = 31.9395253114962$$
$$x_{10} = 96.8657734856853$$
$$x_{11} = 40.317105721069$$
$$x_{12} = 44.5058959258554$$
$$x_{13} = -46.6002910282486$$
$$x_{14} = 16.2315620435473$$
$$x_{15} = -18.3259571459405$$
$$x_{16} = -100.007366139275$$
$$x_{17} = 5.75958653158129$$
$$x_{18} = -57.0722665402146$$
$$x_{19} = -90.5825881785057$$
$$x_{20} = -78.0162175641465$$
$$x_{21} = -91.6297857297023$$
$$x_{22} = -60.2138591938044$$
$$x_{23} = -137.706477982353$$
$$x_{24} = -3.66519142918809$$
$$x_{25} = 46.6002910282486$$
$$x_{26} = -34.0339204138894$$
$$x_{27} = 71.733032256967$$
$$x_{28} = -69.6386371545737$$
$$x_{29} = -97.9129710368819$$
$$x_{30} = -12.0427718387609$$
$$x_{31} = 60.2138591938044$$
$$x_{32} = -19.3731546971371$$
$$x_{33} = 69.6386371545737$$
$$x_{34} = 75.9218224617533$$
$$x_{35} = 78.0162175641465$$
$$x_{36} = -38.2227106186758$$
$$x_{37} = -75.9218224617533$$
$$x_{38} = -25.6563400043166$$
$$x_{39} = 3.66519142918809$$
$$x_{40} = -87.4409955249159$$
$$x_{41} = 53.9306738866248$$
$$x_{42} = 38.2227106186758$$
$$x_{43} = -21.4675497995303$$
$$x_{44} = -71.733032256967$$
$$x_{45} = 2.61799387799149$$
$$x_{46} = 12.0427718387609$$
$$x_{47} = 24.60914245312$$
$$x_{48} = -382.750704962356$$
$$x_{49} = 93.7241808320955$$
$$x_{50} = -63.3554518473942$$
$$x_{51} = -9.94837673636768$$
$$x_{52} = -24.60914245312$$
$$x_{53} = -16.2315620435473$$
$$x_{54} = 9.94837673636768$$
$$x_{55} = -62.3082542961976$$
$$x_{56} = 0.523598775598299$$
$$x_{57} = 49.7418836818384$$
$$x_{58} = 18.3259571459405$$
$$x_{59} = 62.3082542961976$$
$$x_{60} = -56.025068989018$$
$$x_{61} = -49.7418836818384$$
$$x_{62} = 47.6474885794452$$
$$x_{63} = 97.9129710368819$$
$$x_{64} = 84.2994028713261$$
$$x_{65} = 56.025068989018$$
$$x_{66} = 90.5825881785057$$
$$x_{67} = 66.497044500984$$
$$x_{68} = -82.2050077689329$$
$$x_{69} = -40.317105721069$$
$$x_{70} = -93.7241808320955$$
$$x_{71} = 68.5914396033772$$
$$x_{72} = -53.9306738866248$$
$$x_{73} = 25.6563400043166$$
$$x_{74} = -85.3466004225227$$
$$x_{75} = 100.007366139275$$
$$x_{76} = 82.2050077689329$$
$$x_{77} = -2.61799387799149$$
$$x_{78} = -43.4586983746588$$
$$x_{79} = 22.5147473507269$$
$$x_{80} = -84.2994028713261$$
$$x_{81} = -65.4498469497874$$
$$x_{82} = -68.5914396033772$$
$$x_{83} = 88.4881930761125$$
$$x_{84} = -66.497044500984$$
Las raíces dadas
$$x_{48} = -382.750704962356$$
$$x_{23} = -137.706477982353$$
$$x_{16} = -100.007366139275$$
$$x_{29} = -97.9129710368819$$
$$x_{70} = -93.7241808320955$$
$$x_{21} = -91.6297857297023$$
$$x_{19} = -90.5825881785057$$
$$x_{40} = -87.4409955249159$$
$$x_{74} = -85.3466004225227$$
$$x_{80} = -84.2994028713261$$
$$x_{68} = -82.2050077689329$$
$$x_{20} = -78.0162175641465$$
$$x_{37} = -75.9218224617533$$
$$x_{44} = -71.733032256967$$
$$x_{28} = -69.6386371545737$$
$$x_{82} = -68.5914396033772$$
$$x_{84} = -66.497044500984$$
$$x_{81} = -65.4498469497874$$
$$x_{50} = -63.3554518473942$$
$$x_{55} = -62.3082542961976$$
$$x_{22} = -60.2138591938044$$
$$x_{18} = -57.0722665402146$$
$$x_{60} = -56.025068989018$$
$$x_{72} = -53.9306738866248$$
$$x_{61} = -49.7418836818384$$
$$x_{2} = -47.6474885794452$$
$$x_{13} = -46.6002910282486$$
$$x_{78} = -43.4586983746588$$
$$x_{7} = -41.3643032722656$$
$$x_{69} = -40.317105721069$$
$$x_{36} = -38.2227106186758$$
$$x_{26} = -34.0339204138894$$
$$x_{5} = -31.9395253114962$$
$$x_{1} = -27.7507351067098$$
$$x_{38} = -25.6563400043166$$
$$x_{52} = -24.60914245312$$
$$x_{43} = -21.4675497995303$$
$$x_{32} = -19.3731546971371$$
$$x_{15} = -18.3259571459405$$
$$x_{53} = -16.2315620435473$$
$$x_{30} = -12.0427718387609$$
$$x_{51} = -9.94837673636768$$
$$x_{8} = -5.75958653158129$$
$$x_{24} = -3.66519142918809$$
$$x_{77} = -2.61799387799149$$
$$x_{56} = 0.523598775598299$$
$$x_{45} = 2.61799387799149$$
$$x_{39} = 3.66519142918809$$
$$x_{17} = 5.75958653158129$$
$$x_{54} = 9.94837673636768$$
$$x_{46} = 12.0427718387609$$
$$x_{14} = 16.2315620435473$$
$$x_{58} = 18.3259571459405$$
$$x_{79} = 22.5147473507269$$
$$x_{47} = 24.60914245312$$
$$x_{73} = 25.6563400043166$$
$$x_{3} = 27.7507351067098$$
$$x_{9} = 31.9395253114962$$
$$x_{6} = 34.0339204138894$$
$$x_{42} = 38.2227106186758$$
$$x_{11} = 40.317105721069$$
$$x_{12} = 44.5058959258554$$
$$x_{25} = 46.6002910282486$$
$$x_{62} = 47.6474885794452$$
$$x_{57} = 49.7418836818384$$
$$x_{41} = 53.9306738866248$$
$$x_{65} = 56.025068989018$$
$$x_{31} = 60.2138591938044$$
$$x_{59} = 62.3082542961976$$
$$x_{67} = 66.497044500984$$
$$x_{71} = 68.5914396033772$$
$$x_{33} = 69.6386371545737$$
$$x_{27} = 71.733032256967$$
$$x_{34} = 75.9218224617533$$
$$x_{35} = 78.0162175641465$$
$$x_{76} = 82.2050077689329$$
$$x_{64} = 84.2994028713261$$
$$x_{83} = 88.4881930761125$$
$$x_{66} = 90.5825881785057$$
$$x_{4} = 91.6297857297023$$
$$x_{49} = 93.7241808320955$$
$$x_{10} = 96.8657734856853$$
$$x_{63} = 97.9129710368819$$
$$x_{75} = 100.007366139275$$
son puntos de cambio del signo de desigualdad en las soluciones.
Primero definámonos con el signo hasta el punto extremo izquierdo:
$$x_{0} < x_{48}$$
Consideremos, por ejemplo, el punto
$$x_{0} = x_{48} - \frac{1}{10}$$
=
$$-382.750704962356 + - \frac{1}{10}$$
=
$$-382.850704962356$$
lo sustituimos en la expresión
$$\left|{\sin{\left(x \right)}}\right| < \frac{1}{2}$$
$$\left|{\sin{\left(-382.850704962356 \right)}}\right| < \frac{1}{2}$$
significa que una de las soluciones de nuestra ecuación será con:
$$x < -382.750704962356$$
Recibiremos otras soluciones de la desigualdad pasando al polo siguiente etc.
etc.
Respuesta:
$$x < -382.750704962356$$
$$x > -137.706477982353 \wedge x < -100.007366139275$$
$$x > -97.9129710368819 \wedge x < -93.7241808320955$$
$$x > -91.6297857297023 \wedge x < -90.5825881785057$$
$$x > -87.4409955249159 \wedge x < -85.3466004225227$$
$$x > -84.2994028713261 \wedge x < -82.2050077689329$$
$$x > -78.0162175641465 \wedge x < -75.9218224617533$$
$$x > -71.733032256967 \wedge x < -69.6386371545737$$
$$x > -68.5914396033772 \wedge x < -66.497044500984$$
$$x > -65.4498469497874 \wedge x < -63.3554518473942$$
$$x > -62.3082542961976 \wedge x < -60.2138591938044$$
$$x > -57.0722665402146 \wedge x < -56.025068989018$$
$$x > -53.9306738866248 \wedge x < -49.7418836818384$$
$$x > -47.6474885794452 \wedge x < -46.6002910282486$$
$$x > -43.4586983746588 \wedge x < -41.3643032722656$$
$$x > -40.317105721069 \wedge x < -38.2227106186758$$
$$x > -34.0339204138894 \wedge x < -31.9395253114962$$
$$x > -27.7507351067098 \wedge x < -25.6563400043166$$
$$x > -24.60914245312 \wedge x < -21.4675497995303$$
$$x > -19.3731546971371 \wedge x < -18.3259571459405$$
$$x > -16.2315620435473 \wedge x < -12.0427718387609$$
$$x > -9.94837673636768 \wedge x < -5.75958653158129$$
$$x > -3.66519142918809 \wedge x < -2.61799387799149$$
$$x > 0.523598775598299 \wedge x < 2.61799387799149$$
$$x > 3.66519142918809 \wedge x < 5.75958653158129$$
$$x > 9.94837673636768 \wedge x < 12.0427718387609$$
$$x > 16.2315620435473 \wedge x < 18.3259571459405$$
$$x > 22.5147473507269 \wedge x < 24.60914245312$$
$$x > 25.6563400043166 \wedge x < 27.7507351067098$$
$$x > 31.9395253114962 \wedge x < 34.0339204138894$$
$$x > 38.2227106186758 \wedge x < 40.317105721069$$
$$x > 44.5058959258554 \wedge x < 46.6002910282486$$
$$x > 47.6474885794452 \wedge x < 49.7418836818384$$
$$x > 53.9306738866248 \wedge x < 56.025068989018$$
$$x > 60.2138591938044 \wedge x < 62.3082542961976$$
$$x > 66.497044500984 \wedge x < 68.5914396033772$$
$$x > 69.6386371545737 \wedge x < 71.733032256967$$
$$x > 75.9218224617533 \wedge x < 78.0162175641465$$
$$x > 82.2050077689329 \wedge x < 84.2994028713261$$
$$x > 88.4881930761125 \wedge x < 90.5825881785057$$
$$x > 91.6297857297023 \wedge x < 93.7241808320955$$
$$x > 96.8657734856853 \wedge x < 97.9129710368819$$
$$x > 100.007366139275$$
$$\left|{\sin{\left(x \right)}}\right| < \frac{1}{2}$$
Para resolver esta desigualdad primero hay que resolver la ecuación correspondiente:
$$\left|{\sin{\left(x \right)}}\right| = \frac{1}{2}$$
Resolvemos:
Tenemos la ecuación
$$\left|{\sin{\left(x \right)}}\right| = \frac{1}{2}$$
cambiamos
$$\left|{\sin{\left(x \right)}}\right| - \frac{1}{2} = 0$$
$$\left|{\sin{\left(x \right)}}\right| - \frac{1}{2} = 0$$
Sustituimos
$$w = \left|{\sin{\left(x \right)}}\right|$$
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
$$\left|{\sin{\left(x \right)}}\right| = w$$
sustituimos w:
$$x_{1} = -27.7507351067098$$
$$x_{2} = -47.6474885794452$$
$$x_{3} = 27.7507351067098$$
$$x_{4} = 91.6297857297023$$
$$x_{5} = -31.9395253114962$$
$$x_{6} = 34.0339204138894$$
$$x_{7} = -41.3643032722656$$
$$x_{8} = -5.75958653158129$$
$$x_{9} = 31.9395253114962$$
$$x_{10} = 96.8657734856853$$
$$x_{11} = 40.317105721069$$
$$x_{12} = 44.5058959258554$$
$$x_{13} = -46.6002910282486$$
$$x_{14} = 16.2315620435473$$
$$x_{15} = -18.3259571459405$$
$$x_{16} = -100.007366139275$$
$$x_{17} = 5.75958653158129$$
$$x_{18} = -57.0722665402146$$
$$x_{19} = -90.5825881785057$$
$$x_{20} = -78.0162175641465$$
$$x_{21} = -91.6297857297023$$
$$x_{22} = -60.2138591938044$$
$$x_{23} = -137.706477982353$$
$$x_{24} = -3.66519142918809$$
$$x_{25} = 46.6002910282486$$
$$x_{26} = -34.0339204138894$$
$$x_{27} = 71.733032256967$$
$$x_{28} = -69.6386371545737$$
$$x_{29} = -97.9129710368819$$
$$x_{30} = -12.0427718387609$$
$$x_{31} = 60.2138591938044$$
$$x_{32} = -19.3731546971371$$
$$x_{33} = 69.6386371545737$$
$$x_{34} = 75.9218224617533$$
$$x_{35} = 78.0162175641465$$
$$x_{36} = -38.2227106186758$$
$$x_{37} = -75.9218224617533$$
$$x_{38} = -25.6563400043166$$
$$x_{39} = 3.66519142918809$$
$$x_{40} = -87.4409955249159$$
$$x_{41} = 53.9306738866248$$
$$x_{42} = 38.2227106186758$$
$$x_{43} = -21.4675497995303$$
$$x_{44} = -71.733032256967$$
$$x_{45} = 2.61799387799149$$
$$x_{46} = 12.0427718387609$$
$$x_{47} = 24.60914245312$$
$$x_{48} = -382.750704962356$$
$$x_{49} = 93.7241808320955$$
$$x_{50} = -63.3554518473942$$
$$x_{51} = -9.94837673636768$$
$$x_{52} = -24.60914245312$$
$$x_{53} = -16.2315620435473$$
$$x_{54} = 9.94837673636768$$
$$x_{55} = -62.3082542961976$$
$$x_{56} = 0.523598775598299$$
$$x_{57} = 49.7418836818384$$
$$x_{58} = 18.3259571459405$$
$$x_{59} = 62.3082542961976$$
$$x_{60} = -56.025068989018$$
$$x_{61} = -49.7418836818384$$
$$x_{62} = 47.6474885794452$$
$$x_{63} = 97.9129710368819$$
$$x_{64} = 84.2994028713261$$
$$x_{65} = 56.025068989018$$
$$x_{66} = 90.5825881785057$$
$$x_{67} = 66.497044500984$$
$$x_{68} = -82.2050077689329$$
$$x_{69} = -40.317105721069$$
$$x_{70} = -93.7241808320955$$
$$x_{71} = 68.5914396033772$$
$$x_{72} = -53.9306738866248$$
$$x_{73} = 25.6563400043166$$
$$x_{74} = -85.3466004225227$$
$$x_{75} = 100.007366139275$$
$$x_{76} = 82.2050077689329$$
$$x_{77} = -2.61799387799149$$
$$x_{78} = -43.4586983746588$$
$$x_{79} = 22.5147473507269$$
$$x_{80} = -84.2994028713261$$
$$x_{81} = -65.4498469497874$$
$$x_{82} = -68.5914396033772$$
$$x_{83} = 88.4881930761125$$
$$x_{84} = -66.497044500984$$
$$x_{1} = -27.7507351067098$$
$$x_{2} = -47.6474885794452$$
$$x_{3} = 27.7507351067098$$
$$x_{4} = 91.6297857297023$$
$$x_{5} = -31.9395253114962$$
$$x_{6} = 34.0339204138894$$
$$x_{7} = -41.3643032722656$$
$$x_{8} = -5.75958653158129$$
$$x_{9} = 31.9395253114962$$
$$x_{10} = 96.8657734856853$$
$$x_{11} = 40.317105721069$$
$$x_{12} = 44.5058959258554$$
$$x_{13} = -46.6002910282486$$
$$x_{14} = 16.2315620435473$$
$$x_{15} = -18.3259571459405$$
$$x_{16} = -100.007366139275$$
$$x_{17} = 5.75958653158129$$
$$x_{18} = -57.0722665402146$$
$$x_{19} = -90.5825881785057$$
$$x_{20} = -78.0162175641465$$
$$x_{21} = -91.6297857297023$$
$$x_{22} = -60.2138591938044$$
$$x_{23} = -137.706477982353$$
$$x_{24} = -3.66519142918809$$
$$x_{25} = 46.6002910282486$$
$$x_{26} = -34.0339204138894$$
$$x_{27} = 71.733032256967$$
$$x_{28} = -69.6386371545737$$
$$x_{29} = -97.9129710368819$$
$$x_{30} = -12.0427718387609$$
$$x_{31} = 60.2138591938044$$
$$x_{32} = -19.3731546971371$$
$$x_{33} = 69.6386371545737$$
$$x_{34} = 75.9218224617533$$
$$x_{35} = 78.0162175641465$$
$$x_{36} = -38.2227106186758$$
$$x_{37} = -75.9218224617533$$
$$x_{38} = -25.6563400043166$$
$$x_{39} = 3.66519142918809$$
$$x_{40} = -87.4409955249159$$
$$x_{41} = 53.9306738866248$$
$$x_{42} = 38.2227106186758$$
$$x_{43} = -21.4675497995303$$
$$x_{44} = -71.733032256967$$
$$x_{45} = 2.61799387799149$$
$$x_{46} = 12.0427718387609$$
$$x_{47} = 24.60914245312$$
$$x_{48} = -382.750704962356$$
$$x_{49} = 93.7241808320955$$
$$x_{50} = -63.3554518473942$$
$$x_{51} = -9.94837673636768$$
$$x_{52} = -24.60914245312$$
$$x_{53} = -16.2315620435473$$
$$x_{54} = 9.94837673636768$$
$$x_{55} = -62.3082542961976$$
$$x_{56} = 0.523598775598299$$
$$x_{57} = 49.7418836818384$$
$$x_{58} = 18.3259571459405$$
$$x_{59} = 62.3082542961976$$
$$x_{60} = -56.025068989018$$
$$x_{61} = -49.7418836818384$$
$$x_{62} = 47.6474885794452$$
$$x_{63} = 97.9129710368819$$
$$x_{64} = 84.2994028713261$$
$$x_{65} = 56.025068989018$$
$$x_{66} = 90.5825881785057$$
$$x_{67} = 66.497044500984$$
$$x_{68} = -82.2050077689329$$
$$x_{69} = -40.317105721069$$
$$x_{70} = -93.7241808320955$$
$$x_{71} = 68.5914396033772$$
$$x_{72} = -53.9306738866248$$
$$x_{73} = 25.6563400043166$$
$$x_{74} = -85.3466004225227$$
$$x_{75} = 100.007366139275$$
$$x_{76} = 82.2050077689329$$
$$x_{77} = -2.61799387799149$$
$$x_{78} = -43.4586983746588$$
$$x_{79} = 22.5147473507269$$
$$x_{80} = -84.2994028713261$$
$$x_{81} = -65.4498469497874$$
$$x_{82} = -68.5914396033772$$
$$x_{83} = 88.4881930761125$$
$$x_{84} = -66.497044500984$$
Las raíces dadas
$$x_{48} = -382.750704962356$$
$$x_{23} = -137.706477982353$$
$$x_{16} = -100.007366139275$$
$$x_{29} = -97.9129710368819$$
$$x_{70} = -93.7241808320955$$
$$x_{21} = -91.6297857297023$$
$$x_{19} = -90.5825881785057$$
$$x_{40} = -87.4409955249159$$
$$x_{74} = -85.3466004225227$$
$$x_{80} = -84.2994028713261$$
$$x_{68} = -82.2050077689329$$
$$x_{20} = -78.0162175641465$$
$$x_{37} = -75.9218224617533$$
$$x_{44} = -71.733032256967$$
$$x_{28} = -69.6386371545737$$
$$x_{82} = -68.5914396033772$$
$$x_{84} = -66.497044500984$$
$$x_{81} = -65.4498469497874$$
$$x_{50} = -63.3554518473942$$
$$x_{55} = -62.3082542961976$$
$$x_{22} = -60.2138591938044$$
$$x_{18} = -57.0722665402146$$
$$x_{60} = -56.025068989018$$
$$x_{72} = -53.9306738866248$$
$$x_{61} = -49.7418836818384$$
$$x_{2} = -47.6474885794452$$
$$x_{13} = -46.6002910282486$$
$$x_{78} = -43.4586983746588$$
$$x_{7} = -41.3643032722656$$
$$x_{69} = -40.317105721069$$
$$x_{36} = -38.2227106186758$$
$$x_{26} = -34.0339204138894$$
$$x_{5} = -31.9395253114962$$
$$x_{1} = -27.7507351067098$$
$$x_{38} = -25.6563400043166$$
$$x_{52} = -24.60914245312$$
$$x_{43} = -21.4675497995303$$
$$x_{32} = -19.3731546971371$$
$$x_{15} = -18.3259571459405$$
$$x_{53} = -16.2315620435473$$
$$x_{30} = -12.0427718387609$$
$$x_{51} = -9.94837673636768$$
$$x_{8} = -5.75958653158129$$
$$x_{24} = -3.66519142918809$$
$$x_{77} = -2.61799387799149$$
$$x_{56} = 0.523598775598299$$
$$x_{45} = 2.61799387799149$$
$$x_{39} = 3.66519142918809$$
$$x_{17} = 5.75958653158129$$
$$x_{54} = 9.94837673636768$$
$$x_{46} = 12.0427718387609$$
$$x_{14} = 16.2315620435473$$
$$x_{58} = 18.3259571459405$$
$$x_{79} = 22.5147473507269$$
$$x_{47} = 24.60914245312$$
$$x_{73} = 25.6563400043166$$
$$x_{3} = 27.7507351067098$$
$$x_{9} = 31.9395253114962$$
$$x_{6} = 34.0339204138894$$
$$x_{42} = 38.2227106186758$$
$$x_{11} = 40.317105721069$$
$$x_{12} = 44.5058959258554$$
$$x_{25} = 46.6002910282486$$
$$x_{62} = 47.6474885794452$$
$$x_{57} = 49.7418836818384$$
$$x_{41} = 53.9306738866248$$
$$x_{65} = 56.025068989018$$
$$x_{31} = 60.2138591938044$$
$$x_{59} = 62.3082542961976$$
$$x_{67} = 66.497044500984$$
$$x_{71} = 68.5914396033772$$
$$x_{33} = 69.6386371545737$$
$$x_{27} = 71.733032256967$$
$$x_{34} = 75.9218224617533$$
$$x_{35} = 78.0162175641465$$
$$x_{76} = 82.2050077689329$$
$$x_{64} = 84.2994028713261$$
$$x_{83} = 88.4881930761125$$
$$x_{66} = 90.5825881785057$$
$$x_{4} = 91.6297857297023$$
$$x_{49} = 93.7241808320955$$
$$x_{10} = 96.8657734856853$$
$$x_{63} = 97.9129710368819$$
$$x_{75} = 100.007366139275$$
son puntos de cambio del signo de desigualdad en las soluciones.
Primero definámonos con el signo hasta el punto extremo izquierdo:
$$x_{0} < x_{48}$$
Consideremos, por ejemplo, el punto
$$x_{0} = x_{48} - \frac{1}{10}$$
=
$$-382.750704962356 + - \frac{1}{10}$$
=
$$-382.850704962356$$
lo sustituimos en la expresión
$$\left|{\sin{\left(x \right)}}\right| < \frac{1}{2}$$
$$\left|{\sin{\left(-382.850704962356 \right)}}\right| < \frac{1}{2}$$
0.411043807676265 < 1/2
significa que una de las soluciones de nuestra ecuación será con:
$$x < -382.750704962356$$
_____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____
\ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
-------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-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------
x48 x23 x16 x29 x70 x21 x19 x40 x74 x80 x68 x20 x37 x44 x28 x82 x84 x81 x50 x55 x22 x18 x60 x72 x61 x2 x13 x78 x7 x69 x36 x26 x5 x1 x38 x52 x43 x32 x15 x53 x30 x51 x8 x24 x77 x56 x45 x39 x17 x54 x46 x14 x58 x79 x47 x73 x3 x9 x6 x42 x11 x12 x25 x62 x57 x41 x65 x31 x59 x67 x71 x33 x27 x34 x35 x76 x64 x83 x66 x4 x49 x10 x63 x75Recibiremos otras soluciones de la desigualdad pasando al polo siguiente etc.
etc.
Respuesta:
$$x < -382.750704962356$$
$$x > -137.706477982353 \wedge x < -100.007366139275$$
$$x > -97.9129710368819 \wedge x < -93.7241808320955$$
$$x > -91.6297857297023 \wedge x < -90.5825881785057$$
$$x > -87.4409955249159 \wedge x < -85.3466004225227$$
$$x > -84.2994028713261 \wedge x < -82.2050077689329$$
$$x > -78.0162175641465 \wedge x < -75.9218224617533$$
$$x > -71.733032256967 \wedge x < -69.6386371545737$$
$$x > -68.5914396033772 \wedge x < -66.497044500984$$
$$x > -65.4498469497874 \wedge x < -63.3554518473942$$
$$x > -62.3082542961976 \wedge x < -60.2138591938044$$
$$x > -57.0722665402146 \wedge x < -56.025068989018$$
$$x > -53.9306738866248 \wedge x < -49.7418836818384$$
$$x > -47.6474885794452 \wedge x < -46.6002910282486$$
$$x > -43.4586983746588 \wedge x < -41.3643032722656$$
$$x > -40.317105721069 \wedge x < -38.2227106186758$$
$$x > -34.0339204138894 \wedge x < -31.9395253114962$$
$$x > -27.7507351067098 \wedge x < -25.6563400043166$$
$$x > -24.60914245312 \wedge x < -21.4675497995303$$
$$x > -19.3731546971371 \wedge x < -18.3259571459405$$
$$x > -16.2315620435473 \wedge x < -12.0427718387609$$
$$x > -9.94837673636768 \wedge x < -5.75958653158129$$
$$x > -3.66519142918809 \wedge x < -2.61799387799149$$
$$x > 0.523598775598299 \wedge x < 2.61799387799149$$
$$x > 3.66519142918809 \wedge x < 5.75958653158129$$
$$x > 9.94837673636768 \wedge x < 12.0427718387609$$
$$x > 16.2315620435473 \wedge x < 18.3259571459405$$
$$x > 22.5147473507269 \wedge x < 24.60914245312$$
$$x > 25.6563400043166 \wedge x < 27.7507351067098$$
$$x > 31.9395253114962 \wedge x < 34.0339204138894$$
$$x > 38.2227106186758 \wedge x < 40.317105721069$$
$$x > 44.5058959258554 \wedge x < 46.6002910282486$$
$$x > 47.6474885794452 \wedge x < 49.7418836818384$$
$$x > 53.9306738866248 \wedge x < 56.025068989018$$
$$x > 60.2138591938044 \wedge x < 62.3082542961976$$
$$x > 66.497044500984 \wedge x < 68.5914396033772$$
$$x > 69.6386371545737 \wedge x < 71.733032256967$$
$$x > 75.9218224617533 \wedge x < 78.0162175641465$$
$$x > 82.2050077689329 \wedge x < 84.2994028713261$$
$$x > 88.4881930761125 \wedge x < 90.5825881785057$$
$$x > 91.6297857297023 \wedge x < 93.7241808320955$$
$$x > 96.8657734856853 \wedge x < 97.9129710368819$$
$$x > 100.007366139275$$
Solución de la desigualdad en el gráfico
Gráfico