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- Sistemas de desigualdades paso a paso
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-
¿Cómo usar?
- Desigualdades:
-
4+12x>7+13x
-
x-4x^2/x-1>0
-
(x-9)*(x-1)>0
-
x^2-2x+5<0
-
Expresiones idénticas
- 5sinx+cos dos x≥-2
- 5 seno de x más coseno de 2x≥ menos 2
- 5 seno de x más coseno de dos x≥ menos 2
-
Expresiones semejantes
- 5sinx-cos2x≥-2
- 5sinx+cos2x≥+2
5sinx+cos2x≥-2 desigualdades
Solución
Ha introducido
[src]
5*sin(x) + cos(2*x) >= -2
$$5 \sin{\left(x \right)} + \cos{\left(2 x \right)} \geq -2$$
5*sin(x) + cos(2*x) >= -2
Solución detallada
Se da la desigualdad:
$$5 \sin{\left(x \right)} + \cos{\left(2 x \right)} \geq -2$$
Para resolver esta desigualdad primero hay que resolver la ecuación correspondiente:
$$5 \sin{\left(x \right)} + \cos{\left(2 x \right)} = -2$$
Resolvemos:
Tenemos la ecuación
$$5 \sin{\left(x \right)} + \cos{\left(2 x \right)} = -2$$
cambiamos
$$5 \sin{\left(x \right)} + \cos{\left(2 x \right)} + 2 = 0$$
$$- 2 \sin^{2}{\left(x \right)} + 5 \sin{\left(x \right)} + 3 = 0$$
Sustituimos
$$w = \sin{\left(x \right)}$$
Es la ecuación de la forma
La ecuación cuadrática puede ser resuelta
con la ayuda del discriminante.
Las raíces de la ecuación cuadrática:
$$w_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$w_{2} = \frac{- \sqrt{D} - b}{2 a}$$
donde D = b^2 - 4*a*c es el discriminante.
Como
$$a = -2$$
$$b = 5$$
$$c = 3$$
, entonces
Como D > 0 la ecuación tiene dos raíces.
o
$$w_{1} = - \frac{1}{2}$$
$$w_{2} = 3$$
hacemos cambio inverso
$$\sin{\left(x \right)} = w$$
Tenemos la ecuación
$$\sin{\left(x \right)} = w$$
es la ecuación trigonométrica más simple
Esta ecuación se reorganiza en
$$x = 2 \pi n + \operatorname{asin}{\left(w \right)}$$
$$x = 2 \pi n - \operatorname{asin}{\left(w \right)} + \pi$$
O
$$x = 2 \pi n + \operatorname{asin}{\left(w \right)}$$
$$x = 2 \pi n - \operatorname{asin}{\left(w \right)} + \pi$$
, donde n es cualquier número entero
sustituimos w:
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(w_{1} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(- \frac{1}{2} \right)}$$
$$x_{1} = 2 \pi n - \frac{\pi}{6}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(w_{2} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(3 \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(3 \right)}$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(- \frac{1}{2} \right)} + \pi$$
$$x_{3} = 2 \pi n + \frac{7 \pi}{6}$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(w_{2} \right)} + \pi$$
$$x_{4} = 2 \pi n + \pi - \operatorname{asin}{\left(3 \right)}$$
$$x_{4} = 2 \pi n + \pi - \operatorname{asin}{\left(3 \right)}$$
$$x_{1} = 100.007366139275$$
$$x_{2} = -82.2050077689329$$
$$x_{3} = 68.5914396033772$$
$$x_{4} = 87.4409955249159$$
$$x_{5} = -45867.7763411866$$
$$x_{6} = -71.733032256967$$
$$x_{7} = 41.3643032722656$$
$$x_{8} = 74.8746249105567$$
$$x_{9} = 12.0427718387609$$
$$x_{10} = -65.4498469497874$$
$$x_{11} = 37.1755130674792$$
$$x_{12} = 110.479341651241$$
$$x_{13} = 22.5147473507269$$
$$x_{14} = 30.8923277602996$$
$$x_{15} = -15.1843644923507$$
$$x_{16} = 81.1578102177363$$
$$x_{17} = 62.3082542961976$$
$$x_{18} = -59.1666616426078$$
$$x_{19} = 93.7241808320955$$
$$x_{20} = 47.6474885794452$$
$$x_{21} = 56.025068989018$$
$$x_{22} = 91.6297857297023$$
$$x_{23} = 79.0634151153431$$
$$x_{24} = 18.3259571459405$$
$$x_{25} = 66.497044500984$$
$$x_{26} = -75.9218224617533$$
$$x_{27} = -239.284640448423$$
$$x_{28} = -31.9395253114962$$
$$x_{29} = -46.6002910282486$$
$$x_{30} = -101.054563690472$$
$$x_{31} = 28.7979326579064$$
$$x_{32} = 53.9306738866248$$
$$x_{33} = -90.5825881785057$$
$$x_{34} = -8.90117918517108$$
$$x_{35} = -6.80678408277789$$
$$x_{36} = -69.6386371545737$$
$$x_{37} = 60.2138591938044$$
$$x_{38} = -44.5058959258554$$
$$x_{39} = -94.7713783832921$$
$$x_{40} = -27.7507351067098$$
$$x_{41} = -25.6563400043166$$
$$x_{42} = -34.0339204138894$$
$$x_{43} = -50.789081233035$$
$$x_{44} = -13.0899693899575$$
$$x_{45} = 3.66519142918809$$
$$x_{46} = -88.4881930761125$$
$$x_{47} = 85.3466004225227$$
$$x_{48} = -84.2994028713261$$
$$x_{49} = 16.2315620435473$$
$$x_{50} = -52.8834763354282$$
$$x_{51} = -643.502895210309$$
$$x_{52} = -40.317105721069$$
$$x_{53} = -0.523598775598299$$
$$x_{54} = 97.9129710368819$$
$$x_{55} = 49.7418836818384$$
$$x_{56} = 35.081117965086$$
$$x_{57} = 791.15774992903$$
$$x_{58} = -96.8657734856853$$
$$x_{59} = 24.60914245312$$
$$x_{60} = 5.75958653158129$$
$$x_{61} = -21.4675497995303$$
$$x_{62} = -63.3554518473942$$
$$x_{63} = -78.0162175641465$$
$$x_{64} = -38.2227106186758$$
$$x_{65} = 72.7802298081635$$
$$x_{66} = 43.4586983746588$$
$$x_{67} = -2.61799387799149$$
$$x_{68} = 9.94837673636768$$
$$x_{69} = -57.0722665402146$$
$$x_{70} = 112.573736753634$$
$$x_{71} = -19.3731546971371$$
$$x_{1} = 100.007366139275$$
$$x_{2} = -82.2050077689329$$
$$x_{3} = 68.5914396033772$$
$$x_{4} = 87.4409955249159$$
$$x_{5} = -45867.7763411866$$
$$x_{6} = -71.733032256967$$
$$x_{7} = 41.3643032722656$$
$$x_{8} = 74.8746249105567$$
$$x_{9} = 12.0427718387609$$
$$x_{10} = -65.4498469497874$$
$$x_{11} = 37.1755130674792$$
$$x_{12} = 110.479341651241$$
$$x_{13} = 22.5147473507269$$
$$x_{14} = 30.8923277602996$$
$$x_{15} = -15.1843644923507$$
$$x_{16} = 81.1578102177363$$
$$x_{17} = 62.3082542961976$$
$$x_{18} = -59.1666616426078$$
$$x_{19} = 93.7241808320955$$
$$x_{20} = 47.6474885794452$$
$$x_{21} = 56.025068989018$$
$$x_{22} = 91.6297857297023$$
$$x_{23} = 79.0634151153431$$
$$x_{24} = 18.3259571459405$$
$$x_{25} = 66.497044500984$$
$$x_{26} = -75.9218224617533$$
$$x_{27} = -239.284640448423$$
$$x_{28} = -31.9395253114962$$
$$x_{29} = -46.6002910282486$$
$$x_{30} = -101.054563690472$$
$$x_{31} = 28.7979326579064$$
$$x_{32} = 53.9306738866248$$
$$x_{33} = -90.5825881785057$$
$$x_{34} = -8.90117918517108$$
$$x_{35} = -6.80678408277789$$
$$x_{36} = -69.6386371545737$$
$$x_{37} = 60.2138591938044$$
$$x_{38} = -44.5058959258554$$
$$x_{39} = -94.7713783832921$$
$$x_{40} = -27.7507351067098$$
$$x_{41} = -25.6563400043166$$
$$x_{42} = -34.0339204138894$$
$$x_{43} = -50.789081233035$$
$$x_{44} = -13.0899693899575$$
$$x_{45} = 3.66519142918809$$
$$x_{46} = -88.4881930761125$$
$$x_{47} = 85.3466004225227$$
$$x_{48} = -84.2994028713261$$
$$x_{49} = 16.2315620435473$$
$$x_{50} = -52.8834763354282$$
$$x_{51} = -643.502895210309$$
$$x_{52} = -40.317105721069$$
$$x_{53} = -0.523598775598299$$
$$x_{54} = 97.9129710368819$$
$$x_{55} = 49.7418836818384$$
$$x_{56} = 35.081117965086$$
$$x_{57} = 791.15774992903$$
$$x_{58} = -96.8657734856853$$
$$x_{59} = 24.60914245312$$
$$x_{60} = 5.75958653158129$$
$$x_{61} = -21.4675497995303$$
$$x_{62} = -63.3554518473942$$
$$x_{63} = -78.0162175641465$$
$$x_{64} = -38.2227106186758$$
$$x_{65} = 72.7802298081635$$
$$x_{66} = 43.4586983746588$$
$$x_{67} = -2.61799387799149$$
$$x_{68} = 9.94837673636768$$
$$x_{69} = -57.0722665402146$$
$$x_{70} = 112.573736753634$$
$$x_{71} = -19.3731546971371$$
Las raíces dadas
$$x_{5} = -45867.7763411866$$
$$x_{51} = -643.502895210309$$
$$x_{27} = -239.284640448423$$
$$x_{30} = -101.054563690472$$
$$x_{58} = -96.8657734856853$$
$$x_{39} = -94.7713783832921$$
$$x_{33} = -90.5825881785057$$
$$x_{46} = -88.4881930761125$$
$$x_{48} = -84.2994028713261$$
$$x_{2} = -82.2050077689329$$
$$x_{63} = -78.0162175641465$$
$$x_{26} = -75.9218224617533$$
$$x_{6} = -71.733032256967$$
$$x_{36} = -69.6386371545737$$
$$x_{10} = -65.4498469497874$$
$$x_{62} = -63.3554518473942$$
$$x_{18} = -59.1666616426078$$
$$x_{69} = -57.0722665402146$$
$$x_{50} = -52.8834763354282$$
$$x_{43} = -50.789081233035$$
$$x_{29} = -46.6002910282486$$
$$x_{38} = -44.5058959258554$$
$$x_{52} = -40.317105721069$$
$$x_{64} = -38.2227106186758$$
$$x_{42} = -34.0339204138894$$
$$x_{28} = -31.9395253114962$$
$$x_{40} = -27.7507351067098$$
$$x_{41} = -25.6563400043166$$
$$x_{61} = -21.4675497995303$$
$$x_{71} = -19.3731546971371$$
$$x_{15} = -15.1843644923507$$
$$x_{44} = -13.0899693899575$$
$$x_{34} = -8.90117918517108$$
$$x_{35} = -6.80678408277789$$
$$x_{67} = -2.61799387799149$$
$$x_{53} = -0.523598775598299$$
$$x_{45} = 3.66519142918809$$
$$x_{60} = 5.75958653158129$$
$$x_{68} = 9.94837673636768$$
$$x_{9} = 12.0427718387609$$
$$x_{49} = 16.2315620435473$$
$$x_{24} = 18.3259571459405$$
$$x_{13} = 22.5147473507269$$
$$x_{59} = 24.60914245312$$
$$x_{31} = 28.7979326579064$$
$$x_{14} = 30.8923277602996$$
$$x_{56} = 35.081117965086$$
$$x_{11} = 37.1755130674792$$
$$x_{7} = 41.3643032722656$$
$$x_{66} = 43.4586983746588$$
$$x_{20} = 47.6474885794452$$
$$x_{55} = 49.7418836818384$$
$$x_{32} = 53.9306738866248$$
$$x_{21} = 56.025068989018$$
$$x_{37} = 60.2138591938044$$
$$x_{17} = 62.3082542961976$$
$$x_{25} = 66.497044500984$$
$$x_{3} = 68.5914396033772$$
$$x_{65} = 72.7802298081635$$
$$x_{8} = 74.8746249105567$$
$$x_{23} = 79.0634151153431$$
$$x_{16} = 81.1578102177363$$
$$x_{47} = 85.3466004225227$$
$$x_{4} = 87.4409955249159$$
$$x_{22} = 91.6297857297023$$
$$x_{19} = 93.7241808320955$$
$$x_{54} = 97.9129710368819$$
$$x_{1} = 100.007366139275$$
$$x_{12} = 110.479341651241$$
$$x_{70} = 112.573736753634$$
$$x_{57} = 791.15774992903$$
son puntos de cambio del signo de desigualdad en las soluciones.
Primero definámonos con el signo hasta el punto extremo izquierdo:
$$x_{0} \leq x_{5}$$
Consideremos, por ejemplo, el punto
$$x_{0} = x_{5} - \frac{1}{10}$$
=
$$-45867.7763411866 + - \frac{1}{10}$$
=
$$-45867.8763411866$$
lo sustituimos en la expresión
$$5 \sin{\left(x \right)} + \cos{\left(2 x \right)} \geq -2$$
$$5 \sin{\left(-45867.8763411866 \right)} + \cos{\left(\left(-45867.8763411866\right) 2 \right)} \geq -2$$
pero
Entonces
$$x \leq -45867.7763411866$$
no se cumple
significa que una de las soluciones de nuestra ecuación será con:
$$x \geq -45867.7763411866 \wedge x \leq -643.502895210309$$
Recibiremos otras soluciones de la desigualdad pasando al polo siguiente etc.
etc.
Respuesta:
$$x \geq -45867.7763411866 \wedge x \leq -643.502895210309$$
$$x \geq -239.284640448423 \wedge x \leq -101.054563690472$$
$$x \geq -96.8657734856853 \wedge x \leq -94.7713783832921$$
$$x \geq -90.5825881785057 \wedge x \leq -88.4881930761125$$
$$x \geq -84.2994028713261 \wedge x \leq -82.2050077689329$$
$$x \geq -78.0162175641465 \wedge x \leq -75.9218224617533$$
$$x \geq -71.733032256967 \wedge x \leq -69.6386371545737$$
$$x \geq -65.4498469497874 \wedge x \leq -63.3554518473942$$
$$x \geq -59.1666616426078 \wedge x \leq -57.0722665402146$$
$$x \geq -52.8834763354282 \wedge x \leq -50.789081233035$$
$$x \geq -46.6002910282486 \wedge x \leq -44.5058959258554$$
$$x \geq -40.317105721069 \wedge x \leq -38.2227106186758$$
$$x \geq -34.0339204138894 \wedge x \leq -31.9395253114962$$
$$x \geq -27.7507351067098 \wedge x \leq -25.6563400043166$$
$$x \geq -21.4675497995303 \wedge x \leq -19.3731546971371$$
$$x \geq -15.1843644923507 \wedge x \leq -13.0899693899575$$
$$x \geq -8.90117918517108 \wedge x \leq -6.80678408277789$$
$$x \geq -2.61799387799149 \wedge x \leq -0.523598775598299$$
$$x \geq 3.66519142918809 \wedge x \leq 5.75958653158129$$
$$x \geq 9.94837673636768 \wedge x \leq 12.0427718387609$$
$$x \geq 16.2315620435473 \wedge x \leq 18.3259571459405$$
$$x \geq 22.5147473507269 \wedge x \leq 24.60914245312$$
$$x \geq 28.7979326579064 \wedge x \leq 30.8923277602996$$
$$x \geq 35.081117965086 \wedge x \leq 37.1755130674792$$
$$x \geq 41.3643032722656 \wedge x \leq 43.4586983746588$$
$$x \geq 47.6474885794452 \wedge x \leq 49.7418836818384$$
$$x \geq 53.9306738866248 \wedge x \leq 56.025068989018$$
$$x \geq 60.2138591938044 \wedge x \leq 62.3082542961976$$
$$x \geq 66.497044500984 \wedge x \leq 68.5914396033772$$
$$x \geq 72.7802298081635 \wedge x \leq 74.8746249105567$$
$$x \geq 79.0634151153431 \wedge x \leq 81.1578102177363$$
$$x \geq 85.3466004225227 \wedge x \leq 87.4409955249159$$
$$x \geq 91.6297857297023 \wedge x \leq 93.7241808320955$$
$$x \geq 97.9129710368819 \wedge x \leq 100.007366139275$$
$$x \geq 110.479341651241 \wedge x \leq 112.573736753634$$
$$x \geq 791.15774992903$$
$$5 \sin{\left(x \right)} + \cos{\left(2 x \right)} \geq -2$$
Para resolver esta desigualdad primero hay que resolver la ecuación correspondiente:
$$5 \sin{\left(x \right)} + \cos{\left(2 x \right)} = -2$$
Resolvemos:
Tenemos la ecuación
$$5 \sin{\left(x \right)} + \cos{\left(2 x \right)} = -2$$
cambiamos
$$5 \sin{\left(x \right)} + \cos{\left(2 x \right)} + 2 = 0$$
$$- 2 \sin^{2}{\left(x \right)} + 5 \sin{\left(x \right)} + 3 = 0$$
Sustituimos
$$w = \sin{\left(x \right)}$$
Es la ecuación de la forma
a*w^2 + b*w + c = 0
La ecuación cuadrática puede ser resuelta
con la ayuda del discriminante.
Las raíces de la ecuación cuadrática:
$$w_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$w_{2} = \frac{- \sqrt{D} - b}{2 a}$$
donde D = b^2 - 4*a*c es el discriminante.
Como
$$a = -2$$
$$b = 5$$
$$c = 3$$
, entonces
D = b^2 - 4 * a * c =
(5)^2 - 4 * (-2) * (3) = 49
Como D > 0 la ecuación tiene dos raíces.
w1 = (-b + sqrt(D)) / (2*a)
w2 = (-b - sqrt(D)) / (2*a)
o
$$w_{1} = - \frac{1}{2}$$
$$w_{2} = 3$$
hacemos cambio inverso
$$\sin{\left(x \right)} = w$$
Tenemos la ecuación
$$\sin{\left(x \right)} = w$$
es la ecuación trigonométrica más simple
Esta ecuación se reorganiza en
$$x = 2 \pi n + \operatorname{asin}{\left(w \right)}$$
$$x = 2 \pi n - \operatorname{asin}{\left(w \right)} + \pi$$
O
$$x = 2 \pi n + \operatorname{asin}{\left(w \right)}$$
$$x = 2 \pi n - \operatorname{asin}{\left(w \right)} + \pi$$
, donde n es cualquier número entero
sustituimos w:
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(w_{1} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(- \frac{1}{2} \right)}$$
$$x_{1} = 2 \pi n - \frac{\pi}{6}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(w_{2} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(3 \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(3 \right)}$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(- \frac{1}{2} \right)} + \pi$$
$$x_{3} = 2 \pi n + \frac{7 \pi}{6}$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(w_{2} \right)} + \pi$$
$$x_{4} = 2 \pi n + \pi - \operatorname{asin}{\left(3 \right)}$$
$$x_{4} = 2 \pi n + \pi - \operatorname{asin}{\left(3 \right)}$$
$$x_{1} = 100.007366139275$$
$$x_{2} = -82.2050077689329$$
$$x_{3} = 68.5914396033772$$
$$x_{4} = 87.4409955249159$$
$$x_{5} = -45867.7763411866$$
$$x_{6} = -71.733032256967$$
$$x_{7} = 41.3643032722656$$
$$x_{8} = 74.8746249105567$$
$$x_{9} = 12.0427718387609$$
$$x_{10} = -65.4498469497874$$
$$x_{11} = 37.1755130674792$$
$$x_{12} = 110.479341651241$$
$$x_{13} = 22.5147473507269$$
$$x_{14} = 30.8923277602996$$
$$x_{15} = -15.1843644923507$$
$$x_{16} = 81.1578102177363$$
$$x_{17} = 62.3082542961976$$
$$x_{18} = -59.1666616426078$$
$$x_{19} = 93.7241808320955$$
$$x_{20} = 47.6474885794452$$
$$x_{21} = 56.025068989018$$
$$x_{22} = 91.6297857297023$$
$$x_{23} = 79.0634151153431$$
$$x_{24} = 18.3259571459405$$
$$x_{25} = 66.497044500984$$
$$x_{26} = -75.9218224617533$$
$$x_{27} = -239.284640448423$$
$$x_{28} = -31.9395253114962$$
$$x_{29} = -46.6002910282486$$
$$x_{30} = -101.054563690472$$
$$x_{31} = 28.7979326579064$$
$$x_{32} = 53.9306738866248$$
$$x_{33} = -90.5825881785057$$
$$x_{34} = -8.90117918517108$$
$$x_{35} = -6.80678408277789$$
$$x_{36} = -69.6386371545737$$
$$x_{37} = 60.2138591938044$$
$$x_{38} = -44.5058959258554$$
$$x_{39} = -94.7713783832921$$
$$x_{40} = -27.7507351067098$$
$$x_{41} = -25.6563400043166$$
$$x_{42} = -34.0339204138894$$
$$x_{43} = -50.789081233035$$
$$x_{44} = -13.0899693899575$$
$$x_{45} = 3.66519142918809$$
$$x_{46} = -88.4881930761125$$
$$x_{47} = 85.3466004225227$$
$$x_{48} = -84.2994028713261$$
$$x_{49} = 16.2315620435473$$
$$x_{50} = -52.8834763354282$$
$$x_{51} = -643.502895210309$$
$$x_{52} = -40.317105721069$$
$$x_{53} = -0.523598775598299$$
$$x_{54} = 97.9129710368819$$
$$x_{55} = 49.7418836818384$$
$$x_{56} = 35.081117965086$$
$$x_{57} = 791.15774992903$$
$$x_{58} = -96.8657734856853$$
$$x_{59} = 24.60914245312$$
$$x_{60} = 5.75958653158129$$
$$x_{61} = -21.4675497995303$$
$$x_{62} = -63.3554518473942$$
$$x_{63} = -78.0162175641465$$
$$x_{64} = -38.2227106186758$$
$$x_{65} = 72.7802298081635$$
$$x_{66} = 43.4586983746588$$
$$x_{67} = -2.61799387799149$$
$$x_{68} = 9.94837673636768$$
$$x_{69} = -57.0722665402146$$
$$x_{70} = 112.573736753634$$
$$x_{71} = -19.3731546971371$$
$$x_{1} = 100.007366139275$$
$$x_{2} = -82.2050077689329$$
$$x_{3} = 68.5914396033772$$
$$x_{4} = 87.4409955249159$$
$$x_{5} = -45867.7763411866$$
$$x_{6} = -71.733032256967$$
$$x_{7} = 41.3643032722656$$
$$x_{8} = 74.8746249105567$$
$$x_{9} = 12.0427718387609$$
$$x_{10} = -65.4498469497874$$
$$x_{11} = 37.1755130674792$$
$$x_{12} = 110.479341651241$$
$$x_{13} = 22.5147473507269$$
$$x_{14} = 30.8923277602996$$
$$x_{15} = -15.1843644923507$$
$$x_{16} = 81.1578102177363$$
$$x_{17} = 62.3082542961976$$
$$x_{18} = -59.1666616426078$$
$$x_{19} = 93.7241808320955$$
$$x_{20} = 47.6474885794452$$
$$x_{21} = 56.025068989018$$
$$x_{22} = 91.6297857297023$$
$$x_{23} = 79.0634151153431$$
$$x_{24} = 18.3259571459405$$
$$x_{25} = 66.497044500984$$
$$x_{26} = -75.9218224617533$$
$$x_{27} = -239.284640448423$$
$$x_{28} = -31.9395253114962$$
$$x_{29} = -46.6002910282486$$
$$x_{30} = -101.054563690472$$
$$x_{31} = 28.7979326579064$$
$$x_{32} = 53.9306738866248$$
$$x_{33} = -90.5825881785057$$
$$x_{34} = -8.90117918517108$$
$$x_{35} = -6.80678408277789$$
$$x_{36} = -69.6386371545737$$
$$x_{37} = 60.2138591938044$$
$$x_{38} = -44.5058959258554$$
$$x_{39} = -94.7713783832921$$
$$x_{40} = -27.7507351067098$$
$$x_{41} = -25.6563400043166$$
$$x_{42} = -34.0339204138894$$
$$x_{43} = -50.789081233035$$
$$x_{44} = -13.0899693899575$$
$$x_{45} = 3.66519142918809$$
$$x_{46} = -88.4881930761125$$
$$x_{47} = 85.3466004225227$$
$$x_{48} = -84.2994028713261$$
$$x_{49} = 16.2315620435473$$
$$x_{50} = -52.8834763354282$$
$$x_{51} = -643.502895210309$$
$$x_{52} = -40.317105721069$$
$$x_{53} = -0.523598775598299$$
$$x_{54} = 97.9129710368819$$
$$x_{55} = 49.7418836818384$$
$$x_{56} = 35.081117965086$$
$$x_{57} = 791.15774992903$$
$$x_{58} = -96.8657734856853$$
$$x_{59} = 24.60914245312$$
$$x_{60} = 5.75958653158129$$
$$x_{61} = -21.4675497995303$$
$$x_{62} = -63.3554518473942$$
$$x_{63} = -78.0162175641465$$
$$x_{64} = -38.2227106186758$$
$$x_{65} = 72.7802298081635$$
$$x_{66} = 43.4586983746588$$
$$x_{67} = -2.61799387799149$$
$$x_{68} = 9.94837673636768$$
$$x_{69} = -57.0722665402146$$
$$x_{70} = 112.573736753634$$
$$x_{71} = -19.3731546971371$$
Las raíces dadas
$$x_{5} = -45867.7763411866$$
$$x_{51} = -643.502895210309$$
$$x_{27} = -239.284640448423$$
$$x_{30} = -101.054563690472$$
$$x_{58} = -96.8657734856853$$
$$x_{39} = -94.7713783832921$$
$$x_{33} = -90.5825881785057$$
$$x_{46} = -88.4881930761125$$
$$x_{48} = -84.2994028713261$$
$$x_{2} = -82.2050077689329$$
$$x_{63} = -78.0162175641465$$
$$x_{26} = -75.9218224617533$$
$$x_{6} = -71.733032256967$$
$$x_{36} = -69.6386371545737$$
$$x_{10} = -65.4498469497874$$
$$x_{62} = -63.3554518473942$$
$$x_{18} = -59.1666616426078$$
$$x_{69} = -57.0722665402146$$
$$x_{50} = -52.8834763354282$$
$$x_{43} = -50.789081233035$$
$$x_{29} = -46.6002910282486$$
$$x_{38} = -44.5058959258554$$
$$x_{52} = -40.317105721069$$
$$x_{64} = -38.2227106186758$$
$$x_{42} = -34.0339204138894$$
$$x_{28} = -31.9395253114962$$
$$x_{40} = -27.7507351067098$$
$$x_{41} = -25.6563400043166$$
$$x_{61} = -21.4675497995303$$
$$x_{71} = -19.3731546971371$$
$$x_{15} = -15.1843644923507$$
$$x_{44} = -13.0899693899575$$
$$x_{34} = -8.90117918517108$$
$$x_{35} = -6.80678408277789$$
$$x_{67} = -2.61799387799149$$
$$x_{53} = -0.523598775598299$$
$$x_{45} = 3.66519142918809$$
$$x_{60} = 5.75958653158129$$
$$x_{68} = 9.94837673636768$$
$$x_{9} = 12.0427718387609$$
$$x_{49} = 16.2315620435473$$
$$x_{24} = 18.3259571459405$$
$$x_{13} = 22.5147473507269$$
$$x_{59} = 24.60914245312$$
$$x_{31} = 28.7979326579064$$
$$x_{14} = 30.8923277602996$$
$$x_{56} = 35.081117965086$$
$$x_{11} = 37.1755130674792$$
$$x_{7} = 41.3643032722656$$
$$x_{66} = 43.4586983746588$$
$$x_{20} = 47.6474885794452$$
$$x_{55} = 49.7418836818384$$
$$x_{32} = 53.9306738866248$$
$$x_{21} = 56.025068989018$$
$$x_{37} = 60.2138591938044$$
$$x_{17} = 62.3082542961976$$
$$x_{25} = 66.497044500984$$
$$x_{3} = 68.5914396033772$$
$$x_{65} = 72.7802298081635$$
$$x_{8} = 74.8746249105567$$
$$x_{23} = 79.0634151153431$$
$$x_{16} = 81.1578102177363$$
$$x_{47} = 85.3466004225227$$
$$x_{4} = 87.4409955249159$$
$$x_{22} = 91.6297857297023$$
$$x_{19} = 93.7241808320955$$
$$x_{54} = 97.9129710368819$$
$$x_{1} = 100.007366139275$$
$$x_{12} = 110.479341651241$$
$$x_{70} = 112.573736753634$$
$$x_{57} = 791.15774992903$$
son puntos de cambio del signo de desigualdad en las soluciones.
Primero definámonos con el signo hasta el punto extremo izquierdo:
$$x_{0} \leq x_{5}$$
Consideremos, por ejemplo, el punto
$$x_{0} = x_{5} - \frac{1}{10}$$
=
$$-45867.7763411866 + - \frac{1}{10}$$
=
$$-45867.8763411866$$
lo sustituimos en la expresión
$$5 \sin{\left(x \right)} + \cos{\left(2 x \right)} \geq -2$$
$$5 \sin{\left(-45867.8763411866 \right)} + \cos{\left(\left(-45867.8763411866\right) 2 \right)} \geq -2$$
-2.60182118651354 >= -2
pero
-2.60182118651354 < -2
Entonces
$$x \leq -45867.7763411866$$
no se cumple
significa que una de las soluciones de nuestra ecuación será con:
$$x \geq -45867.7763411866 \wedge x \leq -643.502895210309$$
_____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____
/ \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
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x5 x51 x27 x30 x58 x39 x33 x46 x48 x2 x63 x26 x6 x36 x10 x62 x18 x69 x50 x43 x29 x38 x52 x64 x42 x28 x40 x41 x61 x71 x15 x44 x34 x35 x67 x53 x45 x60 x68 x9 x49 x24 x13 x59 x31 x14 x56 x11 x7 x66 x20 x55 x32 x21 x37 x17 x25 x3 x65 x8 x23 x16 x47 x4 x22 x19 x54 x1 x12 x70 x57Recibiremos otras soluciones de la desigualdad pasando al polo siguiente etc.
etc.
Respuesta:
$$x \geq -45867.7763411866 \wedge x \leq -643.502895210309$$
$$x \geq -239.284640448423 \wedge x \leq -101.054563690472$$
$$x \geq -96.8657734856853 \wedge x \leq -94.7713783832921$$
$$x \geq -90.5825881785057 \wedge x \leq -88.4881930761125$$
$$x \geq -84.2994028713261 \wedge x \leq -82.2050077689329$$
$$x \geq -78.0162175641465 \wedge x \leq -75.9218224617533$$
$$x \geq -71.733032256967 \wedge x \leq -69.6386371545737$$
$$x \geq -65.4498469497874 \wedge x \leq -63.3554518473942$$
$$x \geq -59.1666616426078 \wedge x \leq -57.0722665402146$$
$$x \geq -52.8834763354282 \wedge x \leq -50.789081233035$$
$$x \geq -46.6002910282486 \wedge x \leq -44.5058959258554$$
$$x \geq -40.317105721069 \wedge x \leq -38.2227106186758$$
$$x \geq -34.0339204138894 \wedge x \leq -31.9395253114962$$
$$x \geq -27.7507351067098 \wedge x \leq -25.6563400043166$$
$$x \geq -21.4675497995303 \wedge x \leq -19.3731546971371$$
$$x \geq -15.1843644923507 \wedge x \leq -13.0899693899575$$
$$x \geq -8.90117918517108 \wedge x \leq -6.80678408277789$$
$$x \geq -2.61799387799149 \wedge x \leq -0.523598775598299$$
$$x \geq 3.66519142918809 \wedge x \leq 5.75958653158129$$
$$x \geq 9.94837673636768 \wedge x \leq 12.0427718387609$$
$$x \geq 16.2315620435473 \wedge x \leq 18.3259571459405$$
$$x \geq 22.5147473507269 \wedge x \leq 24.60914245312$$
$$x \geq 28.7979326579064 \wedge x \leq 30.8923277602996$$
$$x \geq 35.081117965086 \wedge x \leq 37.1755130674792$$
$$x \geq 41.3643032722656 \wedge x \leq 43.4586983746588$$
$$x \geq 47.6474885794452 \wedge x \leq 49.7418836818384$$
$$x \geq 53.9306738866248 \wedge x \leq 56.025068989018$$
$$x \geq 60.2138591938044 \wedge x \leq 62.3082542961976$$
$$x \geq 66.497044500984 \wedge x \leq 68.5914396033772$$
$$x \geq 72.7802298081635 \wedge x \leq 74.8746249105567$$
$$x \geq 79.0634151153431 \wedge x \leq 81.1578102177363$$
$$x \geq 85.3466004225227 \wedge x \leq 87.4409955249159$$
$$x \geq 91.6297857297023 \wedge x \leq 93.7241808320955$$
$$x \geq 97.9129710368819 \wedge x \leq 100.007366139275$$
$$x \geq 110.479341651241 \wedge x \leq 112.573736753634$$
$$x \geq 791.15774992903$$
Solución de la desigualdad en el gráfico
Respuesta rápida
[src]
/ / 7*pi\ /11*pi \\ Or|And|0 <= x, x <= ----|, And|----- <= x, x <= 2*pi|| \ \ 6 / \ 6 //
$$\left(0 \leq x \wedge x \leq \frac{7 \pi}{6}\right) \vee \left(\frac{11 \pi}{6} \leq x \wedge x \leq 2 \pi\right)$$
((0 <= x)∧(x <= 7*pi/6))∨((11*pi/6 <= x)∧(x <= 2*pi))
Respuesta rápida 2
[src]
7*pi 11*pi
[0, ----] U [-----, 2*pi]
6 6
$$x\ in\ \left[0, \frac{7 \pi}{6}\right] \cup \left[\frac{11 \pi}{6}, 2 \pi\right]$$
x in Union(Interval(0, 7*pi/6), Interval(11*pi/6, 2*pi))