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:
-
3^x-1-4*3^0,5x-1+1>0
-
(x-1)/(x-6)>0
- x^2+64>=0
-
2x-x^2<0
-
Expresiones idénticas
- cos dos x+5sinx<=-2
- coseno de 2x más 5 seno de x menos o igual a menos 2
- coseno de dos x más 5 seno de x menos o igual a menos 2
-
Expresiones semejantes
- cos2x+5sinx<=+2
- cos2x-5sinx<=-2
cos2x+5sinx<=-2 desigualdades
Solución
Ha introducido
[src]
cos(2*x) + 5*sin(x) <= -2
$$5 \sin{\left(x \right)} + \cos{\left(2 x \right)} \leq -2$$
5*sin(x) + cos(2*x) <= -2
Solución detallada
Se da la desigualdad:
$$5 \sin{\left(x \right)} + \cos{\left(2 x \right)} \leq -2$$
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:
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(w_{1} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(- \frac{1}{2} \right)}$$
$$x_{1} = 2 \pi n - \frac{\pi}{6}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(w_{2} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(3 \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(3 \right)}$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(- \frac{1}{2} \right)} + \pi$$
$$x_{3} = 2 \pi n + \frac{7 \pi}{6}$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(w_{2} \right)} + \pi$$
$$x_{4} = 2 \pi n + \pi - \operatorname{asin}{\left(3 \right)}$$
$$x_{4} = 2 \pi n + \pi - \operatorname{asin}{\left(3 \right)}$$
$$x_{1} = -75.9218224617533$$
$$x_{2} = -90.5825881785057$$
$$x_{3} = 30.8923277602996$$
$$x_{4} = -96.8657734856853$$
$$x_{5} = -65.4498469497874$$
$$x_{6} = -15.1843644923507$$
$$x_{7} = -38.2227106186758$$
$$x_{8} = -84.2994028713261$$
$$x_{9} = -45867.7763411866$$
$$x_{10} = 9.94837673636768$$
$$x_{11} = -34.0339204138894$$
$$x_{12} = 110.479341651241$$
$$x_{13} = 66.497044500984$$
$$x_{14} = -21.4675497995303$$
$$x_{15} = -57.0722665402146$$
$$x_{16} = -40.317105721069$$
$$x_{17} = 35.081117965086$$
$$x_{18} = -2.61799387799149$$
$$x_{19} = 18.3259571459405$$
$$x_{20} = -0.523598775598299$$
$$x_{21} = -44.5058959258554$$
$$x_{22} = 53.9306738866248$$
$$x_{23} = -94.7713783832921$$
$$x_{24} = -82.2050077689329$$
$$x_{25} = -52.8834763354282$$
$$x_{26} = -25.6563400043166$$
$$x_{27} = -6.80678408277789$$
$$x_{28} = -643.502895210309$$
$$x_{29} = 16.2315620435473$$
$$x_{30} = 41.3643032722656$$
$$x_{31} = 79.0634151153431$$
$$x_{32} = -101.054563690472$$
$$x_{33} = 74.8746249105567$$
$$x_{34} = 37.1755130674792$$
$$x_{35} = -46.6002910282486$$
$$x_{36} = -78.0162175641465$$
$$x_{37} = 22.5147473507269$$
$$x_{38} = 12.0427718387609$$
$$x_{39} = 47.6474885794452$$
$$x_{40} = 24.60914245312$$
$$x_{41} = 112.573736753634$$
$$x_{42} = 91.6297857297023$$
$$x_{43} = -88.4881930761125$$
$$x_{44} = -8.90117918517108$$
$$x_{45} = -50.789081233035$$
$$x_{46} = 72.7802298081635$$
$$x_{47} = 49.7418836818384$$
$$x_{48} = 60.2138591938044$$
$$x_{49} = 43.4586983746588$$
$$x_{50} = 28.7979326579064$$
$$x_{51} = 62.3082542961976$$
$$x_{52} = -239.284640448423$$
$$x_{53} = -59.1666616426078$$
$$x_{54} = 68.5914396033772$$
$$x_{55} = -71.733032256967$$
$$x_{56} = -13.0899693899575$$
$$x_{57} = 100.007366139275$$
$$x_{58} = 791.15774992903$$
$$x_{59} = -19.3731546971371$$
$$x_{60} = -69.6386371545737$$
$$x_{61} = 93.7241808320955$$
$$x_{62} = 3.66519142918809$$
$$x_{63} = 97.9129710368819$$
$$x_{64} = -27.7507351067098$$
$$x_{65} = 81.1578102177363$$
$$x_{66} = -63.3554518473942$$
$$x_{67} = 56.025068989018$$
$$x_{68} = 87.4409955249159$$
$$x_{69} = -31.9395253114962$$
$$x_{70} = 5.75958653158129$$
$$x_{71} = 85.3466004225227$$
$$x_{1} = -75.9218224617533$$
$$x_{2} = -90.5825881785057$$
$$x_{3} = 30.8923277602996$$
$$x_{4} = -96.8657734856853$$
$$x_{5} = -65.4498469497874$$
$$x_{6} = -15.1843644923507$$
$$x_{7} = -38.2227106186758$$
$$x_{8} = -84.2994028713261$$
$$x_{9} = -45867.7763411866$$
$$x_{10} = 9.94837673636768$$
$$x_{11} = -34.0339204138894$$
$$x_{12} = 110.479341651241$$
$$x_{13} = 66.497044500984$$
$$x_{14} = -21.4675497995303$$
$$x_{15} = -57.0722665402146$$
$$x_{16} = -40.317105721069$$
$$x_{17} = 35.081117965086$$
$$x_{18} = -2.61799387799149$$
$$x_{19} = 18.3259571459405$$
$$x_{20} = -0.523598775598299$$
$$x_{21} = -44.5058959258554$$
$$x_{22} = 53.9306738866248$$
$$x_{23} = -94.7713783832921$$
$$x_{24} = -82.2050077689329$$
$$x_{25} = -52.8834763354282$$
$$x_{26} = -25.6563400043166$$
$$x_{27} = -6.80678408277789$$
$$x_{28} = -643.502895210309$$
$$x_{29} = 16.2315620435473$$
$$x_{30} = 41.3643032722656$$
$$x_{31} = 79.0634151153431$$
$$x_{32} = -101.054563690472$$
$$x_{33} = 74.8746249105567$$
$$x_{34} = 37.1755130674792$$
$$x_{35} = -46.6002910282486$$
$$x_{36} = -78.0162175641465$$
$$x_{37} = 22.5147473507269$$
$$x_{38} = 12.0427718387609$$
$$x_{39} = 47.6474885794452$$
$$x_{40} = 24.60914245312$$
$$x_{41} = 112.573736753634$$
$$x_{42} = 91.6297857297023$$
$$x_{43} = -88.4881930761125$$
$$x_{44} = -8.90117918517108$$
$$x_{45} = -50.789081233035$$
$$x_{46} = 72.7802298081635$$
$$x_{47} = 49.7418836818384$$
$$x_{48} = 60.2138591938044$$
$$x_{49} = 43.4586983746588$$
$$x_{50} = 28.7979326579064$$
$$x_{51} = 62.3082542961976$$
$$x_{52} = -239.284640448423$$
$$x_{53} = -59.1666616426078$$
$$x_{54} = 68.5914396033772$$
$$x_{55} = -71.733032256967$$
$$x_{56} = -13.0899693899575$$
$$x_{57} = 100.007366139275$$
$$x_{58} = 791.15774992903$$
$$x_{59} = -19.3731546971371$$
$$x_{60} = -69.6386371545737$$
$$x_{61} = 93.7241808320955$$
$$x_{62} = 3.66519142918809$$
$$x_{63} = 97.9129710368819$$
$$x_{64} = -27.7507351067098$$
$$x_{65} = 81.1578102177363$$
$$x_{66} = -63.3554518473942$$
$$x_{67} = 56.025068989018$$
$$x_{68} = 87.4409955249159$$
$$x_{69} = -31.9395253114962$$
$$x_{70} = 5.75958653158129$$
$$x_{71} = 85.3466004225227$$
Las raíces dadas
$$x_{9} = -45867.7763411866$$
$$x_{28} = -643.502895210309$$
$$x_{52} = -239.284640448423$$
$$x_{32} = -101.054563690472$$
$$x_{4} = -96.8657734856853$$
$$x_{23} = -94.7713783832921$$
$$x_{2} = -90.5825881785057$$
$$x_{43} = -88.4881930761125$$
$$x_{8} = -84.2994028713261$$
$$x_{24} = -82.2050077689329$$
$$x_{36} = -78.0162175641465$$
$$x_{1} = -75.9218224617533$$
$$x_{55} = -71.733032256967$$
$$x_{60} = -69.6386371545737$$
$$x_{5} = -65.4498469497874$$
$$x_{66} = -63.3554518473942$$
$$x_{53} = -59.1666616426078$$
$$x_{15} = -57.0722665402146$$
$$x_{25} = -52.8834763354282$$
$$x_{45} = -50.789081233035$$
$$x_{35} = -46.6002910282486$$
$$x_{21} = -44.5058959258554$$
$$x_{16} = -40.317105721069$$
$$x_{7} = -38.2227106186758$$
$$x_{11} = -34.0339204138894$$
$$x_{69} = -31.9395253114962$$
$$x_{64} = -27.7507351067098$$
$$x_{26} = -25.6563400043166$$
$$x_{14} = -21.4675497995303$$
$$x_{59} = -19.3731546971371$$
$$x_{6} = -15.1843644923507$$
$$x_{56} = -13.0899693899575$$
$$x_{44} = -8.90117918517108$$
$$x_{27} = -6.80678408277789$$
$$x_{18} = -2.61799387799149$$
$$x_{20} = -0.523598775598299$$
$$x_{62} = 3.66519142918809$$
$$x_{70} = 5.75958653158129$$
$$x_{10} = 9.94837673636768$$
$$x_{38} = 12.0427718387609$$
$$x_{29} = 16.2315620435473$$
$$x_{19} = 18.3259571459405$$
$$x_{37} = 22.5147473507269$$
$$x_{40} = 24.60914245312$$
$$x_{50} = 28.7979326579064$$
$$x_{3} = 30.8923277602996$$
$$x_{17} = 35.081117965086$$
$$x_{34} = 37.1755130674792$$
$$x_{30} = 41.3643032722656$$
$$x_{49} = 43.4586983746588$$
$$x_{39} = 47.6474885794452$$
$$x_{47} = 49.7418836818384$$
$$x_{22} = 53.9306738866248$$
$$x_{67} = 56.025068989018$$
$$x_{48} = 60.2138591938044$$
$$x_{51} = 62.3082542961976$$
$$x_{13} = 66.497044500984$$
$$x_{54} = 68.5914396033772$$
$$x_{46} = 72.7802298081635$$
$$x_{33} = 74.8746249105567$$
$$x_{31} = 79.0634151153431$$
$$x_{65} = 81.1578102177363$$
$$x_{71} = 85.3466004225227$$
$$x_{68} = 87.4409955249159$$
$$x_{42} = 91.6297857297023$$
$$x_{61} = 93.7241808320955$$
$$x_{63} = 97.9129710368819$$
$$x_{57} = 100.007366139275$$
$$x_{12} = 110.479341651241$$
$$x_{41} = 112.573736753634$$
$$x_{58} = 791.15774992903$$
son puntos de cambio del signo de desigualdad en las soluciones.
Primero definámonos con el signo hasta el punto extremo izquierdo:
$$x_{0} \leq x_{9}$$
Consideremos, por ejemplo, el punto
$$x_{0} = x_{9} - \frac{1}{10}$$
=
$$-45867.7763411866 + - \frac{1}{10}$$
=
$$-45867.8763411866$$
lo sustituimos en la expresión
$$5 \sin{\left(x \right)} + \cos{\left(2 x \right)} \leq -2$$
$$5 \sin{\left(-45867.8763411866 \right)} + \cos{\left(\left(-45867.8763411866\right) 2 \right)} \leq -2$$
significa que una de las soluciones de nuestra ecuación será con:
$$x \leq -45867.7763411866$$
Recibiremos otras soluciones de la desigualdad pasando al polo siguiente etc.
etc.
Respuesta:
$$x \leq -45867.7763411866$$
$$x \geq -643.502895210309 \wedge x \leq -239.284640448423$$
$$x \geq -101.054563690472 \wedge x \leq -96.8657734856853$$
$$x \geq -94.7713783832921 \wedge x \leq -90.5825881785057$$
$$x \geq -88.4881930761125 \wedge x \leq -84.2994028713261$$
$$x \geq -82.2050077689329 \wedge x \leq -78.0162175641465$$
$$x \geq -75.9218224617533 \wedge x \leq -71.733032256967$$
$$x \geq -69.6386371545737 \wedge x \leq -65.4498469497874$$
$$x \geq -63.3554518473942 \wedge x \leq -59.1666616426078$$
$$x \geq -57.0722665402146 \wedge x \leq -52.8834763354282$$
$$x \geq -50.789081233035 \wedge x \leq -46.6002910282486$$
$$x \geq -44.5058959258554 \wedge x \leq -40.317105721069$$
$$x \geq -38.2227106186758 \wedge x \leq -34.0339204138894$$
$$x \geq -31.9395253114962 \wedge x \leq -27.7507351067098$$
$$x \geq -25.6563400043166 \wedge x \leq -21.4675497995303$$
$$x \geq -19.3731546971371 \wedge x \leq -15.1843644923507$$
$$x \geq -13.0899693899575 \wedge x \leq -8.90117918517108$$
$$x \geq -6.80678408277789 \wedge x \leq -2.61799387799149$$
$$x \geq -0.523598775598299 \wedge x \leq 3.66519142918809$$
$$x \geq 5.75958653158129 \wedge x \leq 9.94837673636768$$
$$x \geq 12.0427718387609 \wedge x \leq 16.2315620435473$$
$$x \geq 18.3259571459405 \wedge x \leq 22.5147473507269$$
$$x \geq 24.60914245312 \wedge x \leq 28.7979326579064$$
$$x \geq 30.8923277602996 \wedge x \leq 35.081117965086$$
$$x \geq 37.1755130674792 \wedge x \leq 41.3643032722656$$
$$x \geq 43.4586983746588 \wedge x \leq 47.6474885794452$$
$$x \geq 49.7418836818384 \wedge x \leq 53.9306738866248$$
$$x \geq 56.025068989018 \wedge x \leq 60.2138591938044$$
$$x \geq 62.3082542961976 \wedge x \leq 66.497044500984$$
$$x \geq 68.5914396033772 \wedge x \leq 72.7802298081635$$
$$x \geq 74.8746249105567 \wedge x \leq 79.0634151153431$$
$$x \geq 81.1578102177363 \wedge x \leq 85.3466004225227$$
$$x \geq 87.4409955249159 \wedge x \leq 91.6297857297023$$
$$x \geq 93.7241808320955 \wedge x \leq 97.9129710368819$$
$$x \geq 100.007366139275 \wedge x \leq 110.479341651241$$
$$x \geq 112.573736753634 \wedge x \leq 791.15774992903$$
$$5 \sin{\left(x \right)} + \cos{\left(2 x \right)} \leq -2$$
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:
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(w_{1} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(- \frac{1}{2} \right)}$$
$$x_{1} = 2 \pi n - \frac{\pi}{6}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(w_{2} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(3 \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(3 \right)}$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(- \frac{1}{2} \right)} + \pi$$
$$x_{3} = 2 \pi n + \frac{7 \pi}{6}$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(w_{2} \right)} + \pi$$
$$x_{4} = 2 \pi n + \pi - \operatorname{asin}{\left(3 \right)}$$
$$x_{4} = 2 \pi n + \pi - \operatorname{asin}{\left(3 \right)}$$
$$x_{1} = -75.9218224617533$$
$$x_{2} = -90.5825881785057$$
$$x_{3} = 30.8923277602996$$
$$x_{4} = -96.8657734856853$$
$$x_{5} = -65.4498469497874$$
$$x_{6} = -15.1843644923507$$
$$x_{7} = -38.2227106186758$$
$$x_{8} = -84.2994028713261$$
$$x_{9} = -45867.7763411866$$
$$x_{10} = 9.94837673636768$$
$$x_{11} = -34.0339204138894$$
$$x_{12} = 110.479341651241$$
$$x_{13} = 66.497044500984$$
$$x_{14} = -21.4675497995303$$
$$x_{15} = -57.0722665402146$$
$$x_{16} = -40.317105721069$$
$$x_{17} = 35.081117965086$$
$$x_{18} = -2.61799387799149$$
$$x_{19} = 18.3259571459405$$
$$x_{20} = -0.523598775598299$$
$$x_{21} = -44.5058959258554$$
$$x_{22} = 53.9306738866248$$
$$x_{23} = -94.7713783832921$$
$$x_{24} = -82.2050077689329$$
$$x_{25} = -52.8834763354282$$
$$x_{26} = -25.6563400043166$$
$$x_{27} = -6.80678408277789$$
$$x_{28} = -643.502895210309$$
$$x_{29} = 16.2315620435473$$
$$x_{30} = 41.3643032722656$$
$$x_{31} = 79.0634151153431$$
$$x_{32} = -101.054563690472$$
$$x_{33} = 74.8746249105567$$
$$x_{34} = 37.1755130674792$$
$$x_{35} = -46.6002910282486$$
$$x_{36} = -78.0162175641465$$
$$x_{37} = 22.5147473507269$$
$$x_{38} = 12.0427718387609$$
$$x_{39} = 47.6474885794452$$
$$x_{40} = 24.60914245312$$
$$x_{41} = 112.573736753634$$
$$x_{42} = 91.6297857297023$$
$$x_{43} = -88.4881930761125$$
$$x_{44} = -8.90117918517108$$
$$x_{45} = -50.789081233035$$
$$x_{46} = 72.7802298081635$$
$$x_{47} = 49.7418836818384$$
$$x_{48} = 60.2138591938044$$
$$x_{49} = 43.4586983746588$$
$$x_{50} = 28.7979326579064$$
$$x_{51} = 62.3082542961976$$
$$x_{52} = -239.284640448423$$
$$x_{53} = -59.1666616426078$$
$$x_{54} = 68.5914396033772$$
$$x_{55} = -71.733032256967$$
$$x_{56} = -13.0899693899575$$
$$x_{57} = 100.007366139275$$
$$x_{58} = 791.15774992903$$
$$x_{59} = -19.3731546971371$$
$$x_{60} = -69.6386371545737$$
$$x_{61} = 93.7241808320955$$
$$x_{62} = 3.66519142918809$$
$$x_{63} = 97.9129710368819$$
$$x_{64} = -27.7507351067098$$
$$x_{65} = 81.1578102177363$$
$$x_{66} = -63.3554518473942$$
$$x_{67} = 56.025068989018$$
$$x_{68} = 87.4409955249159$$
$$x_{69} = -31.9395253114962$$
$$x_{70} = 5.75958653158129$$
$$x_{71} = 85.3466004225227$$
$$x_{1} = -75.9218224617533$$
$$x_{2} = -90.5825881785057$$
$$x_{3} = 30.8923277602996$$
$$x_{4} = -96.8657734856853$$
$$x_{5} = -65.4498469497874$$
$$x_{6} = -15.1843644923507$$
$$x_{7} = -38.2227106186758$$
$$x_{8} = -84.2994028713261$$
$$x_{9} = -45867.7763411866$$
$$x_{10} = 9.94837673636768$$
$$x_{11} = -34.0339204138894$$
$$x_{12} = 110.479341651241$$
$$x_{13} = 66.497044500984$$
$$x_{14} = -21.4675497995303$$
$$x_{15} = -57.0722665402146$$
$$x_{16} = -40.317105721069$$
$$x_{17} = 35.081117965086$$
$$x_{18} = -2.61799387799149$$
$$x_{19} = 18.3259571459405$$
$$x_{20} = -0.523598775598299$$
$$x_{21} = -44.5058959258554$$
$$x_{22} = 53.9306738866248$$
$$x_{23} = -94.7713783832921$$
$$x_{24} = -82.2050077689329$$
$$x_{25} = -52.8834763354282$$
$$x_{26} = -25.6563400043166$$
$$x_{27} = -6.80678408277789$$
$$x_{28} = -643.502895210309$$
$$x_{29} = 16.2315620435473$$
$$x_{30} = 41.3643032722656$$
$$x_{31} = 79.0634151153431$$
$$x_{32} = -101.054563690472$$
$$x_{33} = 74.8746249105567$$
$$x_{34} = 37.1755130674792$$
$$x_{35} = -46.6002910282486$$
$$x_{36} = -78.0162175641465$$
$$x_{37} = 22.5147473507269$$
$$x_{38} = 12.0427718387609$$
$$x_{39} = 47.6474885794452$$
$$x_{40} = 24.60914245312$$
$$x_{41} = 112.573736753634$$
$$x_{42} = 91.6297857297023$$
$$x_{43} = -88.4881930761125$$
$$x_{44} = -8.90117918517108$$
$$x_{45} = -50.789081233035$$
$$x_{46} = 72.7802298081635$$
$$x_{47} = 49.7418836818384$$
$$x_{48} = 60.2138591938044$$
$$x_{49} = 43.4586983746588$$
$$x_{50} = 28.7979326579064$$
$$x_{51} = 62.3082542961976$$
$$x_{52} = -239.284640448423$$
$$x_{53} = -59.1666616426078$$
$$x_{54} = 68.5914396033772$$
$$x_{55} = -71.733032256967$$
$$x_{56} = -13.0899693899575$$
$$x_{57} = 100.007366139275$$
$$x_{58} = 791.15774992903$$
$$x_{59} = -19.3731546971371$$
$$x_{60} = -69.6386371545737$$
$$x_{61} = 93.7241808320955$$
$$x_{62} = 3.66519142918809$$
$$x_{63} = 97.9129710368819$$
$$x_{64} = -27.7507351067098$$
$$x_{65} = 81.1578102177363$$
$$x_{66} = -63.3554518473942$$
$$x_{67} = 56.025068989018$$
$$x_{68} = 87.4409955249159$$
$$x_{69} = -31.9395253114962$$
$$x_{70} = 5.75958653158129$$
$$x_{71} = 85.3466004225227$$
Las raíces dadas
$$x_{9} = -45867.7763411866$$
$$x_{28} = -643.502895210309$$
$$x_{52} = -239.284640448423$$
$$x_{32} = -101.054563690472$$
$$x_{4} = -96.8657734856853$$
$$x_{23} = -94.7713783832921$$
$$x_{2} = -90.5825881785057$$
$$x_{43} = -88.4881930761125$$
$$x_{8} = -84.2994028713261$$
$$x_{24} = -82.2050077689329$$
$$x_{36} = -78.0162175641465$$
$$x_{1} = -75.9218224617533$$
$$x_{55} = -71.733032256967$$
$$x_{60} = -69.6386371545737$$
$$x_{5} = -65.4498469497874$$
$$x_{66} = -63.3554518473942$$
$$x_{53} = -59.1666616426078$$
$$x_{15} = -57.0722665402146$$
$$x_{25} = -52.8834763354282$$
$$x_{45} = -50.789081233035$$
$$x_{35} = -46.6002910282486$$
$$x_{21} = -44.5058959258554$$
$$x_{16} = -40.317105721069$$
$$x_{7} = -38.2227106186758$$
$$x_{11} = -34.0339204138894$$
$$x_{69} = -31.9395253114962$$
$$x_{64} = -27.7507351067098$$
$$x_{26} = -25.6563400043166$$
$$x_{14} = -21.4675497995303$$
$$x_{59} = -19.3731546971371$$
$$x_{6} = -15.1843644923507$$
$$x_{56} = -13.0899693899575$$
$$x_{44} = -8.90117918517108$$
$$x_{27} = -6.80678408277789$$
$$x_{18} = -2.61799387799149$$
$$x_{20} = -0.523598775598299$$
$$x_{62} = 3.66519142918809$$
$$x_{70} = 5.75958653158129$$
$$x_{10} = 9.94837673636768$$
$$x_{38} = 12.0427718387609$$
$$x_{29} = 16.2315620435473$$
$$x_{19} = 18.3259571459405$$
$$x_{37} = 22.5147473507269$$
$$x_{40} = 24.60914245312$$
$$x_{50} = 28.7979326579064$$
$$x_{3} = 30.8923277602996$$
$$x_{17} = 35.081117965086$$
$$x_{34} = 37.1755130674792$$
$$x_{30} = 41.3643032722656$$
$$x_{49} = 43.4586983746588$$
$$x_{39} = 47.6474885794452$$
$$x_{47} = 49.7418836818384$$
$$x_{22} = 53.9306738866248$$
$$x_{67} = 56.025068989018$$
$$x_{48} = 60.2138591938044$$
$$x_{51} = 62.3082542961976$$
$$x_{13} = 66.497044500984$$
$$x_{54} = 68.5914396033772$$
$$x_{46} = 72.7802298081635$$
$$x_{33} = 74.8746249105567$$
$$x_{31} = 79.0634151153431$$
$$x_{65} = 81.1578102177363$$
$$x_{71} = 85.3466004225227$$
$$x_{68} = 87.4409955249159$$
$$x_{42} = 91.6297857297023$$
$$x_{61} = 93.7241808320955$$
$$x_{63} = 97.9129710368819$$
$$x_{57} = 100.007366139275$$
$$x_{12} = 110.479341651241$$
$$x_{41} = 112.573736753634$$
$$x_{58} = 791.15774992903$$
son puntos de cambio del signo de desigualdad en las soluciones.
Primero definámonos con el signo hasta el punto extremo izquierdo:
$$x_{0} \leq x_{9}$$
Consideremos, por ejemplo, el punto
$$x_{0} = x_{9} - \frac{1}{10}$$
=
$$-45867.7763411866 + - \frac{1}{10}$$
=
$$-45867.8763411866$$
lo sustituimos en la expresión
$$5 \sin{\left(x \right)} + \cos{\left(2 x \right)} \leq -2$$
$$5 \sin{\left(-45867.8763411866 \right)} + \cos{\left(\left(-45867.8763411866\right) 2 \right)} \leq -2$$
-2.60182118651354 <= -2
significa que una de las soluciones de nuestra ecuación será con:
$$x \leq -45867.7763411866$$
_____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____
\ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \
-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------
x9 x28 x52 x32 x4 x23 x2 x43 x8 x24 x36 x1 x55 x60 x5 x66 x53 x15 x25 x45 x35 x21 x16 x7 x11 x69 x64 x26 x14 x59 x6 x56 x44 x27 x18 x20 x62 x70 x10 x38 x29 x19 x37 x40 x50 x3 x17 x34 x30 x49 x39 x47 x22 x67 x48 x51 x13 x54 x46 x33 x31 x65 x71 x68 x42 x61 x63 x57 x12 x41 x58Recibiremos otras soluciones de la desigualdad pasando al polo siguiente etc.
etc.
Respuesta:
$$x \leq -45867.7763411866$$
$$x \geq -643.502895210309 \wedge x \leq -239.284640448423$$
$$x \geq -101.054563690472 \wedge x \leq -96.8657734856853$$
$$x \geq -94.7713783832921 \wedge x \leq -90.5825881785057$$
$$x \geq -88.4881930761125 \wedge x \leq -84.2994028713261$$
$$x \geq -82.2050077689329 \wedge x \leq -78.0162175641465$$
$$x \geq -75.9218224617533 \wedge x \leq -71.733032256967$$
$$x \geq -69.6386371545737 \wedge x \leq -65.4498469497874$$
$$x \geq -63.3554518473942 \wedge x \leq -59.1666616426078$$
$$x \geq -57.0722665402146 \wedge x \leq -52.8834763354282$$
$$x \geq -50.789081233035 \wedge x \leq -46.6002910282486$$
$$x \geq -44.5058959258554 \wedge x \leq -40.317105721069$$
$$x \geq -38.2227106186758 \wedge x \leq -34.0339204138894$$
$$x \geq -31.9395253114962 \wedge x \leq -27.7507351067098$$
$$x \geq -25.6563400043166 \wedge x \leq -21.4675497995303$$
$$x \geq -19.3731546971371 \wedge x \leq -15.1843644923507$$
$$x \geq -13.0899693899575 \wedge x \leq -8.90117918517108$$
$$x \geq -6.80678408277789 \wedge x \leq -2.61799387799149$$
$$x \geq -0.523598775598299 \wedge x \leq 3.66519142918809$$
$$x \geq 5.75958653158129 \wedge x \leq 9.94837673636768$$
$$x \geq 12.0427718387609 \wedge x \leq 16.2315620435473$$
$$x \geq 18.3259571459405 \wedge x \leq 22.5147473507269$$
$$x \geq 24.60914245312 \wedge x \leq 28.7979326579064$$
$$x \geq 30.8923277602996 \wedge x \leq 35.081117965086$$
$$x \geq 37.1755130674792 \wedge x \leq 41.3643032722656$$
$$x \geq 43.4586983746588 \wedge x \leq 47.6474885794452$$
$$x \geq 49.7418836818384 \wedge x \leq 53.9306738866248$$
$$x \geq 56.025068989018 \wedge x \leq 60.2138591938044$$
$$x \geq 62.3082542961976 \wedge x \leq 66.497044500984$$
$$x \geq 68.5914396033772 \wedge x \leq 72.7802298081635$$
$$x \geq 74.8746249105567 \wedge x \leq 79.0634151153431$$
$$x \geq 81.1578102177363 \wedge x \leq 85.3466004225227$$
$$x \geq 87.4409955249159 \wedge x \leq 91.6297857297023$$
$$x \geq 93.7241808320955 \wedge x \leq 97.9129710368819$$
$$x \geq 100.007366139275 \wedge x \leq 110.479341651241$$
$$x \geq 112.573736753634 \wedge x \leq 791.15774992903$$
Solución de la desigualdad en el gráfico
Respuesta rápida
[src]
/7*pi 11*pi\ And|---- <= x, x <= -----| \ 6 6 /
$$\frac{7 \pi}{6} \leq x \wedge x \leq \frac{11 \pi}{6}$$
(7*pi/6 <= x)∧(x <= 11*pi/6)
Respuesta rápida 2
[src]
7*pi 11*pi [----, -----] 6 6
$$x\ in\ \left[\frac{7 \pi}{6}, \frac{11 \pi}{6}\right]$$
x in Interval(7*pi/6, 11*pi/6)