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:
-
-16/((x+2)^2-5)>=0
-
(x-2)/(x^2+2x-8)>0
-
x^2-10x<0
- 5(y-1,4)-6<4y-1,5
- Derivada de:
-
-2
- Forma canónica:
- -2
- Límite de la función:
-
-2
-
Expresiones idénticas
- cos dos x+5sin>-2
- coseno de 2x más 5 seno de más menos 2
- coseno de dos x más 5 seno de más menos 2
-
Expresiones semejantes
- cos2x+5sin>+2
- cos2x-5sin>-2
cos2x+5sin>-2 desigualdades
Solución
Ha introducido
[src]
cos(2*x) + 5*sin(x) > -2
$$5 \sin{\left(x \right)} + \cos{\left(2 x \right)} > -2$$
5*sin(x) + cos(2*x) > -2
Solución detallada
Se da la desigualdad:
$$5 \sin{\left(x \right)} + \cos{\left(2 x \right)} > -2$$
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:
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(w_{1} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(- \frac{1}{2} \right)}$$
$$x_{1} = 2 \pi n - \frac{\pi}{6}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(w_{2} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(3 \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(3 \right)}$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(- \frac{1}{2} \right)} + \pi$$
$$x_{3} = 2 \pi n + \frac{7 \pi}{6}$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(w_{2} \right)} + \pi$$
$$x_{4} = 2 \pi n + \pi - \operatorname{asin}{\left(3 \right)}$$
$$x_{4} = 2 \pi n + \pi - \operatorname{asin}{\left(3 \right)}$$
$$x_{1} = 47.6474885794452$$
$$x_{2} = 74.8746249105567$$
$$x_{3} = 81.1578102177363$$
$$x_{4} = -27.7507351067098$$
$$x_{5} = 60.2138591938044$$
$$x_{6} = 110.479341651241$$
$$x_{7} = -19.3731546971371$$
$$x_{8} = -50.789081233035$$
$$x_{9} = -21.4675497995303$$
$$x_{10} = 112.573736753634$$
$$x_{11} = -44.5058959258554$$
$$x_{12} = -57.0722665402146$$
$$x_{13} = 43.4586983746588$$
$$x_{14} = 18.3259571459405$$
$$x_{15} = -643.502895210309$$
$$x_{16} = -34.0339204138894$$
$$x_{17} = -38.2227106186758$$
$$x_{18} = -59.1666616426078$$
$$x_{19} = 72.7802298081635$$
$$x_{20} = -13.0899693899575$$
$$x_{21} = 3.66519142918809$$
$$x_{22} = -6.80678408277789$$
$$x_{23} = -52.8834763354282$$
$$x_{24} = -65.4498469497874$$
$$x_{25} = -46.6002910282486$$
$$x_{26} = 87.4409955249159$$
$$x_{27} = -88.4881930761125$$
$$x_{28} = 24.60914245312$$
$$x_{29} = 100.007366139275$$
$$x_{30} = -31.9395253114962$$
$$x_{31} = 79.0634151153431$$
$$x_{32} = -63.3554518473942$$
$$x_{33} = 56.025068989018$$
$$x_{34} = 41.3643032722656$$
$$x_{35} = 91.6297857297023$$
$$x_{36} = 28.7979326579064$$
$$x_{37} = 93.7241808320955$$
$$x_{38} = -69.6386371545737$$
$$x_{39} = -45867.7763411866$$
$$x_{40} = 37.1755130674792$$
$$x_{41} = 53.9306738866248$$
$$x_{42} = 16.2315620435473$$
$$x_{43} = -75.9218224617533$$
$$x_{44} = -94.7713783832921$$
$$x_{45} = -78.0162175641465$$
$$x_{46} = 791.15774992903$$
$$x_{47} = 68.5914396033772$$
$$x_{48} = -82.2050077689329$$
$$x_{49} = -96.8657734856853$$
$$x_{50} = -84.2994028713261$$
$$x_{51} = 9.94837673636768$$
$$x_{52} = -0.523598775598299$$
$$x_{53} = 12.0427718387609$$
$$x_{54} = -2.61799387799149$$
$$x_{55} = 49.7418836818384$$
$$x_{56} = -71.733032256967$$
$$x_{57} = -40.317105721069$$
$$x_{58} = -8.90117918517108$$
$$x_{59} = 66.497044500984$$
$$x_{60} = 35.081117965086$$
$$x_{61} = 85.3466004225227$$
$$x_{62} = 5.75958653158129$$
$$x_{63} = -90.5825881785057$$
$$x_{64} = -239.284640448423$$
$$x_{65} = -25.6563400043166$$
$$x_{66} = 22.5147473507269$$
$$x_{67} = -101.054563690472$$
$$x_{68} = -15.1843644923507$$
$$x_{69} = 62.3082542961976$$
$$x_{70} = 30.8923277602996$$
$$x_{71} = 97.9129710368819$$
$$x_{1} = 47.6474885794452$$
$$x_{2} = 74.8746249105567$$
$$x_{3} = 81.1578102177363$$
$$x_{4} = -27.7507351067098$$
$$x_{5} = 60.2138591938044$$
$$x_{6} = 110.479341651241$$
$$x_{7} = -19.3731546971371$$
$$x_{8} = -50.789081233035$$
$$x_{9} = -21.4675497995303$$
$$x_{10} = 112.573736753634$$
$$x_{11} = -44.5058959258554$$
$$x_{12} = -57.0722665402146$$
$$x_{13} = 43.4586983746588$$
$$x_{14} = 18.3259571459405$$
$$x_{15} = -643.502895210309$$
$$x_{16} = -34.0339204138894$$
$$x_{17} = -38.2227106186758$$
$$x_{18} = -59.1666616426078$$
$$x_{19} = 72.7802298081635$$
$$x_{20} = -13.0899693899575$$
$$x_{21} = 3.66519142918809$$
$$x_{22} = -6.80678408277789$$
$$x_{23} = -52.8834763354282$$
$$x_{24} = -65.4498469497874$$
$$x_{25} = -46.6002910282486$$
$$x_{26} = 87.4409955249159$$
$$x_{27} = -88.4881930761125$$
$$x_{28} = 24.60914245312$$
$$x_{29} = 100.007366139275$$
$$x_{30} = -31.9395253114962$$
$$x_{31} = 79.0634151153431$$
$$x_{32} = -63.3554518473942$$
$$x_{33} = 56.025068989018$$
$$x_{34} = 41.3643032722656$$
$$x_{35} = 91.6297857297023$$
$$x_{36} = 28.7979326579064$$
$$x_{37} = 93.7241808320955$$
$$x_{38} = -69.6386371545737$$
$$x_{39} = -45867.7763411866$$
$$x_{40} = 37.1755130674792$$
$$x_{41} = 53.9306738866248$$
$$x_{42} = 16.2315620435473$$
$$x_{43} = -75.9218224617533$$
$$x_{44} = -94.7713783832921$$
$$x_{45} = -78.0162175641465$$
$$x_{46} = 791.15774992903$$
$$x_{47} = 68.5914396033772$$
$$x_{48} = -82.2050077689329$$
$$x_{49} = -96.8657734856853$$
$$x_{50} = -84.2994028713261$$
$$x_{51} = 9.94837673636768$$
$$x_{52} = -0.523598775598299$$
$$x_{53} = 12.0427718387609$$
$$x_{54} = -2.61799387799149$$
$$x_{55} = 49.7418836818384$$
$$x_{56} = -71.733032256967$$
$$x_{57} = -40.317105721069$$
$$x_{58} = -8.90117918517108$$
$$x_{59} = 66.497044500984$$
$$x_{60} = 35.081117965086$$
$$x_{61} = 85.3466004225227$$
$$x_{62} = 5.75958653158129$$
$$x_{63} = -90.5825881785057$$
$$x_{64} = -239.284640448423$$
$$x_{65} = -25.6563400043166$$
$$x_{66} = 22.5147473507269$$
$$x_{67} = -101.054563690472$$
$$x_{68} = -15.1843644923507$$
$$x_{69} = 62.3082542961976$$
$$x_{70} = 30.8923277602996$$
$$x_{71} = 97.9129710368819$$
Las raíces dadas
$$x_{39} = -45867.7763411866$$
$$x_{15} = -643.502895210309$$
$$x_{64} = -239.284640448423$$
$$x_{67} = -101.054563690472$$
$$x_{49} = -96.8657734856853$$
$$x_{44} = -94.7713783832921$$
$$x_{63} = -90.5825881785057$$
$$x_{27} = -88.4881930761125$$
$$x_{50} = -84.2994028713261$$
$$x_{48} = -82.2050077689329$$
$$x_{45} = -78.0162175641465$$
$$x_{43} = -75.9218224617533$$
$$x_{56} = -71.733032256967$$
$$x_{38} = -69.6386371545737$$
$$x_{24} = -65.4498469497874$$
$$x_{32} = -63.3554518473942$$
$$x_{18} = -59.1666616426078$$
$$x_{12} = -57.0722665402146$$
$$x_{23} = -52.8834763354282$$
$$x_{8} = -50.789081233035$$
$$x_{25} = -46.6002910282486$$
$$x_{11} = -44.5058959258554$$
$$x_{57} = -40.317105721069$$
$$x_{17} = -38.2227106186758$$
$$x_{16} = -34.0339204138894$$
$$x_{30} = -31.9395253114962$$
$$x_{4} = -27.7507351067098$$
$$x_{65} = -25.6563400043166$$
$$x_{9} = -21.4675497995303$$
$$x_{7} = -19.3731546971371$$
$$x_{68} = -15.1843644923507$$
$$x_{20} = -13.0899693899575$$
$$x_{58} = -8.90117918517108$$
$$x_{22} = -6.80678408277789$$
$$x_{54} = -2.61799387799149$$
$$x_{52} = -0.523598775598299$$
$$x_{21} = 3.66519142918809$$
$$x_{62} = 5.75958653158129$$
$$x_{51} = 9.94837673636768$$
$$x_{53} = 12.0427718387609$$
$$x_{42} = 16.2315620435473$$
$$x_{14} = 18.3259571459405$$
$$x_{66} = 22.5147473507269$$
$$x_{28} = 24.60914245312$$
$$x_{36} = 28.7979326579064$$
$$x_{70} = 30.8923277602996$$
$$x_{60} = 35.081117965086$$
$$x_{40} = 37.1755130674792$$
$$x_{34} = 41.3643032722656$$
$$x_{13} = 43.4586983746588$$
$$x_{1} = 47.6474885794452$$
$$x_{55} = 49.7418836818384$$
$$x_{41} = 53.9306738866248$$
$$x_{33} = 56.025068989018$$
$$x_{5} = 60.2138591938044$$
$$x_{69} = 62.3082542961976$$
$$x_{59} = 66.497044500984$$
$$x_{47} = 68.5914396033772$$
$$x_{19} = 72.7802298081635$$
$$x_{2} = 74.8746249105567$$
$$x_{31} = 79.0634151153431$$
$$x_{3} = 81.1578102177363$$
$$x_{61} = 85.3466004225227$$
$$x_{26} = 87.4409955249159$$
$$x_{35} = 91.6297857297023$$
$$x_{37} = 93.7241808320955$$
$$x_{71} = 97.9129710368819$$
$$x_{29} = 100.007366139275$$
$$x_{6} = 110.479341651241$$
$$x_{10} = 112.573736753634$$
$$x_{46} = 791.15774992903$$
son puntos de cambio del signo de desigualdad en las soluciones.
Primero definámonos con el signo hasta el punto extremo izquierdo:
$$x_{0} < x_{39}$$
Consideremos, por ejemplo, el punto
$$x_{0} = x_{39} - \frac{1}{10}$$
=
$$-45867.7763411866 + - \frac{1}{10}$$
=
$$-45867.8763411866$$
lo sustituimos en la expresión
$$5 \sin{\left(x \right)} + \cos{\left(2 x \right)} > -2$$
$$5 \sin{\left(-45867.8763411866 \right)} + \cos{\left(\left(-45867.8763411866\right) 2 \right)} > -2$$
Entonces
$$x < -45867.7763411866$$
no se cumple
significa que una de las soluciones de nuestra ecuación será con:
$$x > -45867.7763411866 \wedge x < -643.502895210309$$
Recibiremos otras soluciones de la desigualdad pasando al polo siguiente etc.
etc.
Respuesta:
$$x > -45867.7763411866 \wedge x < -643.502895210309$$
$$x > -239.284640448423 \wedge x < -101.054563690472$$
$$x > -96.8657734856853 \wedge x < -94.7713783832921$$
$$x > -90.5825881785057 \wedge x < -88.4881930761125$$
$$x > -84.2994028713261 \wedge x < -82.2050077689329$$
$$x > -78.0162175641465 \wedge x < -75.9218224617533$$
$$x > -71.733032256967 \wedge x < -69.6386371545737$$
$$x > -65.4498469497874 \wedge x < -63.3554518473942$$
$$x > -59.1666616426078 \wedge x < -57.0722665402146$$
$$x > -52.8834763354282 \wedge x < -50.789081233035$$
$$x > -46.6002910282486 \wedge x < -44.5058959258554$$
$$x > -40.317105721069 \wedge x < -38.2227106186758$$
$$x > -34.0339204138894 \wedge x < -31.9395253114962$$
$$x > -27.7507351067098 \wedge x < -25.6563400043166$$
$$x > -21.4675497995303 \wedge x < -19.3731546971371$$
$$x > -15.1843644923507 \wedge x < -13.0899693899575$$
$$x > -8.90117918517108 \wedge x < -6.80678408277789$$
$$x > -2.61799387799149 \wedge x < -0.523598775598299$$
$$x > 3.66519142918809 \wedge x < 5.75958653158129$$
$$x > 9.94837673636768 \wedge x < 12.0427718387609$$
$$x > 16.2315620435473 \wedge x < 18.3259571459405$$
$$x > 22.5147473507269 \wedge x < 24.60914245312$$
$$x > 28.7979326579064 \wedge x < 30.8923277602996$$
$$x > 35.081117965086 \wedge x < 37.1755130674792$$
$$x > 41.3643032722656 \wedge x < 43.4586983746588$$
$$x > 47.6474885794452 \wedge x < 49.7418836818384$$
$$x > 53.9306738866248 \wedge x < 56.025068989018$$
$$x > 60.2138591938044 \wedge x < 62.3082542961976$$
$$x > 66.497044500984 \wedge x < 68.5914396033772$$
$$x > 72.7802298081635 \wedge x < 74.8746249105567$$
$$x > 79.0634151153431 \wedge x < 81.1578102177363$$
$$x > 85.3466004225227 \wedge x < 87.4409955249159$$
$$x > 91.6297857297023 \wedge x < 93.7241808320955$$
$$x > 97.9129710368819 \wedge x < 100.007366139275$$
$$x > 110.479341651241 \wedge x < 112.573736753634$$
$$x > 791.15774992903$$
$$5 \sin{\left(x \right)} + \cos{\left(2 x \right)} > -2$$
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:
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(w_{1} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(- \frac{1}{2} \right)}$$
$$x_{1} = 2 \pi n - \frac{\pi}{6}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(w_{2} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(3 \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(3 \right)}$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(- \frac{1}{2} \right)} + \pi$$
$$x_{3} = 2 \pi n + \frac{7 \pi}{6}$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(w_{2} \right)} + \pi$$
$$x_{4} = 2 \pi n + \pi - \operatorname{asin}{\left(3 \right)}$$
$$x_{4} = 2 \pi n + \pi - \operatorname{asin}{\left(3 \right)}$$
$$x_{1} = 47.6474885794452$$
$$x_{2} = 74.8746249105567$$
$$x_{3} = 81.1578102177363$$
$$x_{4} = -27.7507351067098$$
$$x_{5} = 60.2138591938044$$
$$x_{6} = 110.479341651241$$
$$x_{7} = -19.3731546971371$$
$$x_{8} = -50.789081233035$$
$$x_{9} = -21.4675497995303$$
$$x_{10} = 112.573736753634$$
$$x_{11} = -44.5058959258554$$
$$x_{12} = -57.0722665402146$$
$$x_{13} = 43.4586983746588$$
$$x_{14} = 18.3259571459405$$
$$x_{15} = -643.502895210309$$
$$x_{16} = -34.0339204138894$$
$$x_{17} = -38.2227106186758$$
$$x_{18} = -59.1666616426078$$
$$x_{19} = 72.7802298081635$$
$$x_{20} = -13.0899693899575$$
$$x_{21} = 3.66519142918809$$
$$x_{22} = -6.80678408277789$$
$$x_{23} = -52.8834763354282$$
$$x_{24} = -65.4498469497874$$
$$x_{25} = -46.6002910282486$$
$$x_{26} = 87.4409955249159$$
$$x_{27} = -88.4881930761125$$
$$x_{28} = 24.60914245312$$
$$x_{29} = 100.007366139275$$
$$x_{30} = -31.9395253114962$$
$$x_{31} = 79.0634151153431$$
$$x_{32} = -63.3554518473942$$
$$x_{33} = 56.025068989018$$
$$x_{34} = 41.3643032722656$$
$$x_{35} = 91.6297857297023$$
$$x_{36} = 28.7979326579064$$
$$x_{37} = 93.7241808320955$$
$$x_{38} = -69.6386371545737$$
$$x_{39} = -45867.7763411866$$
$$x_{40} = 37.1755130674792$$
$$x_{41} = 53.9306738866248$$
$$x_{42} = 16.2315620435473$$
$$x_{43} = -75.9218224617533$$
$$x_{44} = -94.7713783832921$$
$$x_{45} = -78.0162175641465$$
$$x_{46} = 791.15774992903$$
$$x_{47} = 68.5914396033772$$
$$x_{48} = -82.2050077689329$$
$$x_{49} = -96.8657734856853$$
$$x_{50} = -84.2994028713261$$
$$x_{51} = 9.94837673636768$$
$$x_{52} = -0.523598775598299$$
$$x_{53} = 12.0427718387609$$
$$x_{54} = -2.61799387799149$$
$$x_{55} = 49.7418836818384$$
$$x_{56} = -71.733032256967$$
$$x_{57} = -40.317105721069$$
$$x_{58} = -8.90117918517108$$
$$x_{59} = 66.497044500984$$
$$x_{60} = 35.081117965086$$
$$x_{61} = 85.3466004225227$$
$$x_{62} = 5.75958653158129$$
$$x_{63} = -90.5825881785057$$
$$x_{64} = -239.284640448423$$
$$x_{65} = -25.6563400043166$$
$$x_{66} = 22.5147473507269$$
$$x_{67} = -101.054563690472$$
$$x_{68} = -15.1843644923507$$
$$x_{69} = 62.3082542961976$$
$$x_{70} = 30.8923277602996$$
$$x_{71} = 97.9129710368819$$
$$x_{1} = 47.6474885794452$$
$$x_{2} = 74.8746249105567$$
$$x_{3} = 81.1578102177363$$
$$x_{4} = -27.7507351067098$$
$$x_{5} = 60.2138591938044$$
$$x_{6} = 110.479341651241$$
$$x_{7} = -19.3731546971371$$
$$x_{8} = -50.789081233035$$
$$x_{9} = -21.4675497995303$$
$$x_{10} = 112.573736753634$$
$$x_{11} = -44.5058959258554$$
$$x_{12} = -57.0722665402146$$
$$x_{13} = 43.4586983746588$$
$$x_{14} = 18.3259571459405$$
$$x_{15} = -643.502895210309$$
$$x_{16} = -34.0339204138894$$
$$x_{17} = -38.2227106186758$$
$$x_{18} = -59.1666616426078$$
$$x_{19} = 72.7802298081635$$
$$x_{20} = -13.0899693899575$$
$$x_{21} = 3.66519142918809$$
$$x_{22} = -6.80678408277789$$
$$x_{23} = -52.8834763354282$$
$$x_{24} = -65.4498469497874$$
$$x_{25} = -46.6002910282486$$
$$x_{26} = 87.4409955249159$$
$$x_{27} = -88.4881930761125$$
$$x_{28} = 24.60914245312$$
$$x_{29} = 100.007366139275$$
$$x_{30} = -31.9395253114962$$
$$x_{31} = 79.0634151153431$$
$$x_{32} = -63.3554518473942$$
$$x_{33} = 56.025068989018$$
$$x_{34} = 41.3643032722656$$
$$x_{35} = 91.6297857297023$$
$$x_{36} = 28.7979326579064$$
$$x_{37} = 93.7241808320955$$
$$x_{38} = -69.6386371545737$$
$$x_{39} = -45867.7763411866$$
$$x_{40} = 37.1755130674792$$
$$x_{41} = 53.9306738866248$$
$$x_{42} = 16.2315620435473$$
$$x_{43} = -75.9218224617533$$
$$x_{44} = -94.7713783832921$$
$$x_{45} = -78.0162175641465$$
$$x_{46} = 791.15774992903$$
$$x_{47} = 68.5914396033772$$
$$x_{48} = -82.2050077689329$$
$$x_{49} = -96.8657734856853$$
$$x_{50} = -84.2994028713261$$
$$x_{51} = 9.94837673636768$$
$$x_{52} = -0.523598775598299$$
$$x_{53} = 12.0427718387609$$
$$x_{54} = -2.61799387799149$$
$$x_{55} = 49.7418836818384$$
$$x_{56} = -71.733032256967$$
$$x_{57} = -40.317105721069$$
$$x_{58} = -8.90117918517108$$
$$x_{59} = 66.497044500984$$
$$x_{60} = 35.081117965086$$
$$x_{61} = 85.3466004225227$$
$$x_{62} = 5.75958653158129$$
$$x_{63} = -90.5825881785057$$
$$x_{64} = -239.284640448423$$
$$x_{65} = -25.6563400043166$$
$$x_{66} = 22.5147473507269$$
$$x_{67} = -101.054563690472$$
$$x_{68} = -15.1843644923507$$
$$x_{69} = 62.3082542961976$$
$$x_{70} = 30.8923277602996$$
$$x_{71} = 97.9129710368819$$
Las raíces dadas
$$x_{39} = -45867.7763411866$$
$$x_{15} = -643.502895210309$$
$$x_{64} = -239.284640448423$$
$$x_{67} = -101.054563690472$$
$$x_{49} = -96.8657734856853$$
$$x_{44} = -94.7713783832921$$
$$x_{63} = -90.5825881785057$$
$$x_{27} = -88.4881930761125$$
$$x_{50} = -84.2994028713261$$
$$x_{48} = -82.2050077689329$$
$$x_{45} = -78.0162175641465$$
$$x_{43} = -75.9218224617533$$
$$x_{56} = -71.733032256967$$
$$x_{38} = -69.6386371545737$$
$$x_{24} = -65.4498469497874$$
$$x_{32} = -63.3554518473942$$
$$x_{18} = -59.1666616426078$$
$$x_{12} = -57.0722665402146$$
$$x_{23} = -52.8834763354282$$
$$x_{8} = -50.789081233035$$
$$x_{25} = -46.6002910282486$$
$$x_{11} = -44.5058959258554$$
$$x_{57} = -40.317105721069$$
$$x_{17} = -38.2227106186758$$
$$x_{16} = -34.0339204138894$$
$$x_{30} = -31.9395253114962$$
$$x_{4} = -27.7507351067098$$
$$x_{65} = -25.6563400043166$$
$$x_{9} = -21.4675497995303$$
$$x_{7} = -19.3731546971371$$
$$x_{68} = -15.1843644923507$$
$$x_{20} = -13.0899693899575$$
$$x_{58} = -8.90117918517108$$
$$x_{22} = -6.80678408277789$$
$$x_{54} = -2.61799387799149$$
$$x_{52} = -0.523598775598299$$
$$x_{21} = 3.66519142918809$$
$$x_{62} = 5.75958653158129$$
$$x_{51} = 9.94837673636768$$
$$x_{53} = 12.0427718387609$$
$$x_{42} = 16.2315620435473$$
$$x_{14} = 18.3259571459405$$
$$x_{66} = 22.5147473507269$$
$$x_{28} = 24.60914245312$$
$$x_{36} = 28.7979326579064$$
$$x_{70} = 30.8923277602996$$
$$x_{60} = 35.081117965086$$
$$x_{40} = 37.1755130674792$$
$$x_{34} = 41.3643032722656$$
$$x_{13} = 43.4586983746588$$
$$x_{1} = 47.6474885794452$$
$$x_{55} = 49.7418836818384$$
$$x_{41} = 53.9306738866248$$
$$x_{33} = 56.025068989018$$
$$x_{5} = 60.2138591938044$$
$$x_{69} = 62.3082542961976$$
$$x_{59} = 66.497044500984$$
$$x_{47} = 68.5914396033772$$
$$x_{19} = 72.7802298081635$$
$$x_{2} = 74.8746249105567$$
$$x_{31} = 79.0634151153431$$
$$x_{3} = 81.1578102177363$$
$$x_{61} = 85.3466004225227$$
$$x_{26} = 87.4409955249159$$
$$x_{35} = 91.6297857297023$$
$$x_{37} = 93.7241808320955$$
$$x_{71} = 97.9129710368819$$
$$x_{29} = 100.007366139275$$
$$x_{6} = 110.479341651241$$
$$x_{10} = 112.573736753634$$
$$x_{46} = 791.15774992903$$
son puntos de cambio del signo de desigualdad en las soluciones.
Primero definámonos con el signo hasta el punto extremo izquierdo:
$$x_{0} < x_{39}$$
Consideremos, por ejemplo, el punto
$$x_{0} = x_{39} - \frac{1}{10}$$
=
$$-45867.7763411866 + - \frac{1}{10}$$
=
$$-45867.8763411866$$
lo sustituimos en la expresión
$$5 \sin{\left(x \right)} + \cos{\left(2 x \right)} > -2$$
$$5 \sin{\left(-45867.8763411866 \right)} + \cos{\left(\left(-45867.8763411866\right) 2 \right)} > -2$$
-2.60182118651354 > -2
Entonces
$$x < -45867.7763411866$$
no se cumple
significa que una de las soluciones de nuestra ecuación será con:
$$x > -45867.7763411866 \wedge x < -643.502895210309$$
_____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____
/ \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
-------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-------
x39 x15 x64 x67 x49 x44 x63 x27 x50 x48 x45 x43 x56 x38 x24 x32 x18 x12 x23 x8 x25 x11 x57 x17 x16 x30 x4 x65 x9 x7 x68 x20 x58 x22 x54 x52 x21 x62 x51 x53 x42 x14 x66 x28 x36 x70 x60 x40 x34 x13 x1 x55 x41 x33 x5 x69 x59 x47 x19 x2 x31 x3 x61 x26 x35 x37 x71 x29 x6 x10 x46Recibiremos otras soluciones de la desigualdad pasando al polo siguiente etc.
etc.
Respuesta:
$$x > -45867.7763411866 \wedge x < -643.502895210309$$
$$x > -239.284640448423 \wedge x < -101.054563690472$$
$$x > -96.8657734856853 \wedge x < -94.7713783832921$$
$$x > -90.5825881785057 \wedge x < -88.4881930761125$$
$$x > -84.2994028713261 \wedge x < -82.2050077689329$$
$$x > -78.0162175641465 \wedge x < -75.9218224617533$$
$$x > -71.733032256967 \wedge x < -69.6386371545737$$
$$x > -65.4498469497874 \wedge x < -63.3554518473942$$
$$x > -59.1666616426078 \wedge x < -57.0722665402146$$
$$x > -52.8834763354282 \wedge x < -50.789081233035$$
$$x > -46.6002910282486 \wedge x < -44.5058959258554$$
$$x > -40.317105721069 \wedge x < -38.2227106186758$$
$$x > -34.0339204138894 \wedge x < -31.9395253114962$$
$$x > -27.7507351067098 \wedge x < -25.6563400043166$$
$$x > -21.4675497995303 \wedge x < -19.3731546971371$$
$$x > -15.1843644923507 \wedge x < -13.0899693899575$$
$$x > -8.90117918517108 \wedge x < -6.80678408277789$$
$$x > -2.61799387799149 \wedge x < -0.523598775598299$$
$$x > 3.66519142918809 \wedge x < 5.75958653158129$$
$$x > 9.94837673636768 \wedge x < 12.0427718387609$$
$$x > 16.2315620435473 \wedge x < 18.3259571459405$$
$$x > 22.5147473507269 \wedge x < 24.60914245312$$
$$x > 28.7979326579064 \wedge x < 30.8923277602996$$
$$x > 35.081117965086 \wedge x < 37.1755130674792$$
$$x > 41.3643032722656 \wedge x < 43.4586983746588$$
$$x > 47.6474885794452 \wedge x < 49.7418836818384$$
$$x > 53.9306738866248 \wedge x < 56.025068989018$$
$$x > 60.2138591938044 \wedge x < 62.3082542961976$$
$$x > 66.497044500984 \wedge x < 68.5914396033772$$
$$x > 72.7802298081635 \wedge x < 74.8746249105567$$
$$x > 79.0634151153431 \wedge x < 81.1578102177363$$
$$x > 85.3466004225227 \wedge x < 87.4409955249159$$
$$x > 91.6297857297023 \wedge x < 93.7241808320955$$
$$x > 97.9129710368819 \wedge x < 100.007366139275$$
$$x > 110.479341651241 \wedge x < 112.573736753634$$
$$x > 791.15774992903$$
Solución de la desigualdad en el gráfico
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.Ropen(0, 7*pi/6), Interval.Lopen(11*pi/6, 2*pi))
Respuesta rápida
[src]
/ / 7*pi\ / 11*pi \\ Or|And|0 <= x, x < ----|, And|x <= 2*pi, ----- < x|| \ \ 6 / \ 6 //
$$\left(0 \leq x \wedge x < \frac{7 \pi}{6}\right) \vee \left(x \leq 2 \pi \wedge \frac{11 \pi}{6} < x\right)$$
((0 <= x)∧(x < 7*pi/6))∨((x <= 2*pi)∧(11*pi/6 < x))
Gráfico