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:
-
x^2>1
-
(x-2)/(x-4)>0
-
x+1>0
- (x-7)(x+8)>0
- Derivada de:
-
1/2
- Suma de la serie:
-
1/2
- Límite de la función:
-
1/2
-
Expresiones idénticas
- sinxcos dos x+cosxsin2x> uno /2
- seno de x coseno de 2x más coseno de x seno de 2x más 1 dividir por 2
- seno de x coseno de dos x más coseno de x seno de 2x más uno dividir por 2
- sinxcos2x+cosxsin2x>1 dividir por 2
-
Expresiones semejantes
- sinxcos2x-cosxsin2x>1/2
- 1/(2x-3)>x/(x-4)
- (2*x^2-2*x+1)/(2*x-1)<=1
- (x-1)x(x+1)/(2x-3)>=0
sinxcos2x+cosxsin2x>1/2 desigualdades
Solución
Ha introducido
[src]
sin(x)*cos(2*x) + cos(x)*sin(2*x) > 1/2
$$\sin{\left(x \right)} \cos{\left(2 x \right)} + \sin{\left(2 x \right)} \cos{\left(x \right)} > \frac{1}{2}$$
sin(x)*cos(2*x) + sin(2*x)*cos(x) > 1/2
Solución detallada
Se da la desigualdad:
$$\sin{\left(x \right)} \cos{\left(2 x \right)} + \sin{\left(2 x \right)} \cos{\left(x \right)} > \frac{1}{2}$$
Para resolver esta desigualdad primero hay que resolver la ecuación correspondiente:
$$\sin{\left(x \right)} \cos{\left(2 x \right)} + \sin{\left(2 x \right)} \cos{\left(x \right)} = \frac{1}{2}$$
Resolvemos:
$$x_{1} = 6.45771823237902$$
$$x_{2} = 78.3652834145454$$
$$x_{3} = -13.7881010907552$$
$$x_{4} = 83.9503370209273$$
$$x_{5} = -85.6956662729216$$
$$x_{6} = 65.7989128001862$$
$$x_{7} = -45.9021593274509$$
$$x_{8} = 76.2708883121522$$
$$x_{9} = -77.3180858633488$$
$$x_{10} = 50.4400153826361$$
$$x_{11} = 19.7222205475359$$
$$x_{12} = -22.8638132011257$$
$$x_{13} = -43.8077642250577$$
$$x_{14} = -20.0712863979348$$
$$x_{15} = -83.6012711705284$$
$$x_{16} = 63.704517697793$$
$$x_{17} = -68.2423737529783$$
$$x_{18} = 28.0998009571087$$
$$x_{19} = 34.3829862642883$$
$$x_{20} = 39.9680398706701$$
$$x_{21} = -61.9591884457987$$
$$x_{22} = 30.1941960595019$$
$$x_{23} = -11.693705988362$$
$$x_{24} = -87.7900613753148$$
$$x_{25} = 44.1568300754565$$
$$x_{26} = -97.563905186483$$
$$x_{27} = 74.176493209759$$
$$x_{28} = 4.36332312998582$$
$$x_{29} = -15.8824961931484$$
$$x_{30} = 61.6101225953998$$
$$x_{31} = 86.0447321233204$$
$$x_{32} = -41.7133691226645$$
$$x_{33} = 42.0624349730633$$
$$x_{34} = -4.01425727958696$$
$$x_{35} = 46.2512251778497$$
$$x_{36} = 96.5167076352864$$
$$x_{37} = -81.5068760681352$$
$$x_{38} = 54.6288055874225$$
$$x_{39} = -47.9965544298441$$
$$x_{40} = -24.9582083035189$$
$$x_{41} = 21.8166156499291$$
$$x_{42} = -6.10865238198015$$
$$x_{43} = -37.5245789178781$$
$$x_{44} = -91.9788515801012$$
$$x_{45} = -53.5816080362259$$
$$x_{46} = -73.1292956585624$$
$$x_{47} = -1.22173047639603$$
$$x_{48} = 60.9119908946021$$
$$x_{49} = 37.873644768277$$
$$x_{50} = -9.59931088596881$$
$$x_{51} = -79.412480965742$$
$$x_{52} = -64.0535835481919$$
$$x_{53} = 8.55211333477222$$
$$x_{54} = 10.6465084371654$$
$$x_{55} = -1.91986217719376$$
$$x_{56} = 26.0054058547155$$
$$x_{57} = -34.7320521146872$$
$$x_{58} = -52.1853446346305$$
$$x_{59} = -59.8647933434055$$
$$x_{60} = -94.0732466824944$$
$$x_{61} = -26.3544717051144$$
$$x_{62} = -99.6583002888762$$
$$x_{63} = -70.3367688553715$$
$$x_{64} = 0.174532925199433$$
$$x_{65} = 88.1391272257137$$
$$x_{66} = -50.0909495322373$$
$$x_{67} = -29.1469985083053$$
$$x_{68} = -72.4311639577647$$
$$x_{69} = 72.0820981073658$$
$$x_{70} = 98.6111027376796$$
$$x_{71} = 52.5344104850293$$
$$x_{72} = -55.6760031386191$$
$$x_{73} = -39.6189740202713$$
$$x_{74} = 92.3279174305$$
$$x_{75} = -96.1676417848876$$
$$x_{76} = 103.498024643264$$
$$x_{77} = 94.4223125328932$$
$$x_{78} = -17.9768912955416$$
$$x_{79} = 13.4390352403563$$
$$x_{80} = -89.884456477708$$
$$x_{81} = -66.1479786505851$$
$$x_{82} = 67.8933079025794$$
$$x_{83} = -22.165681500328$$
$$x_{84} = 69.9877030049726$$
$$x_{85} = 2.26892802759263$$
$$x_{86} = 23.9110107523223$$
$$x_{87} = 90.2335223281068$$
$$x_{88} = 32.2885911618951$$
$$x_{89} = -76.6199541625511$$
$$x_{90} = -24.2600766027212$$
$$x_{91} = 101.403629540871$$
$$x_{92} = -57.7703982410123$$
$$x_{93} = -35.4301838154849$$
$$x_{94} = 81.8559419185341$$
$$x_{95} = 17.6278254451427$$
$$x_{96} = 48.3456202802429$$
$$x_{1} = 6.45771823237902$$
$$x_{2} = 78.3652834145454$$
$$x_{3} = -13.7881010907552$$
$$x_{4} = 83.9503370209273$$
$$x_{5} = -85.6956662729216$$
$$x_{6} = 65.7989128001862$$
$$x_{7} = -45.9021593274509$$
$$x_{8} = 76.2708883121522$$
$$x_{9} = -77.3180858633488$$
$$x_{10} = 50.4400153826361$$
$$x_{11} = 19.7222205475359$$
$$x_{12} = -22.8638132011257$$
$$x_{13} = -43.8077642250577$$
$$x_{14} = -20.0712863979348$$
$$x_{15} = -83.6012711705284$$
$$x_{16} = 63.704517697793$$
$$x_{17} = -68.2423737529783$$
$$x_{18} = 28.0998009571087$$
$$x_{19} = 34.3829862642883$$
$$x_{20} = 39.9680398706701$$
$$x_{21} = -61.9591884457987$$
$$x_{22} = 30.1941960595019$$
$$x_{23} = -11.693705988362$$
$$x_{24} = -87.7900613753148$$
$$x_{25} = 44.1568300754565$$
$$x_{26} = -97.563905186483$$
$$x_{27} = 74.176493209759$$
$$x_{28} = 4.36332312998582$$
$$x_{29} = -15.8824961931484$$
$$x_{30} = 61.6101225953998$$
$$x_{31} = 86.0447321233204$$
$$x_{32} = -41.7133691226645$$
$$x_{33} = 42.0624349730633$$
$$x_{34} = -4.01425727958696$$
$$x_{35} = 46.2512251778497$$
$$x_{36} = 96.5167076352864$$
$$x_{37} = -81.5068760681352$$
$$x_{38} = 54.6288055874225$$
$$x_{39} = -47.9965544298441$$
$$x_{40} = -24.9582083035189$$
$$x_{41} = 21.8166156499291$$
$$x_{42} = -6.10865238198015$$
$$x_{43} = -37.5245789178781$$
$$x_{44} = -91.9788515801012$$
$$x_{45} = -53.5816080362259$$
$$x_{46} = -73.1292956585624$$
$$x_{47} = -1.22173047639603$$
$$x_{48} = 60.9119908946021$$
$$x_{49} = 37.873644768277$$
$$x_{50} = -9.59931088596881$$
$$x_{51} = -79.412480965742$$
$$x_{52} = -64.0535835481919$$
$$x_{53} = 8.55211333477222$$
$$x_{54} = 10.6465084371654$$
$$x_{55} = -1.91986217719376$$
$$x_{56} = 26.0054058547155$$
$$x_{57} = -34.7320521146872$$
$$x_{58} = -52.1853446346305$$
$$x_{59} = -59.8647933434055$$
$$x_{60} = -94.0732466824944$$
$$x_{61} = -26.3544717051144$$
$$x_{62} = -99.6583002888762$$
$$x_{63} = -70.3367688553715$$
$$x_{64} = 0.174532925199433$$
$$x_{65} = 88.1391272257137$$
$$x_{66} = -50.0909495322373$$
$$x_{67} = -29.1469985083053$$
$$x_{68} = -72.4311639577647$$
$$x_{69} = 72.0820981073658$$
$$x_{70} = 98.6111027376796$$
$$x_{71} = 52.5344104850293$$
$$x_{72} = -55.6760031386191$$
$$x_{73} = -39.6189740202713$$
$$x_{74} = 92.3279174305$$
$$x_{75} = -96.1676417848876$$
$$x_{76} = 103.498024643264$$
$$x_{77} = 94.4223125328932$$
$$x_{78} = -17.9768912955416$$
$$x_{79} = 13.4390352403563$$
$$x_{80} = -89.884456477708$$
$$x_{81} = -66.1479786505851$$
$$x_{82} = 67.8933079025794$$
$$x_{83} = -22.165681500328$$
$$x_{84} = 69.9877030049726$$
$$x_{85} = 2.26892802759263$$
$$x_{86} = 23.9110107523223$$
$$x_{87} = 90.2335223281068$$
$$x_{88} = 32.2885911618951$$
$$x_{89} = -76.6199541625511$$
$$x_{90} = -24.2600766027212$$
$$x_{91} = 101.403629540871$$
$$x_{92} = -57.7703982410123$$
$$x_{93} = -35.4301838154849$$
$$x_{94} = 81.8559419185341$$
$$x_{95} = 17.6278254451427$$
$$x_{96} = 48.3456202802429$$
Las raíces dadas
$$x_{62} = -99.6583002888762$$
$$x_{26} = -97.563905186483$$
$$x_{75} = -96.1676417848876$$
$$x_{60} = -94.0732466824944$$
$$x_{44} = -91.9788515801012$$
$$x_{80} = -89.884456477708$$
$$x_{24} = -87.7900613753148$$
$$x_{5} = -85.6956662729216$$
$$x_{15} = -83.6012711705284$$
$$x_{37} = -81.5068760681352$$
$$x_{51} = -79.412480965742$$
$$x_{9} = -77.3180858633488$$
$$x_{89} = -76.6199541625511$$
$$x_{46} = -73.1292956585624$$
$$x_{68} = -72.4311639577647$$
$$x_{63} = -70.3367688553715$$
$$x_{17} = -68.2423737529783$$
$$x_{81} = -66.1479786505851$$
$$x_{52} = -64.0535835481919$$
$$x_{21} = -61.9591884457987$$
$$x_{59} = -59.8647933434055$$
$$x_{92} = -57.7703982410123$$
$$x_{72} = -55.6760031386191$$
$$x_{45} = -53.5816080362259$$
$$x_{58} = -52.1853446346305$$
$$x_{66} = -50.0909495322373$$
$$x_{39} = -47.9965544298441$$
$$x_{7} = -45.9021593274509$$
$$x_{13} = -43.8077642250577$$
$$x_{32} = -41.7133691226645$$
$$x_{73} = -39.6189740202713$$
$$x_{43} = -37.5245789178781$$
$$x_{93} = -35.4301838154849$$
$$x_{57} = -34.7320521146872$$
$$x_{67} = -29.1469985083053$$
$$x_{61} = -26.3544717051144$$
$$x_{40} = -24.9582083035189$$
$$x_{90} = -24.2600766027212$$
$$x_{12} = -22.8638132011257$$
$$x_{83} = -22.165681500328$$
$$x_{14} = -20.0712863979348$$
$$x_{78} = -17.9768912955416$$
$$x_{29} = -15.8824961931484$$
$$x_{3} = -13.7881010907552$$
$$x_{23} = -11.693705988362$$
$$x_{50} = -9.59931088596881$$
$$x_{42} = -6.10865238198015$$
$$x_{34} = -4.01425727958696$$
$$x_{55} = -1.91986217719376$$
$$x_{47} = -1.22173047639603$$
$$x_{64} = 0.174532925199433$$
$$x_{85} = 2.26892802759263$$
$$x_{28} = 4.36332312998582$$
$$x_{1} = 6.45771823237902$$
$$x_{53} = 8.55211333477222$$
$$x_{54} = 10.6465084371654$$
$$x_{79} = 13.4390352403563$$
$$x_{95} = 17.6278254451427$$
$$x_{11} = 19.7222205475359$$
$$x_{41} = 21.8166156499291$$
$$x_{86} = 23.9110107523223$$
$$x_{56} = 26.0054058547155$$
$$x_{18} = 28.0998009571087$$
$$x_{22} = 30.1941960595019$$
$$x_{88} = 32.2885911618951$$
$$x_{19} = 34.3829862642883$$
$$x_{49} = 37.873644768277$$
$$x_{20} = 39.9680398706701$$
$$x_{33} = 42.0624349730633$$
$$x_{25} = 44.1568300754565$$
$$x_{35} = 46.2512251778497$$
$$x_{96} = 48.3456202802429$$
$$x_{10} = 50.4400153826361$$
$$x_{71} = 52.5344104850293$$
$$x_{38} = 54.6288055874225$$
$$x_{48} = 60.9119908946021$$
$$x_{30} = 61.6101225953998$$
$$x_{16} = 63.704517697793$$
$$x_{6} = 65.7989128001862$$
$$x_{82} = 67.8933079025794$$
$$x_{84} = 69.9877030049726$$
$$x_{69} = 72.0820981073658$$
$$x_{27} = 74.176493209759$$
$$x_{8} = 76.2708883121522$$
$$x_{2} = 78.3652834145454$$
$$x_{94} = 81.8559419185341$$
$$x_{4} = 83.9503370209273$$
$$x_{31} = 86.0447321233204$$
$$x_{65} = 88.1391272257137$$
$$x_{87} = 90.2335223281068$$
$$x_{74} = 92.3279174305$$
$$x_{77} = 94.4223125328932$$
$$x_{36} = 96.5167076352864$$
$$x_{70} = 98.6111027376796$$
$$x_{91} = 101.403629540871$$
$$x_{76} = 103.498024643264$$
son puntos de cambio del signo de desigualdad en las soluciones.
Primero definámonos con el signo hasta el punto extremo izquierdo:
$$x_{0} < x_{62}$$
Consideremos, por ejemplo, el punto
$$x_{0} = x_{62} - \frac{1}{10}$$
=
$$-99.6583002888762 + - \frac{1}{10}$$
=
$$-99.7583002888762$$
lo sustituimos en la expresión
$$\sin{\left(x \right)} \cos{\left(2 x \right)} + \sin{\left(2 x \right)} \cos{\left(x \right)} > \frac{1}{2}$$
$$\sin{\left(-99.7583002888762 \right)} \cos{\left(\left(-99.7583002888762\right) 2 \right)} + \sin{\left(\left(-99.7583002888762\right) 2 \right)} \cos{\left(-99.7583002888762 \right)} > \frac{1}{2}$$
significa que una de las soluciones de nuestra ecuación será con:
$$x < -99.6583002888762$$
Recibiremos otras soluciones de la desigualdad pasando al polo siguiente etc.
etc.
Respuesta:
$$x < -99.6583002888762$$
$$x > -97.563905186483 \wedge x < -96.1676417848876$$
$$x > -94.0732466824944 \wedge x < -91.9788515801012$$
$$x > -89.884456477708 \wedge x < -87.7900613753148$$
$$x > -85.6956662729216 \wedge x < -83.6012711705284$$
$$x > -81.5068760681352 \wedge x < -79.412480965742$$
$$x > -77.3180858633488 \wedge x < -76.6199541625511$$
$$x > -73.1292956585624 \wedge x < -72.4311639577647$$
$$x > -70.3367688553715 \wedge x < -68.2423737529783$$
$$x > -66.1479786505851 \wedge x < -64.0535835481919$$
$$x > -61.9591884457987 \wedge x < -59.8647933434055$$
$$x > -57.7703982410123 \wedge x < -55.6760031386191$$
$$x > -53.5816080362259 \wedge x < -52.1853446346305$$
$$x > -50.0909495322373 \wedge x < -47.9965544298441$$
$$x > -45.9021593274509 \wedge x < -43.8077642250577$$
$$x > -41.7133691226645 \wedge x < -39.6189740202713$$
$$x > -37.5245789178781 \wedge x < -35.4301838154849$$
$$x > -34.7320521146872 \wedge x < -29.1469985083053$$
$$x > -26.3544717051144 \wedge x < -24.9582083035189$$
$$x > -24.2600766027212 \wedge x < -22.8638132011257$$
$$x > -22.165681500328 \wedge x < -20.0712863979348$$
$$x > -17.9768912955416 \wedge x < -15.8824961931484$$
$$x > -13.7881010907552 \wedge x < -11.693705988362$$
$$x > -9.59931088596881 \wedge x < -6.10865238198015$$
$$x > -4.01425727958696 \wedge x < -1.91986217719376$$
$$x > -1.22173047639603 \wedge x < 0.174532925199433$$
$$x > 2.26892802759263 \wedge x < 4.36332312998582$$
$$x > 6.45771823237902 \wedge x < 8.55211333477222$$
$$x > 10.6465084371654 \wedge x < 13.4390352403563$$
$$x > 17.6278254451427 \wedge x < 19.7222205475359$$
$$x > 21.8166156499291 \wedge x < 23.9110107523223$$
$$x > 26.0054058547155 \wedge x < 28.0998009571087$$
$$x > 30.1941960595019 \wedge x < 32.2885911618951$$
$$x > 34.3829862642883 \wedge x < 37.873644768277$$
$$x > 39.9680398706701 \wedge x < 42.0624349730633$$
$$x > 44.1568300754565 \wedge x < 46.2512251778497$$
$$x > 48.3456202802429 \wedge x < 50.4400153826361$$
$$x > 52.5344104850293 \wedge x < 54.6288055874225$$
$$x > 60.9119908946021 \wedge x < 61.6101225953998$$
$$x > 63.704517697793 \wedge x < 65.7989128001862$$
$$x > 67.8933079025794 \wedge x < 69.9877030049726$$
$$x > 72.0820981073658 \wedge x < 74.176493209759$$
$$x > 76.2708883121522 \wedge x < 78.3652834145454$$
$$x > 81.8559419185341 \wedge x < 83.9503370209273$$
$$x > 86.0447321233204 \wedge x < 88.1391272257137$$
$$x > 90.2335223281068 \wedge x < 92.3279174305$$
$$x > 94.4223125328932 \wedge x < 96.5167076352864$$
$$x > 98.6111027376796 \wedge x < 101.403629540871$$
$$x > 103.498024643264$$
$$\sin{\left(x \right)} \cos{\left(2 x \right)} + \sin{\left(2 x \right)} \cos{\left(x \right)} > \frac{1}{2}$$
Para resolver esta desigualdad primero hay que resolver la ecuación correspondiente:
$$\sin{\left(x \right)} \cos{\left(2 x \right)} + \sin{\left(2 x \right)} \cos{\left(x \right)} = \frac{1}{2}$$
Resolvemos:
$$x_{1} = 6.45771823237902$$
$$x_{2} = 78.3652834145454$$
$$x_{3} = -13.7881010907552$$
$$x_{4} = 83.9503370209273$$
$$x_{5} = -85.6956662729216$$
$$x_{6} = 65.7989128001862$$
$$x_{7} = -45.9021593274509$$
$$x_{8} = 76.2708883121522$$
$$x_{9} = -77.3180858633488$$
$$x_{10} = 50.4400153826361$$
$$x_{11} = 19.7222205475359$$
$$x_{12} = -22.8638132011257$$
$$x_{13} = -43.8077642250577$$
$$x_{14} = -20.0712863979348$$
$$x_{15} = -83.6012711705284$$
$$x_{16} = 63.704517697793$$
$$x_{17} = -68.2423737529783$$
$$x_{18} = 28.0998009571087$$
$$x_{19} = 34.3829862642883$$
$$x_{20} = 39.9680398706701$$
$$x_{21} = -61.9591884457987$$
$$x_{22} = 30.1941960595019$$
$$x_{23} = -11.693705988362$$
$$x_{24} = -87.7900613753148$$
$$x_{25} = 44.1568300754565$$
$$x_{26} = -97.563905186483$$
$$x_{27} = 74.176493209759$$
$$x_{28} = 4.36332312998582$$
$$x_{29} = -15.8824961931484$$
$$x_{30} = 61.6101225953998$$
$$x_{31} = 86.0447321233204$$
$$x_{32} = -41.7133691226645$$
$$x_{33} = 42.0624349730633$$
$$x_{34} = -4.01425727958696$$
$$x_{35} = 46.2512251778497$$
$$x_{36} = 96.5167076352864$$
$$x_{37} = -81.5068760681352$$
$$x_{38} = 54.6288055874225$$
$$x_{39} = -47.9965544298441$$
$$x_{40} = -24.9582083035189$$
$$x_{41} = 21.8166156499291$$
$$x_{42} = -6.10865238198015$$
$$x_{43} = -37.5245789178781$$
$$x_{44} = -91.9788515801012$$
$$x_{45} = -53.5816080362259$$
$$x_{46} = -73.1292956585624$$
$$x_{47} = -1.22173047639603$$
$$x_{48} = 60.9119908946021$$
$$x_{49} = 37.873644768277$$
$$x_{50} = -9.59931088596881$$
$$x_{51} = -79.412480965742$$
$$x_{52} = -64.0535835481919$$
$$x_{53} = 8.55211333477222$$
$$x_{54} = 10.6465084371654$$
$$x_{55} = -1.91986217719376$$
$$x_{56} = 26.0054058547155$$
$$x_{57} = -34.7320521146872$$
$$x_{58} = -52.1853446346305$$
$$x_{59} = -59.8647933434055$$
$$x_{60} = -94.0732466824944$$
$$x_{61} = -26.3544717051144$$
$$x_{62} = -99.6583002888762$$
$$x_{63} = -70.3367688553715$$
$$x_{64} = 0.174532925199433$$
$$x_{65} = 88.1391272257137$$
$$x_{66} = -50.0909495322373$$
$$x_{67} = -29.1469985083053$$
$$x_{68} = -72.4311639577647$$
$$x_{69} = 72.0820981073658$$
$$x_{70} = 98.6111027376796$$
$$x_{71} = 52.5344104850293$$
$$x_{72} = -55.6760031386191$$
$$x_{73} = -39.6189740202713$$
$$x_{74} = 92.3279174305$$
$$x_{75} = -96.1676417848876$$
$$x_{76} = 103.498024643264$$
$$x_{77} = 94.4223125328932$$
$$x_{78} = -17.9768912955416$$
$$x_{79} = 13.4390352403563$$
$$x_{80} = -89.884456477708$$
$$x_{81} = -66.1479786505851$$
$$x_{82} = 67.8933079025794$$
$$x_{83} = -22.165681500328$$
$$x_{84} = 69.9877030049726$$
$$x_{85} = 2.26892802759263$$
$$x_{86} = 23.9110107523223$$
$$x_{87} = 90.2335223281068$$
$$x_{88} = 32.2885911618951$$
$$x_{89} = -76.6199541625511$$
$$x_{90} = -24.2600766027212$$
$$x_{91} = 101.403629540871$$
$$x_{92} = -57.7703982410123$$
$$x_{93} = -35.4301838154849$$
$$x_{94} = 81.8559419185341$$
$$x_{95} = 17.6278254451427$$
$$x_{96} = 48.3456202802429$$
$$x_{1} = 6.45771823237902$$
$$x_{2} = 78.3652834145454$$
$$x_{3} = -13.7881010907552$$
$$x_{4} = 83.9503370209273$$
$$x_{5} = -85.6956662729216$$
$$x_{6} = 65.7989128001862$$
$$x_{7} = -45.9021593274509$$
$$x_{8} = 76.2708883121522$$
$$x_{9} = -77.3180858633488$$
$$x_{10} = 50.4400153826361$$
$$x_{11} = 19.7222205475359$$
$$x_{12} = -22.8638132011257$$
$$x_{13} = -43.8077642250577$$
$$x_{14} = -20.0712863979348$$
$$x_{15} = -83.6012711705284$$
$$x_{16} = 63.704517697793$$
$$x_{17} = -68.2423737529783$$
$$x_{18} = 28.0998009571087$$
$$x_{19} = 34.3829862642883$$
$$x_{20} = 39.9680398706701$$
$$x_{21} = -61.9591884457987$$
$$x_{22} = 30.1941960595019$$
$$x_{23} = -11.693705988362$$
$$x_{24} = -87.7900613753148$$
$$x_{25} = 44.1568300754565$$
$$x_{26} = -97.563905186483$$
$$x_{27} = 74.176493209759$$
$$x_{28} = 4.36332312998582$$
$$x_{29} = -15.8824961931484$$
$$x_{30} = 61.6101225953998$$
$$x_{31} = 86.0447321233204$$
$$x_{32} = -41.7133691226645$$
$$x_{33} = 42.0624349730633$$
$$x_{34} = -4.01425727958696$$
$$x_{35} = 46.2512251778497$$
$$x_{36} = 96.5167076352864$$
$$x_{37} = -81.5068760681352$$
$$x_{38} = 54.6288055874225$$
$$x_{39} = -47.9965544298441$$
$$x_{40} = -24.9582083035189$$
$$x_{41} = 21.8166156499291$$
$$x_{42} = -6.10865238198015$$
$$x_{43} = -37.5245789178781$$
$$x_{44} = -91.9788515801012$$
$$x_{45} = -53.5816080362259$$
$$x_{46} = -73.1292956585624$$
$$x_{47} = -1.22173047639603$$
$$x_{48} = 60.9119908946021$$
$$x_{49} = 37.873644768277$$
$$x_{50} = -9.59931088596881$$
$$x_{51} = -79.412480965742$$
$$x_{52} = -64.0535835481919$$
$$x_{53} = 8.55211333477222$$
$$x_{54} = 10.6465084371654$$
$$x_{55} = -1.91986217719376$$
$$x_{56} = 26.0054058547155$$
$$x_{57} = -34.7320521146872$$
$$x_{58} = -52.1853446346305$$
$$x_{59} = -59.8647933434055$$
$$x_{60} = -94.0732466824944$$
$$x_{61} = -26.3544717051144$$
$$x_{62} = -99.6583002888762$$
$$x_{63} = -70.3367688553715$$
$$x_{64} = 0.174532925199433$$
$$x_{65} = 88.1391272257137$$
$$x_{66} = -50.0909495322373$$
$$x_{67} = -29.1469985083053$$
$$x_{68} = -72.4311639577647$$
$$x_{69} = 72.0820981073658$$
$$x_{70} = 98.6111027376796$$
$$x_{71} = 52.5344104850293$$
$$x_{72} = -55.6760031386191$$
$$x_{73} = -39.6189740202713$$
$$x_{74} = 92.3279174305$$
$$x_{75} = -96.1676417848876$$
$$x_{76} = 103.498024643264$$
$$x_{77} = 94.4223125328932$$
$$x_{78} = -17.9768912955416$$
$$x_{79} = 13.4390352403563$$
$$x_{80} = -89.884456477708$$
$$x_{81} = -66.1479786505851$$
$$x_{82} = 67.8933079025794$$
$$x_{83} = -22.165681500328$$
$$x_{84} = 69.9877030049726$$
$$x_{85} = 2.26892802759263$$
$$x_{86} = 23.9110107523223$$
$$x_{87} = 90.2335223281068$$
$$x_{88} = 32.2885911618951$$
$$x_{89} = -76.6199541625511$$
$$x_{90} = -24.2600766027212$$
$$x_{91} = 101.403629540871$$
$$x_{92} = -57.7703982410123$$
$$x_{93} = -35.4301838154849$$
$$x_{94} = 81.8559419185341$$
$$x_{95} = 17.6278254451427$$
$$x_{96} = 48.3456202802429$$
Las raíces dadas
$$x_{62} = -99.6583002888762$$
$$x_{26} = -97.563905186483$$
$$x_{75} = -96.1676417848876$$
$$x_{60} = -94.0732466824944$$
$$x_{44} = -91.9788515801012$$
$$x_{80} = -89.884456477708$$
$$x_{24} = -87.7900613753148$$
$$x_{5} = -85.6956662729216$$
$$x_{15} = -83.6012711705284$$
$$x_{37} = -81.5068760681352$$
$$x_{51} = -79.412480965742$$
$$x_{9} = -77.3180858633488$$
$$x_{89} = -76.6199541625511$$
$$x_{46} = -73.1292956585624$$
$$x_{68} = -72.4311639577647$$
$$x_{63} = -70.3367688553715$$
$$x_{17} = -68.2423737529783$$
$$x_{81} = -66.1479786505851$$
$$x_{52} = -64.0535835481919$$
$$x_{21} = -61.9591884457987$$
$$x_{59} = -59.8647933434055$$
$$x_{92} = -57.7703982410123$$
$$x_{72} = -55.6760031386191$$
$$x_{45} = -53.5816080362259$$
$$x_{58} = -52.1853446346305$$
$$x_{66} = -50.0909495322373$$
$$x_{39} = -47.9965544298441$$
$$x_{7} = -45.9021593274509$$
$$x_{13} = -43.8077642250577$$
$$x_{32} = -41.7133691226645$$
$$x_{73} = -39.6189740202713$$
$$x_{43} = -37.5245789178781$$
$$x_{93} = -35.4301838154849$$
$$x_{57} = -34.7320521146872$$
$$x_{67} = -29.1469985083053$$
$$x_{61} = -26.3544717051144$$
$$x_{40} = -24.9582083035189$$
$$x_{90} = -24.2600766027212$$
$$x_{12} = -22.8638132011257$$
$$x_{83} = -22.165681500328$$
$$x_{14} = -20.0712863979348$$
$$x_{78} = -17.9768912955416$$
$$x_{29} = -15.8824961931484$$
$$x_{3} = -13.7881010907552$$
$$x_{23} = -11.693705988362$$
$$x_{50} = -9.59931088596881$$
$$x_{42} = -6.10865238198015$$
$$x_{34} = -4.01425727958696$$
$$x_{55} = -1.91986217719376$$
$$x_{47} = -1.22173047639603$$
$$x_{64} = 0.174532925199433$$
$$x_{85} = 2.26892802759263$$
$$x_{28} = 4.36332312998582$$
$$x_{1} = 6.45771823237902$$
$$x_{53} = 8.55211333477222$$
$$x_{54} = 10.6465084371654$$
$$x_{79} = 13.4390352403563$$
$$x_{95} = 17.6278254451427$$
$$x_{11} = 19.7222205475359$$
$$x_{41} = 21.8166156499291$$
$$x_{86} = 23.9110107523223$$
$$x_{56} = 26.0054058547155$$
$$x_{18} = 28.0998009571087$$
$$x_{22} = 30.1941960595019$$
$$x_{88} = 32.2885911618951$$
$$x_{19} = 34.3829862642883$$
$$x_{49} = 37.873644768277$$
$$x_{20} = 39.9680398706701$$
$$x_{33} = 42.0624349730633$$
$$x_{25} = 44.1568300754565$$
$$x_{35} = 46.2512251778497$$
$$x_{96} = 48.3456202802429$$
$$x_{10} = 50.4400153826361$$
$$x_{71} = 52.5344104850293$$
$$x_{38} = 54.6288055874225$$
$$x_{48} = 60.9119908946021$$
$$x_{30} = 61.6101225953998$$
$$x_{16} = 63.704517697793$$
$$x_{6} = 65.7989128001862$$
$$x_{82} = 67.8933079025794$$
$$x_{84} = 69.9877030049726$$
$$x_{69} = 72.0820981073658$$
$$x_{27} = 74.176493209759$$
$$x_{8} = 76.2708883121522$$
$$x_{2} = 78.3652834145454$$
$$x_{94} = 81.8559419185341$$
$$x_{4} = 83.9503370209273$$
$$x_{31} = 86.0447321233204$$
$$x_{65} = 88.1391272257137$$
$$x_{87} = 90.2335223281068$$
$$x_{74} = 92.3279174305$$
$$x_{77} = 94.4223125328932$$
$$x_{36} = 96.5167076352864$$
$$x_{70} = 98.6111027376796$$
$$x_{91} = 101.403629540871$$
$$x_{76} = 103.498024643264$$
son puntos de cambio del signo de desigualdad en las soluciones.
Primero definámonos con el signo hasta el punto extremo izquierdo:
$$x_{0} < x_{62}$$
Consideremos, por ejemplo, el punto
$$x_{0} = x_{62} - \frac{1}{10}$$
=
$$-99.6583002888762 + - \frac{1}{10}$$
=
$$-99.7583002888762$$
lo sustituimos en la expresión
$$\sin{\left(x \right)} \cos{\left(2 x \right)} + \sin{\left(2 x \right)} \cos{\left(x \right)} > \frac{1}{2}$$
$$\sin{\left(-99.7583002888762 \right)} \cos{\left(\left(-99.7583002888762\right) 2 \right)} + \sin{\left(\left(-99.7583002888762\right) 2 \right)} \cos{\left(-99.7583002888762 \right)} > \frac{1}{2}$$
0.733596250863140 > 1/2
significa que una de las soluciones de nuestra ecuación será con:
$$x < -99.6583002888762$$
_____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____
\ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
-------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-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------
x62 x26 x75 x60 x44 x80 x24 x5 x15 x37 x51 x9 x89 x46 x68 x63 x17 x81 x52 x21 x59 x92 x72 x45 x58 x66 x39 x7 x13 x32 x73 x43 x93 x57 x67 x61 x40 x90 x12 x83 x14 x78 x29 x3 x23 x50 x42 x34 x55 x47 x64 x85 x28 x1 x53 x54 x79 x95 x11 x41 x86 x56 x18 x22 x88 x19 x49 x20 x33 x25 x35 x96 x10 x71 x38 x48 x30 x16 x6 x82 x84 x69 x27 x8 x2 x94 x4 x31 x65 x87 x74 x77 x36 x70 x91 x76Recibiremos otras soluciones de la desigualdad pasando al polo siguiente etc.
etc.
Respuesta:
$$x < -99.6583002888762$$
$$x > -97.563905186483 \wedge x < -96.1676417848876$$
$$x > -94.0732466824944 \wedge x < -91.9788515801012$$
$$x > -89.884456477708 \wedge x < -87.7900613753148$$
$$x > -85.6956662729216 \wedge x < -83.6012711705284$$
$$x > -81.5068760681352 \wedge x < -79.412480965742$$
$$x > -77.3180858633488 \wedge x < -76.6199541625511$$
$$x > -73.1292956585624 \wedge x < -72.4311639577647$$
$$x > -70.3367688553715 \wedge x < -68.2423737529783$$
$$x > -66.1479786505851 \wedge x < -64.0535835481919$$
$$x > -61.9591884457987 \wedge x < -59.8647933434055$$
$$x > -57.7703982410123 \wedge x < -55.6760031386191$$
$$x > -53.5816080362259 \wedge x < -52.1853446346305$$
$$x > -50.0909495322373 \wedge x < -47.9965544298441$$
$$x > -45.9021593274509 \wedge x < -43.8077642250577$$
$$x > -41.7133691226645 \wedge x < -39.6189740202713$$
$$x > -37.5245789178781 \wedge x < -35.4301838154849$$
$$x > -34.7320521146872 \wedge x < -29.1469985083053$$
$$x > -26.3544717051144 \wedge x < -24.9582083035189$$
$$x > -24.2600766027212 \wedge x < -22.8638132011257$$
$$x > -22.165681500328 \wedge x < -20.0712863979348$$
$$x > -17.9768912955416 \wedge x < -15.8824961931484$$
$$x > -13.7881010907552 \wedge x < -11.693705988362$$
$$x > -9.59931088596881 \wedge x < -6.10865238198015$$
$$x > -4.01425727958696 \wedge x < -1.91986217719376$$
$$x > -1.22173047639603 \wedge x < 0.174532925199433$$
$$x > 2.26892802759263 \wedge x < 4.36332312998582$$
$$x > 6.45771823237902 \wedge x < 8.55211333477222$$
$$x > 10.6465084371654 \wedge x < 13.4390352403563$$
$$x > 17.6278254451427 \wedge x < 19.7222205475359$$
$$x > 21.8166156499291 \wedge x < 23.9110107523223$$
$$x > 26.0054058547155 \wedge x < 28.0998009571087$$
$$x > 30.1941960595019 \wedge x < 32.2885911618951$$
$$x > 34.3829862642883 \wedge x < 37.873644768277$$
$$x > 39.9680398706701 \wedge x < 42.0624349730633$$
$$x > 44.1568300754565 \wedge x < 46.2512251778497$$
$$x > 48.3456202802429 \wedge x < 50.4400153826361$$
$$x > 52.5344104850293 \wedge x < 54.6288055874225$$
$$x > 60.9119908946021 \wedge x < 61.6101225953998$$
$$x > 63.704517697793 \wedge x < 65.7989128001862$$
$$x > 67.8933079025794 \wedge x < 69.9877030049726$$
$$x > 72.0820981073658 \wedge x < 74.176493209759$$
$$x > 76.2708883121522 \wedge x < 78.3652834145454$$
$$x > 81.8559419185341 \wedge x < 83.9503370209273$$
$$x > 86.0447321233204 \wedge x < 88.1391272257137$$
$$x > 90.2335223281068 \wedge x < 92.3279174305$$
$$x > 94.4223125328932 \wedge x < 96.5167076352864$$
$$x > 98.6111027376796 \wedge x < 101.403629540871$$
$$x > 103.498024643264$$
Solución de la desigualdad en el gráfico
Respuesta rápida
[src]
/pi 5*pi\ And|-- < x, x < ----| \18 18 /
$$\frac{\pi}{18} < x \wedge x < \frac{5 \pi}{18}$$
(pi/18 < x)∧(x < 5*pi/18)
Respuesta rápida 2
[src]
pi 5*pi (--, ----) 18 18
$$x\ in\ \left(\frac{\pi}{18}, \frac{5 \pi}{18}\right)$$
x in Interval.open(pi/18, 5*pi/18)