Sr Examen
Otras calculadoras
- Sistemas de desigualdades paso a paso
- Ecuaciones básicas paso a paso
- Sistemas de ecuaciones paso a paso
- Solución de desigualdades trigonométricas en línea
- Resolución de desigualdades exponenciales en línea
- Resolución de desigualdades con un módulo en línea
-
¿Cómo usar?
- Desigualdades:
-
4+12x>7+13x
-
x-4x^2/x-1>0
-
x^2-17x+72>0
-
(x-9)*(x-1)>0
- Integral de d{x}:
- |cos(x)|
- Gráfico de la función y =:
- |cos(x)|
-
Expresiones idénticas
- |cos(x)|>√ tres / dos
- módulo de coseno de (x)| más √3 dividir por 2
- módulo de coseno de (x)| más √ tres dividir por dos
- |cosx|>√3/2
- |cos(x)|>√3 dividir por 2
-
Expresiones semejantes
- |sinx|+|cosx|>1
- |sinx|>|cosx|
- |cosx|<(1/2)
- |cosx|>√3/2
-
Expresiones con funciones
- Coseno cos
- cos(x)<-0,5
- cos(x)+sin(x)<0
- cos2x<√2/2
- cos(x)>x^2+1
- cos^2(x)>3/4
- Módulo |
- |2-x|-|6+2x|<3
- |x^3-3x+1|<=x^3+x^2-1
- |x+2|-|x-3|>=2x-1
- |x^3+2x^2-2|>=2+3x
- |x-3|<1
|cos(x)|>√3/2 desigualdades
Solución
Ha introducido
[src]
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|cos(x)| > -----
2
$$\left|{\cos{\left(x \right)}}\right| > \frac{\sqrt{3}}{2}$$
Abs(cos(x)) > sqrt(3)/2
Solución detallada
Se da la desigualdad:
$$\left|{\cos{\left(x \right)}}\right| > \frac{\sqrt{3}}{2}$$
Para resolver esta desigualdad primero hay que resolver la ecuación correspondiente:
$$\left|{\cos{\left(x \right)}}\right| = \frac{\sqrt{3}}{2}$$
Resolvemos:
Tenemos la ecuación
$$\left|{\cos{\left(x \right)}}\right| = \frac{\sqrt{3}}{2}$$
cambiamos
$$\left|{\cos{\left(x \right)}}\right| - 1 - \frac{\sqrt{3}}{2} = 0$$
$$\left|{\cos{\left(x \right)}}\right| - 1 - \frac{\sqrt{3}}{2} = 0$$
Sustituimos
$$w = \left|{\cos{\left(x \right)}}\right|$$
Abrimos los paréntesis en el miembro izquierdo de la ecuación
Transportamos los términos libres (sin w)
del miembro izquierdo al derecho, obtenemos:
$$w - \frac{\sqrt{3}}{2} = 1$$
Dividamos ambos miembros de la ecuación en (w - sqrt(3)/2)/w
Obtenemos la respuesta: w = 1 + sqrt(3)/2
hacemos cambio inverso
$$\left|{\cos{\left(x \right)}}\right| = w$$
sustituimos w:
$$x_{1} = 69.6386371545737$$
$$x_{2} = 25.6563400043166$$
$$x_{3} = 22.5147473507269$$
$$x_{4} = -25.6563400043166$$
$$x_{5} = -60.2138591938044$$
$$x_{6} = 41.3643032722656$$
$$x_{7} = 84.2994028713261$$
$$x_{8} = 60.2138591938044$$
$$x_{9} = 31.9395253114962$$
$$x_{10} = 85.3466004225227$$
$$x_{11} = 53.9306738866248$$
$$x_{12} = 18.3259571459405$$
$$x_{13} = 44.5058959258554$$
$$x_{14} = 91.6297857297023$$
$$x_{15} = -91.6297857297023$$
$$x_{16} = -103.148958792865$$
$$x_{17} = 5.75958653158129$$
$$x_{18} = 9.94837673636768$$
$$x_{19} = 49.7418836818384$$
$$x_{20} = -49.7418836818384$$
$$x_{21} = 74.8746249105567$$
$$x_{22} = 2.61799387799149$$
$$x_{23} = 8.90117918517108$$
$$x_{24} = -65.4498469497874$$
$$x_{25} = 27.7507351067098$$
$$x_{26} = -85.3466004225227$$
$$x_{27} = -90.5825881785057$$
$$x_{28} = -79.0634151153431$$
$$x_{29} = -31.9395253114962$$
$$x_{30} = -97.9129710368819$$
$$x_{31} = -18.3259571459405$$
$$x_{32} = -43.4586983746588$$
$$x_{33} = -82.2050077689329$$
$$x_{34} = 93.7241808320955$$
$$x_{35} = -19.3731546971371$$
$$x_{36} = 88.4881930761125$$
$$x_{37} = -62.3082542961976$$
$$x_{38} = 66.497044500984$$
$$x_{39} = -78.0162175641465$$
$$x_{40} = -9.94837673636768$$
$$x_{41} = 16.2315620435473$$
$$x_{42} = 46.6002910282486$$
$$x_{43} = -5.75958653158129$$
$$x_{44} = -100.007366139275$$
$$x_{45} = 52.8834763354282$$
$$x_{46} = -87.4409955249159$$
$$x_{47} = -93.7241808320955$$
$$x_{48} = -40.317105721069$$
$$x_{49} = 4868.94501428858$$
$$x_{50} = 78.0162175641465$$
$$x_{51} = -21.4675497995303$$
$$x_{52} = -57.0722665402146$$
$$x_{53} = 100.007366139275$$
$$x_{54} = 82.2050077689329$$
$$x_{55} = -75.9218224617533$$
$$x_{56} = -53.9306738866248$$
$$x_{57} = -27.7507351067098$$
$$x_{58} = -2.61799387799149$$
$$x_{59} = 63.3554518473942$$
$$x_{60} = 0.523598775598299$$
$$x_{61} = 3.66519142918809$$
$$x_{62} = -16.2315620435473$$
$$x_{63} = -56.025068989018$$
$$x_{64} = -84.2994028713261$$
$$x_{65} = 68.5914396033772$$
$$x_{66} = 24.60914245312$$
$$x_{67} = 47.6474885794452$$
$$x_{68} = -34.0339204138894$$
$$x_{69} = 75.9218224617533$$
$$x_{70} = 97.9129710368819$$
$$x_{71} = 12.0427718387609$$
$$x_{72} = -46.6002910282486$$
$$x_{73} = 30.8923277602996$$
$$x_{74} = -35.081117965086$$
$$x_{75} = -63.3554518473942$$
$$x_{76} = 19.3731546971371$$
$$x_{77} = 62.3082542961976$$
$$x_{78} = -68.5914396033772$$
$$x_{79} = 71.733032256967$$
$$x_{80} = -12.0427718387609$$
$$x_{81} = -24.60914245312$$
$$x_{82} = 90.5825881785057$$
$$x_{83} = 6.80678408277789$$
$$x_{84} = -69.6386371545737$$
$$x_{85} = -71.733032256967$$
$$x_{86} = 38.2227106186758$$
$$x_{87} = -13.0899693899575$$
$$x_{88} = 40.317105721069$$
$$x_{89} = -41.3643032722656$$
$$x_{90} = 56.025068989018$$
$$x_{91} = -47.6474885794452$$
$$x_{92} = -3.66519142918809$$
$$x_{93} = 96.8657734856853$$
$$x_{94} = 34.0339204138894$$
$$x_{95} = -38.2227106186758$$
$$x_{1} = 69.6386371545737$$
$$x_{2} = 25.6563400043166$$
$$x_{3} = 22.5147473507269$$
$$x_{4} = -25.6563400043166$$
$$x_{5} = -60.2138591938044$$
$$x_{6} = 41.3643032722656$$
$$x_{7} = 84.2994028713261$$
$$x_{8} = 60.2138591938044$$
$$x_{9} = 31.9395253114962$$
$$x_{10} = 85.3466004225227$$
$$x_{11} = 53.9306738866248$$
$$x_{12} = 18.3259571459405$$
$$x_{13} = 44.5058959258554$$
$$x_{14} = 91.6297857297023$$
$$x_{15} = -91.6297857297023$$
$$x_{16} = -103.148958792865$$
$$x_{17} = 5.75958653158129$$
$$x_{18} = 9.94837673636768$$
$$x_{19} = 49.7418836818384$$
$$x_{20} = -49.7418836818384$$
$$x_{21} = 74.8746249105567$$
$$x_{22} = 2.61799387799149$$
$$x_{23} = 8.90117918517108$$
$$x_{24} = -65.4498469497874$$
$$x_{25} = 27.7507351067098$$
$$x_{26} = -85.3466004225227$$
$$x_{27} = -90.5825881785057$$
$$x_{28} = -79.0634151153431$$
$$x_{29} = -31.9395253114962$$
$$x_{30} = -97.9129710368819$$
$$x_{31} = -18.3259571459405$$
$$x_{32} = -43.4586983746588$$
$$x_{33} = -82.2050077689329$$
$$x_{34} = 93.7241808320955$$
$$x_{35} = -19.3731546971371$$
$$x_{36} = 88.4881930761125$$
$$x_{37} = -62.3082542961976$$
$$x_{38} = 66.497044500984$$
$$x_{39} = -78.0162175641465$$
$$x_{40} = -9.94837673636768$$
$$x_{41} = 16.2315620435473$$
$$x_{42} = 46.6002910282486$$
$$x_{43} = -5.75958653158129$$
$$x_{44} = -100.007366139275$$
$$x_{45} = 52.8834763354282$$
$$x_{46} = -87.4409955249159$$
$$x_{47} = -93.7241808320955$$
$$x_{48} = -40.317105721069$$
$$x_{49} = 4868.94501428858$$
$$x_{50} = 78.0162175641465$$
$$x_{51} = -21.4675497995303$$
$$x_{52} = -57.0722665402146$$
$$x_{53} = 100.007366139275$$
$$x_{54} = 82.2050077689329$$
$$x_{55} = -75.9218224617533$$
$$x_{56} = -53.9306738866248$$
$$x_{57} = -27.7507351067098$$
$$x_{58} = -2.61799387799149$$
$$x_{59} = 63.3554518473942$$
$$x_{60} = 0.523598775598299$$
$$x_{61} = 3.66519142918809$$
$$x_{62} = -16.2315620435473$$
$$x_{63} = -56.025068989018$$
$$x_{64} = -84.2994028713261$$
$$x_{65} = 68.5914396033772$$
$$x_{66} = 24.60914245312$$
$$x_{67} = 47.6474885794452$$
$$x_{68} = -34.0339204138894$$
$$x_{69} = 75.9218224617533$$
$$x_{70} = 97.9129710368819$$
$$x_{71} = 12.0427718387609$$
$$x_{72} = -46.6002910282486$$
$$x_{73} = 30.8923277602996$$
$$x_{74} = -35.081117965086$$
$$x_{75} = -63.3554518473942$$
$$x_{76} = 19.3731546971371$$
$$x_{77} = 62.3082542961976$$
$$x_{78} = -68.5914396033772$$
$$x_{79} = 71.733032256967$$
$$x_{80} = -12.0427718387609$$
$$x_{81} = -24.60914245312$$
$$x_{82} = 90.5825881785057$$
$$x_{83} = 6.80678408277789$$
$$x_{84} = -69.6386371545737$$
$$x_{85} = -71.733032256967$$
$$x_{86} = 38.2227106186758$$
$$x_{87} = -13.0899693899575$$
$$x_{88} = 40.317105721069$$
$$x_{89} = -41.3643032722656$$
$$x_{90} = 56.025068989018$$
$$x_{91} = -47.6474885794452$$
$$x_{92} = -3.66519142918809$$
$$x_{93} = 96.8657734856853$$
$$x_{94} = 34.0339204138894$$
$$x_{95} = -38.2227106186758$$
Las raíces dadas
$$x_{16} = -103.148958792865$$
$$x_{44} = -100.007366139275$$
$$x_{30} = -97.9129710368819$$
$$x_{47} = -93.7241808320955$$
$$x_{15} = -91.6297857297023$$
$$x_{27} = -90.5825881785057$$
$$x_{46} = -87.4409955249159$$
$$x_{26} = -85.3466004225227$$
$$x_{64} = -84.2994028713261$$
$$x_{33} = -82.2050077689329$$
$$x_{28} = -79.0634151153431$$
$$x_{39} = -78.0162175641465$$
$$x_{55} = -75.9218224617533$$
$$x_{85} = -71.733032256967$$
$$x_{84} = -69.6386371545737$$
$$x_{78} = -68.5914396033772$$
$$x_{24} = -65.4498469497874$$
$$x_{75} = -63.3554518473942$$
$$x_{37} = -62.3082542961976$$
$$x_{5} = -60.2138591938044$$
$$x_{52} = -57.0722665402146$$
$$x_{63} = -56.025068989018$$
$$x_{56} = -53.9306738866248$$
$$x_{20} = -49.7418836818384$$
$$x_{91} = -47.6474885794452$$
$$x_{72} = -46.6002910282486$$
$$x_{32} = -43.4586983746588$$
$$x_{89} = -41.3643032722656$$
$$x_{48} = -40.317105721069$$
$$x_{95} = -38.2227106186758$$
$$x_{74} = -35.081117965086$$
$$x_{68} = -34.0339204138894$$
$$x_{29} = -31.9395253114962$$
$$x_{57} = -27.7507351067098$$
$$x_{4} = -25.6563400043166$$
$$x_{81} = -24.60914245312$$
$$x_{51} = -21.4675497995303$$
$$x_{35} = -19.3731546971371$$
$$x_{31} = -18.3259571459405$$
$$x_{62} = -16.2315620435473$$
$$x_{87} = -13.0899693899575$$
$$x_{80} = -12.0427718387609$$
$$x_{40} = -9.94837673636768$$
$$x_{43} = -5.75958653158129$$
$$x_{92} = -3.66519142918809$$
$$x_{58} = -2.61799387799149$$
$$x_{60} = 0.523598775598299$$
$$x_{22} = 2.61799387799149$$
$$x_{61} = 3.66519142918809$$
$$x_{17} = 5.75958653158129$$
$$x_{83} = 6.80678408277789$$
$$x_{23} = 8.90117918517108$$
$$x_{18} = 9.94837673636768$$
$$x_{71} = 12.0427718387609$$
$$x_{41} = 16.2315620435473$$
$$x_{12} = 18.3259571459405$$
$$x_{76} = 19.3731546971371$$
$$x_{3} = 22.5147473507269$$
$$x_{66} = 24.60914245312$$
$$x_{2} = 25.6563400043166$$
$$x_{25} = 27.7507351067098$$
$$x_{73} = 30.8923277602996$$
$$x_{9} = 31.9395253114962$$
$$x_{94} = 34.0339204138894$$
$$x_{86} = 38.2227106186758$$
$$x_{88} = 40.317105721069$$
$$x_{6} = 41.3643032722656$$
$$x_{13} = 44.5058959258554$$
$$x_{42} = 46.6002910282486$$
$$x_{67} = 47.6474885794452$$
$$x_{19} = 49.7418836818384$$
$$x_{45} = 52.8834763354282$$
$$x_{11} = 53.9306738866248$$
$$x_{90} = 56.025068989018$$
$$x_{8} = 60.2138591938044$$
$$x_{77} = 62.3082542961976$$
$$x_{59} = 63.3554518473942$$
$$x_{38} = 66.497044500984$$
$$x_{65} = 68.5914396033772$$
$$x_{1} = 69.6386371545737$$
$$x_{79} = 71.733032256967$$
$$x_{21} = 74.8746249105567$$
$$x_{69} = 75.9218224617533$$
$$x_{50} = 78.0162175641465$$
$$x_{54} = 82.2050077689329$$
$$x_{7} = 84.2994028713261$$
$$x_{10} = 85.3466004225227$$
$$x_{36} = 88.4881930761125$$
$$x_{82} = 90.5825881785057$$
$$x_{14} = 91.6297857297023$$
$$x_{34} = 93.7241808320955$$
$$x_{93} = 96.8657734856853$$
$$x_{70} = 97.9129710368819$$
$$x_{53} = 100.007366139275$$
$$x_{49} = 4868.94501428858$$
son puntos de cambio del signo de desigualdad en las soluciones.
Primero definámonos con el signo hasta el punto extremo izquierdo:
$$x_{0} < x_{16}$$
Consideremos, por ejemplo, el punto
$$x_{0} = x_{16} - \frac{1}{10}$$
=
$$-103.148958792865 + - \frac{1}{10}$$
=
$$-103.248958792865$$
lo sustituimos en la expresión
$$\left|{\cos{\left(x \right)}}\right| > \frac{\sqrt{3}}{2}$$
$$\left|{\cos{\left(-103.248958792865 \right)}}\right| > \frac{\sqrt{3}}{2}$$
significa que una de las soluciones de nuestra ecuación será con:
$$x < -103.148958792865$$
Recibiremos otras soluciones de la desigualdad pasando al polo siguiente etc.
etc.
Respuesta:
$$x < -103.148958792865$$
$$x > -100.007366139275 \wedge x < -97.9129710368819$$
$$x > -93.7241808320955 \wedge x < -91.6297857297023$$
$$x > -90.5825881785057 \wedge x < -87.4409955249159$$
$$x > -85.3466004225227 \wedge x < -84.2994028713261$$
$$x > -82.2050077689329 \wedge x < -79.0634151153431$$
$$x > -78.0162175641465 \wedge x < -75.9218224617533$$
$$x > -71.733032256967 \wedge x < -69.6386371545737$$
$$x > -68.5914396033772 \wedge x < -65.4498469497874$$
$$x > -63.3554518473942 \wedge x < -62.3082542961976$$
$$x > -60.2138591938044 \wedge x < -57.0722665402146$$
$$x > -56.025068989018 \wedge x < -53.9306738866248$$
$$x > -49.7418836818384 \wedge x < -47.6474885794452$$
$$x > -46.6002910282486 \wedge x < -43.4586983746588$$
$$x > -41.3643032722656 \wedge x < -40.317105721069$$
$$x > -38.2227106186758 \wedge x < -35.081117965086$$
$$x > -34.0339204138894 \wedge x < -31.9395253114962$$
$$x > -27.7507351067098 \wedge x < -25.6563400043166$$
$$x > -24.60914245312 \wedge x < -21.4675497995303$$
$$x > -19.3731546971371 \wedge x < -18.3259571459405$$
$$x > -16.2315620435473 \wedge x < -13.0899693899575$$
$$x > -12.0427718387609 \wedge x < -9.94837673636768$$
$$x > -5.75958653158129 \wedge x < -3.66519142918809$$
$$x > -2.61799387799149 \wedge x < 0.523598775598299$$
$$x > 2.61799387799149 \wedge x < 3.66519142918809$$
$$x > 5.75958653158129 \wedge x < 6.80678408277789$$
$$x > 8.90117918517108 \wedge x < 9.94837673636768$$
$$x > 12.0427718387609 \wedge x < 16.2315620435473$$
$$x > 18.3259571459405 \wedge x < 19.3731546971371$$
$$x > 22.5147473507269 \wedge x < 24.60914245312$$
$$x > 25.6563400043166 \wedge x < 27.7507351067098$$
$$x > 30.8923277602996 \wedge x < 31.9395253114962$$
$$x > 34.0339204138894 \wedge x < 38.2227106186758$$
$$x > 40.317105721069 \wedge x < 41.3643032722656$$
$$x > 44.5058959258554 \wedge x < 46.6002910282486$$
$$x > 47.6474885794452 \wedge x < 49.7418836818384$$
$$x > 52.8834763354282 \wedge x < 53.9306738866248$$
$$x > 56.025068989018 \wedge x < 60.2138591938044$$
$$x > 62.3082542961976 \wedge x < 63.3554518473942$$
$$x > 66.497044500984 \wedge x < 68.5914396033772$$
$$x > 69.6386371545737 \wedge x < 71.733032256967$$
$$x > 74.8746249105567 \wedge x < 75.9218224617533$$
$$x > 78.0162175641465 \wedge x < 82.2050077689329$$
$$x > 84.2994028713261 \wedge x < 85.3466004225227$$
$$x > 88.4881930761125 \wedge x < 90.5825881785057$$
$$x > 91.6297857297023 \wedge x < 93.7241808320955$$
$$x > 96.8657734856853 \wedge x < 97.9129710368819$$
$$x > 100.007366139275 \wedge x < 4868.94501428858$$
$$\left|{\cos{\left(x \right)}}\right| > \frac{\sqrt{3}}{2}$$
Para resolver esta desigualdad primero hay que resolver la ecuación correspondiente:
$$\left|{\cos{\left(x \right)}}\right| = \frac{\sqrt{3}}{2}$$
Resolvemos:
Tenemos la ecuación
$$\left|{\cos{\left(x \right)}}\right| = \frac{\sqrt{3}}{2}$$
cambiamos
$$\left|{\cos{\left(x \right)}}\right| - 1 - \frac{\sqrt{3}}{2} = 0$$
$$\left|{\cos{\left(x \right)}}\right| - 1 - \frac{\sqrt{3}}{2} = 0$$
Sustituimos
$$w = \left|{\cos{\left(x \right)}}\right|$$
Abrimos los paréntesis en el miembro izquierdo de la ecuación
-1 + w - sqrt3/2 = 0
Transportamos los términos libres (sin w)
del miembro izquierdo al derecho, obtenemos:
$$w - \frac{\sqrt{3}}{2} = 1$$
Dividamos ambos miembros de la ecuación en (w - sqrt(3)/2)/w
w = 1 / ((w - sqrt(3)/2)/w)
Obtenemos la respuesta: w = 1 + sqrt(3)/2
hacemos cambio inverso
$$\left|{\cos{\left(x \right)}}\right| = w$$
sustituimos w:
$$x_{1} = 69.6386371545737$$
$$x_{2} = 25.6563400043166$$
$$x_{3} = 22.5147473507269$$
$$x_{4} = -25.6563400043166$$
$$x_{5} = -60.2138591938044$$
$$x_{6} = 41.3643032722656$$
$$x_{7} = 84.2994028713261$$
$$x_{8} = 60.2138591938044$$
$$x_{9} = 31.9395253114962$$
$$x_{10} = 85.3466004225227$$
$$x_{11} = 53.9306738866248$$
$$x_{12} = 18.3259571459405$$
$$x_{13} = 44.5058959258554$$
$$x_{14} = 91.6297857297023$$
$$x_{15} = -91.6297857297023$$
$$x_{16} = -103.148958792865$$
$$x_{17} = 5.75958653158129$$
$$x_{18} = 9.94837673636768$$
$$x_{19} = 49.7418836818384$$
$$x_{20} = -49.7418836818384$$
$$x_{21} = 74.8746249105567$$
$$x_{22} = 2.61799387799149$$
$$x_{23} = 8.90117918517108$$
$$x_{24} = -65.4498469497874$$
$$x_{25} = 27.7507351067098$$
$$x_{26} = -85.3466004225227$$
$$x_{27} = -90.5825881785057$$
$$x_{28} = -79.0634151153431$$
$$x_{29} = -31.9395253114962$$
$$x_{30} = -97.9129710368819$$
$$x_{31} = -18.3259571459405$$
$$x_{32} = -43.4586983746588$$
$$x_{33} = -82.2050077689329$$
$$x_{34} = 93.7241808320955$$
$$x_{35} = -19.3731546971371$$
$$x_{36} = 88.4881930761125$$
$$x_{37} = -62.3082542961976$$
$$x_{38} = 66.497044500984$$
$$x_{39} = -78.0162175641465$$
$$x_{40} = -9.94837673636768$$
$$x_{41} = 16.2315620435473$$
$$x_{42} = 46.6002910282486$$
$$x_{43} = -5.75958653158129$$
$$x_{44} = -100.007366139275$$
$$x_{45} = 52.8834763354282$$
$$x_{46} = -87.4409955249159$$
$$x_{47} = -93.7241808320955$$
$$x_{48} = -40.317105721069$$
$$x_{49} = 4868.94501428858$$
$$x_{50} = 78.0162175641465$$
$$x_{51} = -21.4675497995303$$
$$x_{52} = -57.0722665402146$$
$$x_{53} = 100.007366139275$$
$$x_{54} = 82.2050077689329$$
$$x_{55} = -75.9218224617533$$
$$x_{56} = -53.9306738866248$$
$$x_{57} = -27.7507351067098$$
$$x_{58} = -2.61799387799149$$
$$x_{59} = 63.3554518473942$$
$$x_{60} = 0.523598775598299$$
$$x_{61} = 3.66519142918809$$
$$x_{62} = -16.2315620435473$$
$$x_{63} = -56.025068989018$$
$$x_{64} = -84.2994028713261$$
$$x_{65} = 68.5914396033772$$
$$x_{66} = 24.60914245312$$
$$x_{67} = 47.6474885794452$$
$$x_{68} = -34.0339204138894$$
$$x_{69} = 75.9218224617533$$
$$x_{70} = 97.9129710368819$$
$$x_{71} = 12.0427718387609$$
$$x_{72} = -46.6002910282486$$
$$x_{73} = 30.8923277602996$$
$$x_{74} = -35.081117965086$$
$$x_{75} = -63.3554518473942$$
$$x_{76} = 19.3731546971371$$
$$x_{77} = 62.3082542961976$$
$$x_{78} = -68.5914396033772$$
$$x_{79} = 71.733032256967$$
$$x_{80} = -12.0427718387609$$
$$x_{81} = -24.60914245312$$
$$x_{82} = 90.5825881785057$$
$$x_{83} = 6.80678408277789$$
$$x_{84} = -69.6386371545737$$
$$x_{85} = -71.733032256967$$
$$x_{86} = 38.2227106186758$$
$$x_{87} = -13.0899693899575$$
$$x_{88} = 40.317105721069$$
$$x_{89} = -41.3643032722656$$
$$x_{90} = 56.025068989018$$
$$x_{91} = -47.6474885794452$$
$$x_{92} = -3.66519142918809$$
$$x_{93} = 96.8657734856853$$
$$x_{94} = 34.0339204138894$$
$$x_{95} = -38.2227106186758$$
$$x_{1} = 69.6386371545737$$
$$x_{2} = 25.6563400043166$$
$$x_{3} = 22.5147473507269$$
$$x_{4} = -25.6563400043166$$
$$x_{5} = -60.2138591938044$$
$$x_{6} = 41.3643032722656$$
$$x_{7} = 84.2994028713261$$
$$x_{8} = 60.2138591938044$$
$$x_{9} = 31.9395253114962$$
$$x_{10} = 85.3466004225227$$
$$x_{11} = 53.9306738866248$$
$$x_{12} = 18.3259571459405$$
$$x_{13} = 44.5058959258554$$
$$x_{14} = 91.6297857297023$$
$$x_{15} = -91.6297857297023$$
$$x_{16} = -103.148958792865$$
$$x_{17} = 5.75958653158129$$
$$x_{18} = 9.94837673636768$$
$$x_{19} = 49.7418836818384$$
$$x_{20} = -49.7418836818384$$
$$x_{21} = 74.8746249105567$$
$$x_{22} = 2.61799387799149$$
$$x_{23} = 8.90117918517108$$
$$x_{24} = -65.4498469497874$$
$$x_{25} = 27.7507351067098$$
$$x_{26} = -85.3466004225227$$
$$x_{27} = -90.5825881785057$$
$$x_{28} = -79.0634151153431$$
$$x_{29} = -31.9395253114962$$
$$x_{30} = -97.9129710368819$$
$$x_{31} = -18.3259571459405$$
$$x_{32} = -43.4586983746588$$
$$x_{33} = -82.2050077689329$$
$$x_{34} = 93.7241808320955$$
$$x_{35} = -19.3731546971371$$
$$x_{36} = 88.4881930761125$$
$$x_{37} = -62.3082542961976$$
$$x_{38} = 66.497044500984$$
$$x_{39} = -78.0162175641465$$
$$x_{40} = -9.94837673636768$$
$$x_{41} = 16.2315620435473$$
$$x_{42} = 46.6002910282486$$
$$x_{43} = -5.75958653158129$$
$$x_{44} = -100.007366139275$$
$$x_{45} = 52.8834763354282$$
$$x_{46} = -87.4409955249159$$
$$x_{47} = -93.7241808320955$$
$$x_{48} = -40.317105721069$$
$$x_{49} = 4868.94501428858$$
$$x_{50} = 78.0162175641465$$
$$x_{51} = -21.4675497995303$$
$$x_{52} = -57.0722665402146$$
$$x_{53} = 100.007366139275$$
$$x_{54} = 82.2050077689329$$
$$x_{55} = -75.9218224617533$$
$$x_{56} = -53.9306738866248$$
$$x_{57} = -27.7507351067098$$
$$x_{58} = -2.61799387799149$$
$$x_{59} = 63.3554518473942$$
$$x_{60} = 0.523598775598299$$
$$x_{61} = 3.66519142918809$$
$$x_{62} = -16.2315620435473$$
$$x_{63} = -56.025068989018$$
$$x_{64} = -84.2994028713261$$
$$x_{65} = 68.5914396033772$$
$$x_{66} = 24.60914245312$$
$$x_{67} = 47.6474885794452$$
$$x_{68} = -34.0339204138894$$
$$x_{69} = 75.9218224617533$$
$$x_{70} = 97.9129710368819$$
$$x_{71} = 12.0427718387609$$
$$x_{72} = -46.6002910282486$$
$$x_{73} = 30.8923277602996$$
$$x_{74} = -35.081117965086$$
$$x_{75} = -63.3554518473942$$
$$x_{76} = 19.3731546971371$$
$$x_{77} = 62.3082542961976$$
$$x_{78} = -68.5914396033772$$
$$x_{79} = 71.733032256967$$
$$x_{80} = -12.0427718387609$$
$$x_{81} = -24.60914245312$$
$$x_{82} = 90.5825881785057$$
$$x_{83} = 6.80678408277789$$
$$x_{84} = -69.6386371545737$$
$$x_{85} = -71.733032256967$$
$$x_{86} = 38.2227106186758$$
$$x_{87} = -13.0899693899575$$
$$x_{88} = 40.317105721069$$
$$x_{89} = -41.3643032722656$$
$$x_{90} = 56.025068989018$$
$$x_{91} = -47.6474885794452$$
$$x_{92} = -3.66519142918809$$
$$x_{93} = 96.8657734856853$$
$$x_{94} = 34.0339204138894$$
$$x_{95} = -38.2227106186758$$
Las raíces dadas
$$x_{16} = -103.148958792865$$
$$x_{44} = -100.007366139275$$
$$x_{30} = -97.9129710368819$$
$$x_{47} = -93.7241808320955$$
$$x_{15} = -91.6297857297023$$
$$x_{27} = -90.5825881785057$$
$$x_{46} = -87.4409955249159$$
$$x_{26} = -85.3466004225227$$
$$x_{64} = -84.2994028713261$$
$$x_{33} = -82.2050077689329$$
$$x_{28} = -79.0634151153431$$
$$x_{39} = -78.0162175641465$$
$$x_{55} = -75.9218224617533$$
$$x_{85} = -71.733032256967$$
$$x_{84} = -69.6386371545737$$
$$x_{78} = -68.5914396033772$$
$$x_{24} = -65.4498469497874$$
$$x_{75} = -63.3554518473942$$
$$x_{37} = -62.3082542961976$$
$$x_{5} = -60.2138591938044$$
$$x_{52} = -57.0722665402146$$
$$x_{63} = -56.025068989018$$
$$x_{56} = -53.9306738866248$$
$$x_{20} = -49.7418836818384$$
$$x_{91} = -47.6474885794452$$
$$x_{72} = -46.6002910282486$$
$$x_{32} = -43.4586983746588$$
$$x_{89} = -41.3643032722656$$
$$x_{48} = -40.317105721069$$
$$x_{95} = -38.2227106186758$$
$$x_{74} = -35.081117965086$$
$$x_{68} = -34.0339204138894$$
$$x_{29} = -31.9395253114962$$
$$x_{57} = -27.7507351067098$$
$$x_{4} = -25.6563400043166$$
$$x_{81} = -24.60914245312$$
$$x_{51} = -21.4675497995303$$
$$x_{35} = -19.3731546971371$$
$$x_{31} = -18.3259571459405$$
$$x_{62} = -16.2315620435473$$
$$x_{87} = -13.0899693899575$$
$$x_{80} = -12.0427718387609$$
$$x_{40} = -9.94837673636768$$
$$x_{43} = -5.75958653158129$$
$$x_{92} = -3.66519142918809$$
$$x_{58} = -2.61799387799149$$
$$x_{60} = 0.523598775598299$$
$$x_{22} = 2.61799387799149$$
$$x_{61} = 3.66519142918809$$
$$x_{17} = 5.75958653158129$$
$$x_{83} = 6.80678408277789$$
$$x_{23} = 8.90117918517108$$
$$x_{18} = 9.94837673636768$$
$$x_{71} = 12.0427718387609$$
$$x_{41} = 16.2315620435473$$
$$x_{12} = 18.3259571459405$$
$$x_{76} = 19.3731546971371$$
$$x_{3} = 22.5147473507269$$
$$x_{66} = 24.60914245312$$
$$x_{2} = 25.6563400043166$$
$$x_{25} = 27.7507351067098$$
$$x_{73} = 30.8923277602996$$
$$x_{9} = 31.9395253114962$$
$$x_{94} = 34.0339204138894$$
$$x_{86} = 38.2227106186758$$
$$x_{88} = 40.317105721069$$
$$x_{6} = 41.3643032722656$$
$$x_{13} = 44.5058959258554$$
$$x_{42} = 46.6002910282486$$
$$x_{67} = 47.6474885794452$$
$$x_{19} = 49.7418836818384$$
$$x_{45} = 52.8834763354282$$
$$x_{11} = 53.9306738866248$$
$$x_{90} = 56.025068989018$$
$$x_{8} = 60.2138591938044$$
$$x_{77} = 62.3082542961976$$
$$x_{59} = 63.3554518473942$$
$$x_{38} = 66.497044500984$$
$$x_{65} = 68.5914396033772$$
$$x_{1} = 69.6386371545737$$
$$x_{79} = 71.733032256967$$
$$x_{21} = 74.8746249105567$$
$$x_{69} = 75.9218224617533$$
$$x_{50} = 78.0162175641465$$
$$x_{54} = 82.2050077689329$$
$$x_{7} = 84.2994028713261$$
$$x_{10} = 85.3466004225227$$
$$x_{36} = 88.4881930761125$$
$$x_{82} = 90.5825881785057$$
$$x_{14} = 91.6297857297023$$
$$x_{34} = 93.7241808320955$$
$$x_{93} = 96.8657734856853$$
$$x_{70} = 97.9129710368819$$
$$x_{53} = 100.007366139275$$
$$x_{49} = 4868.94501428858$$
son puntos de cambio del signo de desigualdad en las soluciones.
Primero definámonos con el signo hasta el punto extremo izquierdo:
$$x_{0} < x_{16}$$
Consideremos, por ejemplo, el punto
$$x_{0} = x_{16} - \frac{1}{10}$$
=
$$-103.148958792865 + - \frac{1}{10}$$
=
$$-103.248958792865$$
lo sustituimos en la expresión
$$\left|{\cos{\left(x \right)}}\right| > \frac{\sqrt{3}}{2}$$
$$\left|{\cos{\left(-103.248958792865 \right)}}\right| > \frac{\sqrt{3}}{2}$$
___
\/ 3
0.911615592325511 > -----
2
significa que una de las soluciones de nuestra ecuación será con:
$$x < -103.148958792865$$
_____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____
\ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \
-------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-------
x16 x44 x30 x47 x15 x27 x46 x26 x64 x33 x28 x39 x55 x85 x84 x78 x24 x75 x37 x5 x52 x63 x56 x20 x91 x72 x32 x89 x48 x95 x74 x68 x29 x57 x4 x81 x51 x35 x31 x62 x87 x80 x40 x43 x92 x58 x60 x22 x61 x17 x83 x23 x18 x71 x41 x12 x76 x3 x66 x2 x25 x73 x9 x94 x86 x88 x6 x13 x42 x67 x19 x45 x11 x90 x8 x77 x59 x38 x65 x1 x79 x21 x69 x50 x54 x7 x10 x36 x82 x14 x34 x93 x70 x53 x49Recibiremos otras soluciones de la desigualdad pasando al polo siguiente etc.
etc.
Respuesta:
$$x < -103.148958792865$$
$$x > -100.007366139275 \wedge x < -97.9129710368819$$
$$x > -93.7241808320955 \wedge x < -91.6297857297023$$
$$x > -90.5825881785057 \wedge x < -87.4409955249159$$
$$x > -85.3466004225227 \wedge x < -84.2994028713261$$
$$x > -82.2050077689329 \wedge x < -79.0634151153431$$
$$x > -78.0162175641465 \wedge x < -75.9218224617533$$
$$x > -71.733032256967 \wedge x < -69.6386371545737$$
$$x > -68.5914396033772 \wedge x < -65.4498469497874$$
$$x > -63.3554518473942 \wedge x < -62.3082542961976$$
$$x > -60.2138591938044 \wedge x < -57.0722665402146$$
$$x > -56.025068989018 \wedge x < -53.9306738866248$$
$$x > -49.7418836818384 \wedge x < -47.6474885794452$$
$$x > -46.6002910282486 \wedge x < -43.4586983746588$$
$$x > -41.3643032722656 \wedge x < -40.317105721069$$
$$x > -38.2227106186758 \wedge x < -35.081117965086$$
$$x > -34.0339204138894 \wedge x < -31.9395253114962$$
$$x > -27.7507351067098 \wedge x < -25.6563400043166$$
$$x > -24.60914245312 \wedge x < -21.4675497995303$$
$$x > -19.3731546971371 \wedge x < -18.3259571459405$$
$$x > -16.2315620435473 \wedge x < -13.0899693899575$$
$$x > -12.0427718387609 \wedge x < -9.94837673636768$$
$$x > -5.75958653158129 \wedge x < -3.66519142918809$$
$$x > -2.61799387799149 \wedge x < 0.523598775598299$$
$$x > 2.61799387799149 \wedge x < 3.66519142918809$$
$$x > 5.75958653158129 \wedge x < 6.80678408277789$$
$$x > 8.90117918517108 \wedge x < 9.94837673636768$$
$$x > 12.0427718387609 \wedge x < 16.2315620435473$$
$$x > 18.3259571459405 \wedge x < 19.3731546971371$$
$$x > 22.5147473507269 \wedge x < 24.60914245312$$
$$x > 25.6563400043166 \wedge x < 27.7507351067098$$
$$x > 30.8923277602996 \wedge x < 31.9395253114962$$
$$x > 34.0339204138894 \wedge x < 38.2227106186758$$
$$x > 40.317105721069 \wedge x < 41.3643032722656$$
$$x > 44.5058959258554 \wedge x < 46.6002910282486$$
$$x > 47.6474885794452 \wedge x < 49.7418836818384$$
$$x > 52.8834763354282 \wedge x < 53.9306738866248$$
$$x > 56.025068989018 \wedge x < 60.2138591938044$$
$$x > 62.3082542961976 \wedge x < 63.3554518473942$$
$$x > 66.497044500984 \wedge x < 68.5914396033772$$
$$x > 69.6386371545737 \wedge x < 71.733032256967$$
$$x > 74.8746249105567 \wedge x < 75.9218224617533$$
$$x > 78.0162175641465 \wedge x < 82.2050077689329$$
$$x > 84.2994028713261 \wedge x < 85.3466004225227$$
$$x > 88.4881930761125 \wedge x < 90.5825881785057$$
$$x > 91.6297857297023 \wedge x < 93.7241808320955$$
$$x > 96.8657734856853 \wedge x < 97.9129710368819$$
$$x > 100.007366139275 \wedge x < 4868.94501428858$$
Solución de la desigualdad en el gráfico