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)²>x(x-4)
-
(x^2-4)*(x^2-9)>0
-
x^2<0
-
x^2+2>0
-
Expresiones idénticas
- |sin(dos x)|<(sqrt(tres))/2
- módulo de seno de (2x)| menos ( raíz cuadrada de (3)) dividir por 2
- módulo de seno de (dos x)| menos ( raíz cuadrada de (tres)) dividir por 2
- |sin(2x)|<(√(3))/2
- |sin2x|<sqrt3/2
- |sin(2x)|<(sqrt(3)) dividir por 2
-
Expresiones semejantes
- 1-4*sin(x)^2>sqrt(3)/2
- ((sqrt3)-2)(26-9x)<(sqrt3)/(2sqrt3+3)
- (1-4sin(x)^2)>sqrt(3)/2
- sin(3*x-pi/5)<=sqrt(3)/2
-
Expresiones con funciones
- Seno sin
- sin(x)<sort/2
- sin(x)^2=>1/4
- sin(x)>-1/3
- sin(x)>-5
- sin(x)>-sqrt(10)/3
- Raíz cuadrada sqrt
- sqrt(5-13x)*(x^2-x+6)<0
- sqrt(3+x)=<4
- sqrt(3x^2+1)<x-1
- sqrt3tan(x+pi*1/3)<=0
- sqrt(3*x^2-4*x+1)-1-3*x^2+4*x>=0
- Módulo |
- |x-3|-|x+4|>2
- |x-3|+|x+1|>=5
- |x+3|+|x-2|>5
- |x-4|>4x
- |x-4|<2
|sin(2x)|<(sqrt(3))/2 desigualdades
Solución
Ha introducido
[src]
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|sin(2*x)| < -----
2
$$\left|{\sin{\left(2 x \right)}}\right| < \frac{\sqrt{3}}{2}$$
Abs(sin(2*x)) < sqrt(3)/2
Solución detallada
Se da la desigualdad:
$$\left|{\sin{\left(2 x \right)}}\right| < \frac{\sqrt{3}}{2}$$
Para resolver esta desigualdad primero hay que resolver la ecuación correspondiente:
$$\left|{\sin{\left(2 x \right)}}\right| = \frac{\sqrt{3}}{2}$$
Resolvemos:
Tenemos la ecuación
$$\left|{\sin{\left(2 x \right)}}\right| = \frac{\sqrt{3}}{2}$$
cambiamos
$$\left|{\sin{\left(2 x \right)}}\right| - \frac{\sqrt{3}}{2} = 0$$
$$\left|{\sin{\left(2 x \right)}}\right| - \frac{\sqrt{3}}{2} = 0$$
Sustituimos
$$w = \left|{\sin{\left(2 x \right)}}\right|$$
Abrimos los paréntesis en el miembro izquierdo de la ecuación
Dividamos ambos miembros de la ecuación en (w - sqrt(3)/2)/w
Obtenemos la respuesta: w = sqrt(3)/2
hacemos cambio inverso
$$\left|{\sin{\left(2 x \right)}}\right| = w$$
sustituimos w:
$$x_{1} = -39.7935069454707$$
$$x_{2} = 22.5147473507269$$
$$x_{3} = -25.6563400043166$$
$$x_{4} = 70.162235930172$$
$$x_{5} = -60.2138591938044$$
$$x_{6} = 4.18879020478639$$
$$x_{7} = 84.2994028713261$$
$$x_{8} = 60.2138591938044$$
$$x_{9} = 96.342174710087$$
$$x_{10} = 31.9395253114962$$
$$x_{11} = 53.9306738866248$$
$$x_{12} = 24.0855436775217$$
$$x_{13} = -35.6047167406843$$
$$x_{14} = 18.3259571459405$$
$$x_{15} = 44.5058959258554$$
$$x_{16} = 26.1799387799149$$
$$x_{17} = -8.90117918517108$$
$$x_{18} = 63.8790506229925$$
$$x_{19} = 5.75958653158129$$
$$x_{20} = 9.94837673636768$$
$$x_{21} = 49.7418836818384$$
$$x_{22} = 77.4926187885482$$
$$x_{23} = -57.5958653158129$$
$$x_{24} = -49.7418836818384$$
$$x_{25} = 41.8879020478639$$
$$x_{26} = -65.4498469497874$$
$$x_{27} = 27.7507351067098$$
$$x_{28} = -61.7846555205993$$
$$x_{29} = -31.9395253114962$$
$$x_{30} = -97.9129710368819$$
$$x_{31} = -43.4586983746588$$
$$x_{32} = 52.3598775598299$$
$$x_{33} = -82.2050077689329$$
$$x_{34} = 93.7241808320955$$
$$x_{35} = 88.4881930761125$$
$$x_{36} = -24.0855436775217$$
$$x_{37} = 66.497044500984$$
$$x_{38} = -74.8746249105567$$
$$x_{39} = -78.0162175641465$$
$$x_{40} = -9.94837673636768$$
$$x_{41} = 16.2315620435473$$
$$x_{42} = -5.75958653158129$$
$$x_{43} = -100.007366139275$$
$$x_{44} = -41.8879020478639$$
$$x_{45} = -87.4409955249159$$
$$x_{46} = 46.0766922526503$$
$$x_{47} = 85.870199198121$$
$$x_{48} = -93.7241808320955$$
$$x_{49} = 19.8967534727354$$
$$x_{50} = -46.0766922526503$$
$$x_{51} = -52.8834763354282$$
$$x_{52} = 78.0162175641465$$
$$x_{53} = -21.4675497995303$$
$$x_{54} = 100.007366139275$$
$$x_{55} = 82.2050077689329$$
$$x_{56} = -83.7758040957278$$
$$x_{57} = -13.6135681655558$$
$$x_{58} = -75.9218224617533$$
$$x_{59} = -53.9306738866248$$
$$x_{60} = -27.7507351067098$$
$$x_{61} = -19.8967534727354$$
$$x_{62} = 0.523598775598299$$
$$x_{63} = -16.2315620435473$$
$$x_{64} = -56.025068989018$$
$$x_{65} = -85.870199198121$$
$$x_{66} = -63.8790506229925$$
$$x_{67} = 30.3687289847013$$
$$x_{68} = 92.1533845053006$$
$$x_{69} = -34.0339204138894$$
$$x_{70} = 74.3510261349584$$
$$x_{71} = 48.1710873550435$$
$$x_{72} = 8.37758040957278$$
$$x_{73} = 75.9218224617533$$
$$x_{74} = 97.9129710368819$$
$$x_{75} = 12.0427718387609$$
$$x_{76} = -2.0943951023932$$
$$x_{77} = -90.0589894029074$$
$$x_{78} = 62.3082542961976$$
$$x_{79} = 71.733032256967$$
$$x_{80} = 90.0589894029074$$
$$x_{81} = -12.0427718387609$$
$$x_{82} = -110.479341651241$$
$$x_{83} = -68.0678408277789$$
$$x_{84} = -79.5870138909414$$
$$x_{85} = -69.6386371545737$$
$$x_{86} = -71.733032256967$$
$$x_{87} = 38.2227106186758$$
$$x_{88} = 40.317105721069$$
$$x_{89} = 56.025068989018$$
$$x_{90} = -47.6474885794452$$
$$x_{91} = -3.66519142918809$$
$$x_{92} = 34.0339204138894$$
$$x_{93} = -17.8023583703422$$
$$x_{94} = 68.0678408277789$$
$$x_{95} = -96.342174710087$$
$$x_{96} = 2.0943951023932$$
$$x_{97} = -38.2227106186758$$
$$x_{1} = -39.7935069454707$$
$$x_{2} = 22.5147473507269$$
$$x_{3} = -25.6563400043166$$
$$x_{4} = 70.162235930172$$
$$x_{5} = -60.2138591938044$$
$$x_{6} = 4.18879020478639$$
$$x_{7} = 84.2994028713261$$
$$x_{8} = 60.2138591938044$$
$$x_{9} = 96.342174710087$$
$$x_{10} = 31.9395253114962$$
$$x_{11} = 53.9306738866248$$
$$x_{12} = 24.0855436775217$$
$$x_{13} = -35.6047167406843$$
$$x_{14} = 18.3259571459405$$
$$x_{15} = 44.5058959258554$$
$$x_{16} = 26.1799387799149$$
$$x_{17} = -8.90117918517108$$
$$x_{18} = 63.8790506229925$$
$$x_{19} = 5.75958653158129$$
$$x_{20} = 9.94837673636768$$
$$x_{21} = 49.7418836818384$$
$$x_{22} = 77.4926187885482$$
$$x_{23} = -57.5958653158129$$
$$x_{24} = -49.7418836818384$$
$$x_{25} = 41.8879020478639$$
$$x_{26} = -65.4498469497874$$
$$x_{27} = 27.7507351067098$$
$$x_{28} = -61.7846555205993$$
$$x_{29} = -31.9395253114962$$
$$x_{30} = -97.9129710368819$$
$$x_{31} = -43.4586983746588$$
$$x_{32} = 52.3598775598299$$
$$x_{33} = -82.2050077689329$$
$$x_{34} = 93.7241808320955$$
$$x_{35} = 88.4881930761125$$
$$x_{36} = -24.0855436775217$$
$$x_{37} = 66.497044500984$$
$$x_{38} = -74.8746249105567$$
$$x_{39} = -78.0162175641465$$
$$x_{40} = -9.94837673636768$$
$$x_{41} = 16.2315620435473$$
$$x_{42} = -5.75958653158129$$
$$x_{43} = -100.007366139275$$
$$x_{44} = -41.8879020478639$$
$$x_{45} = -87.4409955249159$$
$$x_{46} = 46.0766922526503$$
$$x_{47} = 85.870199198121$$
$$x_{48} = -93.7241808320955$$
$$x_{49} = 19.8967534727354$$
$$x_{50} = -46.0766922526503$$
$$x_{51} = -52.8834763354282$$
$$x_{52} = 78.0162175641465$$
$$x_{53} = -21.4675497995303$$
$$x_{54} = 100.007366139275$$
$$x_{55} = 82.2050077689329$$
$$x_{56} = -83.7758040957278$$
$$x_{57} = -13.6135681655558$$
$$x_{58} = -75.9218224617533$$
$$x_{59} = -53.9306738866248$$
$$x_{60} = -27.7507351067098$$
$$x_{61} = -19.8967534727354$$
$$x_{62} = 0.523598775598299$$
$$x_{63} = -16.2315620435473$$
$$x_{64} = -56.025068989018$$
$$x_{65} = -85.870199198121$$
$$x_{66} = -63.8790506229925$$
$$x_{67} = 30.3687289847013$$
$$x_{68} = 92.1533845053006$$
$$x_{69} = -34.0339204138894$$
$$x_{70} = 74.3510261349584$$
$$x_{71} = 48.1710873550435$$
$$x_{72} = 8.37758040957278$$
$$x_{73} = 75.9218224617533$$
$$x_{74} = 97.9129710368819$$
$$x_{75} = 12.0427718387609$$
$$x_{76} = -2.0943951023932$$
$$x_{77} = -90.0589894029074$$
$$x_{78} = 62.3082542961976$$
$$x_{79} = 71.733032256967$$
$$x_{80} = 90.0589894029074$$
$$x_{81} = -12.0427718387609$$
$$x_{82} = -110.479341651241$$
$$x_{83} = -68.0678408277789$$
$$x_{84} = -79.5870138909414$$
$$x_{85} = -69.6386371545737$$
$$x_{86} = -71.733032256967$$
$$x_{87} = 38.2227106186758$$
$$x_{88} = 40.317105721069$$
$$x_{89} = 56.025068989018$$
$$x_{90} = -47.6474885794452$$
$$x_{91} = -3.66519142918809$$
$$x_{92} = 34.0339204138894$$
$$x_{93} = -17.8023583703422$$
$$x_{94} = 68.0678408277789$$
$$x_{95} = -96.342174710087$$
$$x_{96} = 2.0943951023932$$
$$x_{97} = -38.2227106186758$$
Las raíces dadas
$$x_{82} = -110.479341651241$$
$$x_{43} = -100.007366139275$$
$$x_{30} = -97.9129710368819$$
$$x_{95} = -96.342174710087$$
$$x_{48} = -93.7241808320955$$
$$x_{77} = -90.0589894029074$$
$$x_{45} = -87.4409955249159$$
$$x_{65} = -85.870199198121$$
$$x_{56} = -83.7758040957278$$
$$x_{33} = -82.2050077689329$$
$$x_{84} = -79.5870138909414$$
$$x_{39} = -78.0162175641465$$
$$x_{58} = -75.9218224617533$$
$$x_{38} = -74.8746249105567$$
$$x_{86} = -71.733032256967$$
$$x_{85} = -69.6386371545737$$
$$x_{83} = -68.0678408277789$$
$$x_{26} = -65.4498469497874$$
$$x_{66} = -63.8790506229925$$
$$x_{28} = -61.7846555205993$$
$$x_{5} = -60.2138591938044$$
$$x_{23} = -57.5958653158129$$
$$x_{64} = -56.025068989018$$
$$x_{59} = -53.9306738866248$$
$$x_{51} = -52.8834763354282$$
$$x_{24} = -49.7418836818384$$
$$x_{90} = -47.6474885794452$$
$$x_{50} = -46.0766922526503$$
$$x_{31} = -43.4586983746588$$
$$x_{44} = -41.8879020478639$$
$$x_{1} = -39.7935069454707$$
$$x_{97} = -38.2227106186758$$
$$x_{13} = -35.6047167406843$$
$$x_{69} = -34.0339204138894$$
$$x_{29} = -31.9395253114962$$
$$x_{60} = -27.7507351067098$$
$$x_{3} = -25.6563400043166$$
$$x_{36} = -24.0855436775217$$
$$x_{53} = -21.4675497995303$$
$$x_{61} = -19.8967534727354$$
$$x_{93} = -17.8023583703422$$
$$x_{63} = -16.2315620435473$$
$$x_{57} = -13.6135681655558$$
$$x_{81} = -12.0427718387609$$
$$x_{40} = -9.94837673636768$$
$$x_{17} = -8.90117918517108$$
$$x_{42} = -5.75958653158129$$
$$x_{91} = -3.66519142918809$$
$$x_{76} = -2.0943951023932$$
$$x_{62} = 0.523598775598299$$
$$x_{96} = 2.0943951023932$$
$$x_{6} = 4.18879020478639$$
$$x_{19} = 5.75958653158129$$
$$x_{72} = 8.37758040957278$$
$$x_{20} = 9.94837673636768$$
$$x_{75} = 12.0427718387609$$
$$x_{41} = 16.2315620435473$$
$$x_{14} = 18.3259571459405$$
$$x_{49} = 19.8967534727354$$
$$x_{2} = 22.5147473507269$$
$$x_{12} = 24.0855436775217$$
$$x_{16} = 26.1799387799149$$
$$x_{27} = 27.7507351067098$$
$$x_{67} = 30.3687289847013$$
$$x_{10} = 31.9395253114962$$
$$x_{92} = 34.0339204138894$$
$$x_{87} = 38.2227106186758$$
$$x_{88} = 40.317105721069$$
$$x_{25} = 41.8879020478639$$
$$x_{15} = 44.5058959258554$$
$$x_{46} = 46.0766922526503$$
$$x_{71} = 48.1710873550435$$
$$x_{21} = 49.7418836818384$$
$$x_{32} = 52.3598775598299$$
$$x_{11} = 53.9306738866248$$
$$x_{89} = 56.025068989018$$
$$x_{8} = 60.2138591938044$$
$$x_{78} = 62.3082542961976$$
$$x_{18} = 63.8790506229925$$
$$x_{37} = 66.497044500984$$
$$x_{94} = 68.0678408277789$$
$$x_{4} = 70.162235930172$$
$$x_{79} = 71.733032256967$$
$$x_{70} = 74.3510261349584$$
$$x_{73} = 75.9218224617533$$
$$x_{22} = 77.4926187885482$$
$$x_{52} = 78.0162175641465$$
$$x_{55} = 82.2050077689329$$
$$x_{7} = 84.2994028713261$$
$$x_{47} = 85.870199198121$$
$$x_{35} = 88.4881930761125$$
$$x_{80} = 90.0589894029074$$
$$x_{68} = 92.1533845053006$$
$$x_{34} = 93.7241808320955$$
$$x_{9} = 96.342174710087$$
$$x_{74} = 97.9129710368819$$
$$x_{54} = 100.007366139275$$
son puntos de cambio del signo de desigualdad en las soluciones.
Primero definámonos con el signo hasta el punto extremo izquierdo:
$$x_{0} < x_{82}$$
Consideremos, por ejemplo, el punto
$$x_{0} = x_{82} - \frac{1}{10}$$
=
$$-110.479341651241 + - \frac{1}{10}$$
=
$$-110.579341651241$$
lo sustituimos en la expresión
$$\left|{\sin{\left(2 x \right)}}\right| < \frac{\sqrt{3}}{2}$$
$$\left|{\sin{\left(\left(-110.579341651241\right) 2 \right)}}\right| < \frac{\sqrt{3}}{2}$$
pero
Entonces
$$x < -110.479341651241$$
no se cumple
significa que una de las soluciones de nuestra ecuación será con:
$$x > -110.479341651241 \wedge x < -100.007366139275$$
Recibiremos otras soluciones de la desigualdad pasando al polo siguiente etc.
etc.
Respuesta:
$$x > -110.479341651241 \wedge x < -100.007366139275$$
$$x > -97.9129710368819 \wedge x < -96.342174710087$$
$$x > -93.7241808320955 \wedge x < -90.0589894029074$$
$$x > -87.4409955249159 \wedge x < -85.870199198121$$
$$x > -83.7758040957278 \wedge x < -82.2050077689329$$
$$x > -79.5870138909414 \wedge x < -78.0162175641465$$
$$x > -75.9218224617533 \wedge x < -74.8746249105567$$
$$x > -71.733032256967 \wedge x < -69.6386371545737$$
$$x > -68.0678408277789 \wedge x < -65.4498469497874$$
$$x > -63.8790506229925 \wedge x < -61.7846555205993$$
$$x > -60.2138591938044 \wedge x < -57.5958653158129$$
$$x > -56.025068989018 \wedge x < -53.9306738866248$$
$$x > -52.8834763354282 \wedge x < -49.7418836818384$$
$$x > -47.6474885794452 \wedge x < -46.0766922526503$$
$$x > -43.4586983746588 \wedge x < -41.8879020478639$$
$$x > -39.7935069454707 \wedge x < -38.2227106186758$$
$$x > -35.6047167406843 \wedge x < -34.0339204138894$$
$$x > -31.9395253114962 \wedge x < -27.7507351067098$$
$$x > -25.6563400043166 \wedge x < -24.0855436775217$$
$$x > -21.4675497995303 \wedge x < -19.8967534727354$$
$$x > -17.8023583703422 \wedge x < -16.2315620435473$$
$$x > -13.6135681655558 \wedge x < -12.0427718387609$$
$$x > -9.94837673636768 \wedge x < -8.90117918517108$$
$$x > -5.75958653158129 \wedge x < -3.66519142918809$$
$$x > -2.0943951023932 \wedge x < 0.523598775598299$$
$$x > 2.0943951023932 \wedge x < 4.18879020478639$$
$$x > 5.75958653158129 \wedge x < 8.37758040957278$$
$$x > 9.94837673636768 \wedge x < 12.0427718387609$$
$$x > 16.2315620435473 \wedge x < 18.3259571459405$$
$$x > 19.8967534727354 \wedge x < 22.5147473507269$$
$$x > 24.0855436775217 \wedge x < 26.1799387799149$$
$$x > 27.7507351067098 \wedge x < 30.3687289847013$$
$$x > 31.9395253114962 \wedge x < 34.0339204138894$$
$$x > 38.2227106186758 \wedge x < 40.317105721069$$
$$x > 41.8879020478639 \wedge x < 44.5058959258554$$
$$x > 46.0766922526503 \wedge x < 48.1710873550435$$
$$x > 49.7418836818384 \wedge x < 52.3598775598299$$
$$x > 53.9306738866248 \wedge x < 56.025068989018$$
$$x > 60.2138591938044 \wedge x < 62.3082542961976$$
$$x > 63.8790506229925 \wedge x < 66.497044500984$$
$$x > 68.0678408277789 \wedge x < 70.162235930172$$
$$x > 71.733032256967 \wedge x < 74.3510261349584$$
$$x > 75.9218224617533 \wedge x < 77.4926187885482$$
$$x > 78.0162175641465 \wedge x < 82.2050077689329$$
$$x > 84.2994028713261 \wedge x < 85.870199198121$$
$$x > 88.4881930761125 \wedge x < 90.0589894029074$$
$$x > 92.1533845053006 \wedge x < 93.7241808320955$$
$$x > 96.342174710087 \wedge x < 97.9129710368819$$
$$x > 100.007366139275$$
$$\left|{\sin{\left(2 x \right)}}\right| < \frac{\sqrt{3}}{2}$$
Para resolver esta desigualdad primero hay que resolver la ecuación correspondiente:
$$\left|{\sin{\left(2 x \right)}}\right| = \frac{\sqrt{3}}{2}$$
Resolvemos:
Tenemos la ecuación
$$\left|{\sin{\left(2 x \right)}}\right| = \frac{\sqrt{3}}{2}$$
cambiamos
$$\left|{\sin{\left(2 x \right)}}\right| - \frac{\sqrt{3}}{2} = 0$$
$$\left|{\sin{\left(2 x \right)}}\right| - \frac{\sqrt{3}}{2} = 0$$
Sustituimos
$$w = \left|{\sin{\left(2 x \right)}}\right|$$
Abrimos los paréntesis en el miembro izquierdo de la ecuación
w - sqrt3/2 = 0
Dividamos ambos miembros de la ecuación en (w - sqrt(3)/2)/w
w = 0 / ((w - sqrt(3)/2)/w)
Obtenemos la respuesta: w = sqrt(3)/2
hacemos cambio inverso
$$\left|{\sin{\left(2 x \right)}}\right| = w$$
sustituimos w:
$$x_{1} = -39.7935069454707$$
$$x_{2} = 22.5147473507269$$
$$x_{3} = -25.6563400043166$$
$$x_{4} = 70.162235930172$$
$$x_{5} = -60.2138591938044$$
$$x_{6} = 4.18879020478639$$
$$x_{7} = 84.2994028713261$$
$$x_{8} = 60.2138591938044$$
$$x_{9} = 96.342174710087$$
$$x_{10} = 31.9395253114962$$
$$x_{11} = 53.9306738866248$$
$$x_{12} = 24.0855436775217$$
$$x_{13} = -35.6047167406843$$
$$x_{14} = 18.3259571459405$$
$$x_{15} = 44.5058959258554$$
$$x_{16} = 26.1799387799149$$
$$x_{17} = -8.90117918517108$$
$$x_{18} = 63.8790506229925$$
$$x_{19} = 5.75958653158129$$
$$x_{20} = 9.94837673636768$$
$$x_{21} = 49.7418836818384$$
$$x_{22} = 77.4926187885482$$
$$x_{23} = -57.5958653158129$$
$$x_{24} = -49.7418836818384$$
$$x_{25} = 41.8879020478639$$
$$x_{26} = -65.4498469497874$$
$$x_{27} = 27.7507351067098$$
$$x_{28} = -61.7846555205993$$
$$x_{29} = -31.9395253114962$$
$$x_{30} = -97.9129710368819$$
$$x_{31} = -43.4586983746588$$
$$x_{32} = 52.3598775598299$$
$$x_{33} = -82.2050077689329$$
$$x_{34} = 93.7241808320955$$
$$x_{35} = 88.4881930761125$$
$$x_{36} = -24.0855436775217$$
$$x_{37} = 66.497044500984$$
$$x_{38} = -74.8746249105567$$
$$x_{39} = -78.0162175641465$$
$$x_{40} = -9.94837673636768$$
$$x_{41} = 16.2315620435473$$
$$x_{42} = -5.75958653158129$$
$$x_{43} = -100.007366139275$$
$$x_{44} = -41.8879020478639$$
$$x_{45} = -87.4409955249159$$
$$x_{46} = 46.0766922526503$$
$$x_{47} = 85.870199198121$$
$$x_{48} = -93.7241808320955$$
$$x_{49} = 19.8967534727354$$
$$x_{50} = -46.0766922526503$$
$$x_{51} = -52.8834763354282$$
$$x_{52} = 78.0162175641465$$
$$x_{53} = -21.4675497995303$$
$$x_{54} = 100.007366139275$$
$$x_{55} = 82.2050077689329$$
$$x_{56} = -83.7758040957278$$
$$x_{57} = -13.6135681655558$$
$$x_{58} = -75.9218224617533$$
$$x_{59} = -53.9306738866248$$
$$x_{60} = -27.7507351067098$$
$$x_{61} = -19.8967534727354$$
$$x_{62} = 0.523598775598299$$
$$x_{63} = -16.2315620435473$$
$$x_{64} = -56.025068989018$$
$$x_{65} = -85.870199198121$$
$$x_{66} = -63.8790506229925$$
$$x_{67} = 30.3687289847013$$
$$x_{68} = 92.1533845053006$$
$$x_{69} = -34.0339204138894$$
$$x_{70} = 74.3510261349584$$
$$x_{71} = 48.1710873550435$$
$$x_{72} = 8.37758040957278$$
$$x_{73} = 75.9218224617533$$
$$x_{74} = 97.9129710368819$$
$$x_{75} = 12.0427718387609$$
$$x_{76} = -2.0943951023932$$
$$x_{77} = -90.0589894029074$$
$$x_{78} = 62.3082542961976$$
$$x_{79} = 71.733032256967$$
$$x_{80} = 90.0589894029074$$
$$x_{81} = -12.0427718387609$$
$$x_{82} = -110.479341651241$$
$$x_{83} = -68.0678408277789$$
$$x_{84} = -79.5870138909414$$
$$x_{85} = -69.6386371545737$$
$$x_{86} = -71.733032256967$$
$$x_{87} = 38.2227106186758$$
$$x_{88} = 40.317105721069$$
$$x_{89} = 56.025068989018$$
$$x_{90} = -47.6474885794452$$
$$x_{91} = -3.66519142918809$$
$$x_{92} = 34.0339204138894$$
$$x_{93} = -17.8023583703422$$
$$x_{94} = 68.0678408277789$$
$$x_{95} = -96.342174710087$$
$$x_{96} = 2.0943951023932$$
$$x_{97} = -38.2227106186758$$
$$x_{1} = -39.7935069454707$$
$$x_{2} = 22.5147473507269$$
$$x_{3} = -25.6563400043166$$
$$x_{4} = 70.162235930172$$
$$x_{5} = -60.2138591938044$$
$$x_{6} = 4.18879020478639$$
$$x_{7} = 84.2994028713261$$
$$x_{8} = 60.2138591938044$$
$$x_{9} = 96.342174710087$$
$$x_{10} = 31.9395253114962$$
$$x_{11} = 53.9306738866248$$
$$x_{12} = 24.0855436775217$$
$$x_{13} = -35.6047167406843$$
$$x_{14} = 18.3259571459405$$
$$x_{15} = 44.5058959258554$$
$$x_{16} = 26.1799387799149$$
$$x_{17} = -8.90117918517108$$
$$x_{18} = 63.8790506229925$$
$$x_{19} = 5.75958653158129$$
$$x_{20} = 9.94837673636768$$
$$x_{21} = 49.7418836818384$$
$$x_{22} = 77.4926187885482$$
$$x_{23} = -57.5958653158129$$
$$x_{24} = -49.7418836818384$$
$$x_{25} = 41.8879020478639$$
$$x_{26} = -65.4498469497874$$
$$x_{27} = 27.7507351067098$$
$$x_{28} = -61.7846555205993$$
$$x_{29} = -31.9395253114962$$
$$x_{30} = -97.9129710368819$$
$$x_{31} = -43.4586983746588$$
$$x_{32} = 52.3598775598299$$
$$x_{33} = -82.2050077689329$$
$$x_{34} = 93.7241808320955$$
$$x_{35} = 88.4881930761125$$
$$x_{36} = -24.0855436775217$$
$$x_{37} = 66.497044500984$$
$$x_{38} = -74.8746249105567$$
$$x_{39} = -78.0162175641465$$
$$x_{40} = -9.94837673636768$$
$$x_{41} = 16.2315620435473$$
$$x_{42} = -5.75958653158129$$
$$x_{43} = -100.007366139275$$
$$x_{44} = -41.8879020478639$$
$$x_{45} = -87.4409955249159$$
$$x_{46} = 46.0766922526503$$
$$x_{47} = 85.870199198121$$
$$x_{48} = -93.7241808320955$$
$$x_{49} = 19.8967534727354$$
$$x_{50} = -46.0766922526503$$
$$x_{51} = -52.8834763354282$$
$$x_{52} = 78.0162175641465$$
$$x_{53} = -21.4675497995303$$
$$x_{54} = 100.007366139275$$
$$x_{55} = 82.2050077689329$$
$$x_{56} = -83.7758040957278$$
$$x_{57} = -13.6135681655558$$
$$x_{58} = -75.9218224617533$$
$$x_{59} = -53.9306738866248$$
$$x_{60} = -27.7507351067098$$
$$x_{61} = -19.8967534727354$$
$$x_{62} = 0.523598775598299$$
$$x_{63} = -16.2315620435473$$
$$x_{64} = -56.025068989018$$
$$x_{65} = -85.870199198121$$
$$x_{66} = -63.8790506229925$$
$$x_{67} = 30.3687289847013$$
$$x_{68} = 92.1533845053006$$
$$x_{69} = -34.0339204138894$$
$$x_{70} = 74.3510261349584$$
$$x_{71} = 48.1710873550435$$
$$x_{72} = 8.37758040957278$$
$$x_{73} = 75.9218224617533$$
$$x_{74} = 97.9129710368819$$
$$x_{75} = 12.0427718387609$$
$$x_{76} = -2.0943951023932$$
$$x_{77} = -90.0589894029074$$
$$x_{78} = 62.3082542961976$$
$$x_{79} = 71.733032256967$$
$$x_{80} = 90.0589894029074$$
$$x_{81} = -12.0427718387609$$
$$x_{82} = -110.479341651241$$
$$x_{83} = -68.0678408277789$$
$$x_{84} = -79.5870138909414$$
$$x_{85} = -69.6386371545737$$
$$x_{86} = -71.733032256967$$
$$x_{87} = 38.2227106186758$$
$$x_{88} = 40.317105721069$$
$$x_{89} = 56.025068989018$$
$$x_{90} = -47.6474885794452$$
$$x_{91} = -3.66519142918809$$
$$x_{92} = 34.0339204138894$$
$$x_{93} = -17.8023583703422$$
$$x_{94} = 68.0678408277789$$
$$x_{95} = -96.342174710087$$
$$x_{96} = 2.0943951023932$$
$$x_{97} = -38.2227106186758$$
Las raíces dadas
$$x_{82} = -110.479341651241$$
$$x_{43} = -100.007366139275$$
$$x_{30} = -97.9129710368819$$
$$x_{95} = -96.342174710087$$
$$x_{48} = -93.7241808320955$$
$$x_{77} = -90.0589894029074$$
$$x_{45} = -87.4409955249159$$
$$x_{65} = -85.870199198121$$
$$x_{56} = -83.7758040957278$$
$$x_{33} = -82.2050077689329$$
$$x_{84} = -79.5870138909414$$
$$x_{39} = -78.0162175641465$$
$$x_{58} = -75.9218224617533$$
$$x_{38} = -74.8746249105567$$
$$x_{86} = -71.733032256967$$
$$x_{85} = -69.6386371545737$$
$$x_{83} = -68.0678408277789$$
$$x_{26} = -65.4498469497874$$
$$x_{66} = -63.8790506229925$$
$$x_{28} = -61.7846555205993$$
$$x_{5} = -60.2138591938044$$
$$x_{23} = -57.5958653158129$$
$$x_{64} = -56.025068989018$$
$$x_{59} = -53.9306738866248$$
$$x_{51} = -52.8834763354282$$
$$x_{24} = -49.7418836818384$$
$$x_{90} = -47.6474885794452$$
$$x_{50} = -46.0766922526503$$
$$x_{31} = -43.4586983746588$$
$$x_{44} = -41.8879020478639$$
$$x_{1} = -39.7935069454707$$
$$x_{97} = -38.2227106186758$$
$$x_{13} = -35.6047167406843$$
$$x_{69} = -34.0339204138894$$
$$x_{29} = -31.9395253114962$$
$$x_{60} = -27.7507351067098$$
$$x_{3} = -25.6563400043166$$
$$x_{36} = -24.0855436775217$$
$$x_{53} = -21.4675497995303$$
$$x_{61} = -19.8967534727354$$
$$x_{93} = -17.8023583703422$$
$$x_{63} = -16.2315620435473$$
$$x_{57} = -13.6135681655558$$
$$x_{81} = -12.0427718387609$$
$$x_{40} = -9.94837673636768$$
$$x_{17} = -8.90117918517108$$
$$x_{42} = -5.75958653158129$$
$$x_{91} = -3.66519142918809$$
$$x_{76} = -2.0943951023932$$
$$x_{62} = 0.523598775598299$$
$$x_{96} = 2.0943951023932$$
$$x_{6} = 4.18879020478639$$
$$x_{19} = 5.75958653158129$$
$$x_{72} = 8.37758040957278$$
$$x_{20} = 9.94837673636768$$
$$x_{75} = 12.0427718387609$$
$$x_{41} = 16.2315620435473$$
$$x_{14} = 18.3259571459405$$
$$x_{49} = 19.8967534727354$$
$$x_{2} = 22.5147473507269$$
$$x_{12} = 24.0855436775217$$
$$x_{16} = 26.1799387799149$$
$$x_{27} = 27.7507351067098$$
$$x_{67} = 30.3687289847013$$
$$x_{10} = 31.9395253114962$$
$$x_{92} = 34.0339204138894$$
$$x_{87} = 38.2227106186758$$
$$x_{88} = 40.317105721069$$
$$x_{25} = 41.8879020478639$$
$$x_{15} = 44.5058959258554$$
$$x_{46} = 46.0766922526503$$
$$x_{71} = 48.1710873550435$$
$$x_{21} = 49.7418836818384$$
$$x_{32} = 52.3598775598299$$
$$x_{11} = 53.9306738866248$$
$$x_{89} = 56.025068989018$$
$$x_{8} = 60.2138591938044$$
$$x_{78} = 62.3082542961976$$
$$x_{18} = 63.8790506229925$$
$$x_{37} = 66.497044500984$$
$$x_{94} = 68.0678408277789$$
$$x_{4} = 70.162235930172$$
$$x_{79} = 71.733032256967$$
$$x_{70} = 74.3510261349584$$
$$x_{73} = 75.9218224617533$$
$$x_{22} = 77.4926187885482$$
$$x_{52} = 78.0162175641465$$
$$x_{55} = 82.2050077689329$$
$$x_{7} = 84.2994028713261$$
$$x_{47} = 85.870199198121$$
$$x_{35} = 88.4881930761125$$
$$x_{80} = 90.0589894029074$$
$$x_{68} = 92.1533845053006$$
$$x_{34} = 93.7241808320955$$
$$x_{9} = 96.342174710087$$
$$x_{74} = 97.9129710368819$$
$$x_{54} = 100.007366139275$$
son puntos de cambio del signo de desigualdad en las soluciones.
Primero definámonos con el signo hasta el punto extremo izquierdo:
$$x_{0} < x_{82}$$
Consideremos, por ejemplo, el punto
$$x_{0} = x_{82} - \frac{1}{10}$$
=
$$-110.479341651241 + - \frac{1}{10}$$
=
$$-110.579341651241$$
lo sustituimos en la expresión
$$\left|{\sin{\left(2 x \right)}}\right| < \frac{\sqrt{3}}{2}$$
$$\left|{\sin{\left(\left(-110.579341651241\right) 2 \right)}}\right| < \frac{\sqrt{3}}{2}$$
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\/ 3
0.948097219208118 < -----
2
pero
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\/ 3
0.948097219208118 > -----
2
Entonces
$$x < -110.479341651241$$
no se cumple
significa que una de las soluciones de nuestra ecuación será con:
$$x > -110.479341651241 \wedge x < -100.007366139275$$
_____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____
/ \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
-------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-------ο-------
x82 x43 x30 x95 x48 x77 x45 x65 x56 x33 x84 x39 x58 x38 x86 x85 x83 x26 x66 x28 x5 x23 x64 x59 x51 x24 x90 x50 x31 x44 x1 x97 x13 x69 x29 x60 x3 x36 x53 x61 x93 x63 x57 x81 x40 x17 x42 x91 x76 x62 x96 x6 x19 x72 x20 x75 x41 x14 x49 x2 x12 x16 x27 x67 x10 x92 x87 x88 x25 x15 x46 x71 x21 x32 x11 x89 x8 x78 x18 x37 x94 x4 x79 x70 x73 x22 x52 x55 x7 x47 x35 x80 x68 x34 x9 x74 x54Recibiremos otras soluciones de la desigualdad pasando al polo siguiente etc.
etc.
Respuesta:
$$x > -110.479341651241 \wedge x < -100.007366139275$$
$$x > -97.9129710368819 \wedge x < -96.342174710087$$
$$x > -93.7241808320955 \wedge x < -90.0589894029074$$
$$x > -87.4409955249159 \wedge x < -85.870199198121$$
$$x > -83.7758040957278 \wedge x < -82.2050077689329$$
$$x > -79.5870138909414 \wedge x < -78.0162175641465$$
$$x > -75.9218224617533 \wedge x < -74.8746249105567$$
$$x > -71.733032256967 \wedge x < -69.6386371545737$$
$$x > -68.0678408277789 \wedge x < -65.4498469497874$$
$$x > -63.8790506229925 \wedge x < -61.7846555205993$$
$$x > -60.2138591938044 \wedge x < -57.5958653158129$$
$$x > -56.025068989018 \wedge x < -53.9306738866248$$
$$x > -52.8834763354282 \wedge x < -49.7418836818384$$
$$x > -47.6474885794452 \wedge x < -46.0766922526503$$
$$x > -43.4586983746588 \wedge x < -41.8879020478639$$
$$x > -39.7935069454707 \wedge x < -38.2227106186758$$
$$x > -35.6047167406843 \wedge x < -34.0339204138894$$
$$x > -31.9395253114962 \wedge x < -27.7507351067098$$
$$x > -25.6563400043166 \wedge x < -24.0855436775217$$
$$x > -21.4675497995303 \wedge x < -19.8967534727354$$
$$x > -17.8023583703422 \wedge x < -16.2315620435473$$
$$x > -13.6135681655558 \wedge x < -12.0427718387609$$
$$x > -9.94837673636768 \wedge x < -8.90117918517108$$
$$x > -5.75958653158129 \wedge x < -3.66519142918809$$
$$x > -2.0943951023932 \wedge x < 0.523598775598299$$
$$x > 2.0943951023932 \wedge x < 4.18879020478639$$
$$x > 5.75958653158129 \wedge x < 8.37758040957278$$
$$x > 9.94837673636768 \wedge x < 12.0427718387609$$
$$x > 16.2315620435473 \wedge x < 18.3259571459405$$
$$x > 19.8967534727354 \wedge x < 22.5147473507269$$
$$x > 24.0855436775217 \wedge x < 26.1799387799149$$
$$x > 27.7507351067098 \wedge x < 30.3687289847013$$
$$x > 31.9395253114962 \wedge x < 34.0339204138894$$
$$x > 38.2227106186758 \wedge x < 40.317105721069$$
$$x > 41.8879020478639 \wedge x < 44.5058959258554$$
$$x > 46.0766922526503 \wedge x < 48.1710873550435$$
$$x > 49.7418836818384 \wedge x < 52.3598775598299$$
$$x > 53.9306738866248 \wedge x < 56.025068989018$$
$$x > 60.2138591938044 \wedge x < 62.3082542961976$$
$$x > 63.8790506229925 \wedge x < 66.497044500984$$
$$x > 68.0678408277789 \wedge x < 70.162235930172$$
$$x > 71.733032256967 \wedge x < 74.3510261349584$$
$$x > 75.9218224617533 \wedge x < 77.4926187885482$$
$$x > 78.0162175641465 \wedge x < 82.2050077689329$$
$$x > 84.2994028713261 \wedge x < 85.870199198121$$
$$x > 88.4881930761125 \wedge x < 90.0589894029074$$
$$x > 92.1533845053006 \wedge x < 93.7241808320955$$
$$x > 96.342174710087 \wedge x < 97.9129710368819$$
$$x > 100.007366139275$$
Solución de la desigualdad en el gráfico