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+1)*(x-9)>0
-
x^2+x-6>0
-
(x-1)/(x-4)>0
-
-16/(x+2)^2-5>=0
-
Expresiones idénticas
- abs(sin(3x))<=sqrt(dos)/ dos
- abs( seno de (3x)) menos o igual a raíz cuadrada de (2) dividir por 2
- abs( seno de (3x)) menos o igual a raíz cuadrada de (dos) dividir por dos
- abs(sin(3x))<=√(2)/2
- abssin3x<=sqrt2/2
- abs(sin(3x))<=sqrt(2) dividir por 2
-
Expresiones semejantes
- abs(sin(3*x-pi/10))<=1
-
Expresiones con funciones
- abs
- abs(x-3)*(abs(x-4)+abs(x+5)-abs(2x+1))<=0
- abs(1/(lg0,5/lg((3-x)^2)+2))*(x^2-16)<=0
- abs((2-x)/(3x+1))>1
- abs(2x+1)>1-2abs(x-1)
- abs(x^2-6x)>7
- Seno sin
- sin(x)>0
- sin(x)<sqrt(3)/2
- sin(x)<√3/2
- sinx>-1
- sin(x)>1/2
- Raíz cuadrada sqrt
- sqrt(2+x-x^2)>-1
- sqrt(4-x)>3
- sqrt(x)>=x^2
- sqrt(x^2-9)>4
- sqrt(x^2-x-2)>=2x+3
abs(sin(3x))<=sqrt(2)/2 desigualdades
Solución
Ha introducido
[src]
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|sin(3*x)| <= -----
2
$$\left|{\sin{\left(3 x \right)}}\right| \leq \frac{\sqrt{2}}{2}$$
Abs(sin(3*x)) <= sqrt(2)/2
Solución detallada
Se da la desigualdad:
$$\left|{\sin{\left(3 x \right)}}\right| \leq \frac{\sqrt{2}}{2}$$
Para resolver esta desigualdad primero hay que resolver la ecuación correspondiente:
$$\left|{\sin{\left(3 x \right)}}\right| = \frac{\sqrt{2}}{2}$$
Resolvemos:
Tenemos la ecuación
$$\left|{\sin{\left(3 x \right)}}\right| = \frac{\sqrt{2}}{2}$$
cambiamos
$$\left|{\sin{\left(3 x \right)}}\right| - \frac{\sqrt{2}}{2} = 0$$
$$\left|{\sin{\left(3 x \right)}}\right| - \frac{\sqrt{2}}{2} = 0$$
Sustituimos
$$w = \left|{\sin{\left(3 x \right)}}\right|$$
Abrimos los paréntesis en el miembro izquierdo de la ecuación
Dividamos ambos miembros de la ecuación en (w - sqrt(2)/2)/w
Obtenemos la respuesta: w = sqrt(2)/2
hacemos cambio inverso
$$\left|{\sin{\left(3 x \right)}}\right| = w$$
sustituimos w:
$$x_{1} = 56.2868683768171$$
$$x_{2} = -87.7027949127151$$
$$x_{3} = -71.9948316447661$$
$$x_{4} = 18.0641577581413$$
$$x_{5} = -6.02138591938044$$
$$x_{6} = 25.9181393921158$$
$$x_{7} = -65.7116463375865$$
$$x_{8} = 40.0553063332699$$
$$x_{9} = -47.9092879672443$$
$$x_{10} = -19.1113553093379$$
$$x_{11} = 44.2440965380563$$
$$x_{12} = -15.9697626557481$$
$$x_{13} = -43.720497762458$$
$$x_{14} = -62.0464549083984$$
$$x_{15} = -53.6688744988256$$
$$x_{16} = -3.92699081698724$$
$$x_{17} = 88.2263936883134$$
$$x_{18} = -84.037603483527$$
$$x_{19} = -31.6777259236971$$
$$x_{20} = -81.9432083811338$$
$$x_{21} = 74.0892267471593$$
$$x_{22} = -28.012534494509$$
$$x_{23} = 91.8915851175014$$
$$x_{24} = -13.8753675533549$$
$$x_{25} = -172.525796559639$$
$$x_{26} = 0.261799387799149$$
$$x_{27} = 32.2013246992954$$
$$x_{28} = 100.269165527074$$
$$x_{29} = -93.9859802198946$$
$$x_{30} = 3.92699081698724$$
$$x_{31} = 2.35619449019234$$
$$x_{32} = -75.6600230739542$$
$$x_{33} = 76.1836218495525$$
$$x_{34} = -59.9520598060052$$
$$x_{35} = -37.9609112308767$$
$$x_{36} = -9.68657734856853$$
$$x_{37} = 46.3384916404494$$
$$x_{38} = 64.1408500107916$$
$$x_{39} = 78.2780169519457$$
$$x_{40} = 47.9092879672443$$
$$x_{41} = 1738.08613559855$$
$$x_{42} = -91.8915851175014$$
$$x_{43} = 42.1497014356631$$
$$x_{44} = -50.0036830696375$$
$$x_{45} = 96.0803753222878$$
$$x_{46} = -33.7721210260903$$
$$x_{47} = 71.9948316447661$$
$$x_{48} = 86.1319985859202$$
$$x_{49} = -74.0892267471593$$
$$x_{50} = 66.2352451131848$$
$$x_{51} = 28.012534494509$$
$$x_{52} = 12.30457122656$$
$$x_{53} = 50.0036830696375$$
$$x_{54} = 84.037603483527$$
$$x_{55} = 136.397481043357$$
$$x_{56} = 22.2529479629277$$
$$x_{57} = -21.7293491873294$$
$$x_{58} = -52.0980781720307$$
$$x_{59} = 37.9609112308767$$
$$x_{60} = 68.329640215578$$
$$x_{61} = 6.02138591938044$$
$$x_{62} = 59.9520598060052$$
$$x_{63} = -23.8237442897226$$
$$x_{64} = -11.7809724509617$$
$$x_{65} = -79.8488132787406$$
$$x_{66} = -67.8060414399797$$
$$x_{67} = 8.11578102177363$$
$$x_{68} = 62.0464549083984$$
$$x_{69} = -25.9181393921158$$
$$x_{70} = 20.1585528605345$$
$$x_{71} = -96.0803753222878$$
$$x_{72} = -40.0553063332699$$
$$x_{73} = 54.1924732744239$$
$$x_{74} = 81.9432083811338$$
$$x_{75} = 93.9859802198946$$
$$x_{76} = 34.2957198016886$$
$$x_{77} = 90.3207887907066$$
$$x_{78} = -57.857664703612$$
$$x_{79} = -55.7632696012188$$
$$x_{80} = 52.0980781720307$$
$$x_{81} = 81.4196096055355$$
$$x_{82} = 30.1069295969022$$
$$x_{83} = -69.3768377667746$$
$$x_{84} = -89.7971900151083$$
$$x_{85} = -45.8148928648512$$
$$x_{86} = -99.7455667514759$$
$$x_{87} = -18.0641577581413$$
$$x_{88} = -1.83259571459405$$
$$x_{89} = 15.9697626557481$$
$$x_{90} = -77.7544181763474$$
$$x_{91} = 10.2101761241668$$
$$x_{92} = 98.174770424681$$
$$x_{93} = -69.9004365423729$$
$$x_{94} = 69.9004365423729$$
$$x_{95} = -35.8665161284835$$
$$x_{96} = 24.3473430653209$$
$$x_{1} = 56.2868683768171$$
$$x_{2} = -87.7027949127151$$
$$x_{3} = -71.9948316447661$$
$$x_{4} = 18.0641577581413$$
$$x_{5} = -6.02138591938044$$
$$x_{6} = 25.9181393921158$$
$$x_{7} = -65.7116463375865$$
$$x_{8} = 40.0553063332699$$
$$x_{9} = -47.9092879672443$$
$$x_{10} = -19.1113553093379$$
$$x_{11} = 44.2440965380563$$
$$x_{12} = -15.9697626557481$$
$$x_{13} = -43.720497762458$$
$$x_{14} = -62.0464549083984$$
$$x_{15} = -53.6688744988256$$
$$x_{16} = -3.92699081698724$$
$$x_{17} = 88.2263936883134$$
$$x_{18} = -84.037603483527$$
$$x_{19} = -31.6777259236971$$
$$x_{20} = -81.9432083811338$$
$$x_{21} = 74.0892267471593$$
$$x_{22} = -28.012534494509$$
$$x_{23} = 91.8915851175014$$
$$x_{24} = -13.8753675533549$$
$$x_{25} = -172.525796559639$$
$$x_{26} = 0.261799387799149$$
$$x_{27} = 32.2013246992954$$
$$x_{28} = 100.269165527074$$
$$x_{29} = -93.9859802198946$$
$$x_{30} = 3.92699081698724$$
$$x_{31} = 2.35619449019234$$
$$x_{32} = -75.6600230739542$$
$$x_{33} = 76.1836218495525$$
$$x_{34} = -59.9520598060052$$
$$x_{35} = -37.9609112308767$$
$$x_{36} = -9.68657734856853$$
$$x_{37} = 46.3384916404494$$
$$x_{38} = 64.1408500107916$$
$$x_{39} = 78.2780169519457$$
$$x_{40} = 47.9092879672443$$
$$x_{41} = 1738.08613559855$$
$$x_{42} = -91.8915851175014$$
$$x_{43} = 42.1497014356631$$
$$x_{44} = -50.0036830696375$$
$$x_{45} = 96.0803753222878$$
$$x_{46} = -33.7721210260903$$
$$x_{47} = 71.9948316447661$$
$$x_{48} = 86.1319985859202$$
$$x_{49} = -74.0892267471593$$
$$x_{50} = 66.2352451131848$$
$$x_{51} = 28.012534494509$$
$$x_{52} = 12.30457122656$$
$$x_{53} = 50.0036830696375$$
$$x_{54} = 84.037603483527$$
$$x_{55} = 136.397481043357$$
$$x_{56} = 22.2529479629277$$
$$x_{57} = -21.7293491873294$$
$$x_{58} = -52.0980781720307$$
$$x_{59} = 37.9609112308767$$
$$x_{60} = 68.329640215578$$
$$x_{61} = 6.02138591938044$$
$$x_{62} = 59.9520598060052$$
$$x_{63} = -23.8237442897226$$
$$x_{64} = -11.7809724509617$$
$$x_{65} = -79.8488132787406$$
$$x_{66} = -67.8060414399797$$
$$x_{67} = 8.11578102177363$$
$$x_{68} = 62.0464549083984$$
$$x_{69} = -25.9181393921158$$
$$x_{70} = 20.1585528605345$$
$$x_{71} = -96.0803753222878$$
$$x_{72} = -40.0553063332699$$
$$x_{73} = 54.1924732744239$$
$$x_{74} = 81.9432083811338$$
$$x_{75} = 93.9859802198946$$
$$x_{76} = 34.2957198016886$$
$$x_{77} = 90.3207887907066$$
$$x_{78} = -57.857664703612$$
$$x_{79} = -55.7632696012188$$
$$x_{80} = 52.0980781720307$$
$$x_{81} = 81.4196096055355$$
$$x_{82} = 30.1069295969022$$
$$x_{83} = -69.3768377667746$$
$$x_{84} = -89.7971900151083$$
$$x_{85} = -45.8148928648512$$
$$x_{86} = -99.7455667514759$$
$$x_{87} = -18.0641577581413$$
$$x_{88} = -1.83259571459405$$
$$x_{89} = 15.9697626557481$$
$$x_{90} = -77.7544181763474$$
$$x_{91} = 10.2101761241668$$
$$x_{92} = 98.174770424681$$
$$x_{93} = -69.9004365423729$$
$$x_{94} = 69.9004365423729$$
$$x_{95} = -35.8665161284835$$
$$x_{96} = 24.3473430653209$$
Las raíces dadas
$$x_{25} = -172.525796559639$$
$$x_{86} = -99.7455667514759$$
$$x_{71} = -96.0803753222878$$
$$x_{29} = -93.9859802198946$$
$$x_{42} = -91.8915851175014$$
$$x_{84} = -89.7971900151083$$
$$x_{2} = -87.7027949127151$$
$$x_{18} = -84.037603483527$$
$$x_{20} = -81.9432083811338$$
$$x_{65} = -79.8488132787406$$
$$x_{90} = -77.7544181763474$$
$$x_{32} = -75.6600230739542$$
$$x_{49} = -74.0892267471593$$
$$x_{3} = -71.9948316447661$$
$$x_{93} = -69.9004365423729$$
$$x_{83} = -69.3768377667746$$
$$x_{66} = -67.8060414399797$$
$$x_{7} = -65.7116463375865$$
$$x_{14} = -62.0464549083984$$
$$x_{34} = -59.9520598060052$$
$$x_{78} = -57.857664703612$$
$$x_{79} = -55.7632696012188$$
$$x_{15} = -53.6688744988256$$
$$x_{58} = -52.0980781720307$$
$$x_{44} = -50.0036830696375$$
$$x_{9} = -47.9092879672443$$
$$x_{85} = -45.8148928648512$$
$$x_{13} = -43.720497762458$$
$$x_{72} = -40.0553063332699$$
$$x_{35} = -37.9609112308767$$
$$x_{95} = -35.8665161284835$$
$$x_{46} = -33.7721210260903$$
$$x_{19} = -31.6777259236971$$
$$x_{22} = -28.012534494509$$
$$x_{69} = -25.9181393921158$$
$$x_{63} = -23.8237442897226$$
$$x_{57} = -21.7293491873294$$
$$x_{10} = -19.1113553093379$$
$$x_{87} = -18.0641577581413$$
$$x_{12} = -15.9697626557481$$
$$x_{24} = -13.8753675533549$$
$$x_{64} = -11.7809724509617$$
$$x_{36} = -9.68657734856853$$
$$x_{5} = -6.02138591938044$$
$$x_{16} = -3.92699081698724$$
$$x_{88} = -1.83259571459405$$
$$x_{26} = 0.261799387799149$$
$$x_{31} = 2.35619449019234$$
$$x_{30} = 3.92699081698724$$
$$x_{61} = 6.02138591938044$$
$$x_{67} = 8.11578102177363$$
$$x_{91} = 10.2101761241668$$
$$x_{52} = 12.30457122656$$
$$x_{89} = 15.9697626557481$$
$$x_{4} = 18.0641577581413$$
$$x_{70} = 20.1585528605345$$
$$x_{56} = 22.2529479629277$$
$$x_{96} = 24.3473430653209$$
$$x_{6} = 25.9181393921158$$
$$x_{51} = 28.012534494509$$
$$x_{82} = 30.1069295969022$$
$$x_{27} = 32.2013246992954$$
$$x_{76} = 34.2957198016886$$
$$x_{59} = 37.9609112308767$$
$$x_{8} = 40.0553063332699$$
$$x_{43} = 42.1497014356631$$
$$x_{11} = 44.2440965380563$$
$$x_{37} = 46.3384916404494$$
$$x_{40} = 47.9092879672443$$
$$x_{53} = 50.0036830696375$$
$$x_{80} = 52.0980781720307$$
$$x_{73} = 54.1924732744239$$
$$x_{1} = 56.2868683768171$$
$$x_{62} = 59.9520598060052$$
$$x_{68} = 62.0464549083984$$
$$x_{38} = 64.1408500107916$$
$$x_{50} = 66.2352451131848$$
$$x_{60} = 68.329640215578$$
$$x_{94} = 69.9004365423729$$
$$x_{47} = 71.9948316447661$$
$$x_{21} = 74.0892267471593$$
$$x_{33} = 76.1836218495525$$
$$x_{39} = 78.2780169519457$$
$$x_{81} = 81.4196096055355$$
$$x_{74} = 81.9432083811338$$
$$x_{54} = 84.037603483527$$
$$x_{48} = 86.1319985859202$$
$$x_{17} = 88.2263936883134$$
$$x_{77} = 90.3207887907066$$
$$x_{23} = 91.8915851175014$$
$$x_{75} = 93.9859802198946$$
$$x_{45} = 96.0803753222878$$
$$x_{92} = 98.174770424681$$
$$x_{28} = 100.269165527074$$
$$x_{55} = 136.397481043357$$
$$x_{41} = 1738.08613559855$$
son puntos de cambio del signo de desigualdad en las soluciones.
Primero definámonos con el signo hasta el punto extremo izquierdo:
$$x_{0} \leq x_{25}$$
Consideremos, por ejemplo, el punto
$$x_{0} = x_{25} - \frac{1}{10}$$
=
$$-172.525796559639 + - \frac{1}{10}$$
=
$$-172.625796559639$$
lo sustituimos en la expresión
$$\left|{\sin{\left(3 x \right)}}\right| \leq \frac{\sqrt{2}}{2}$$
$$\left|{\sin{\left(\left(-172.625796559639\right) 3 \right)}}\right| \leq \frac{\sqrt{2}}{2}$$
significa que una de las soluciones de nuestra ecuación será con:
$$x \leq -172.525796559639$$
Recibiremos otras soluciones de la desigualdad pasando al polo siguiente etc.
etc.
Respuesta:
$$x \leq -172.525796559639$$
$$x \geq -99.7455667514759 \wedge x \leq -96.0803753222878$$
$$x \geq -93.9859802198946 \wedge x \leq -91.8915851175014$$
$$x \geq -89.7971900151083 \wedge x \leq -87.7027949127151$$
$$x \geq -84.037603483527 \wedge x \leq -81.9432083811338$$
$$x \geq -79.8488132787406 \wedge x \leq -77.7544181763474$$
$$x \geq -75.6600230739542 \wedge x \leq -74.0892267471593$$
$$x \geq -71.9948316447661 \wedge x \leq -69.9004365423729$$
$$x \geq -69.3768377667746 \wedge x \leq -67.8060414399797$$
$$x \geq -65.7116463375865 \wedge x \leq -62.0464549083984$$
$$x \geq -59.9520598060052 \wedge x \leq -57.857664703612$$
$$x \geq -55.7632696012188 \wedge x \leq -53.6688744988256$$
$$x \geq -52.0980781720307 \wedge x \leq -50.0036830696375$$
$$x \geq -47.9092879672443 \wedge x \leq -45.8148928648512$$
$$x \geq -43.720497762458 \wedge x \leq -40.0553063332699$$
$$x \geq -37.9609112308767 \wedge x \leq -35.8665161284835$$
$$x \geq -33.7721210260903 \wedge x \leq -31.6777259236971$$
$$x \geq -28.012534494509 \wedge x \leq -25.9181393921158$$
$$x \geq -23.8237442897226 \wedge x \leq -21.7293491873294$$
$$x \geq -19.1113553093379 \wedge x \leq -18.0641577581413$$
$$x \geq -15.9697626557481 \wedge x \leq -13.8753675533549$$
$$x \geq -11.7809724509617 \wedge x \leq -9.68657734856853$$
$$x \geq -6.02138591938044 \wedge x \leq -3.92699081698724$$
$$x \geq -1.83259571459405 \wedge x \leq 0.261799387799149$$
$$x \geq 2.35619449019234 \wedge x \leq 3.92699081698724$$
$$x \geq 6.02138591938044 \wedge x \leq 8.11578102177363$$
$$x \geq 10.2101761241668 \wedge x \leq 12.30457122656$$
$$x \geq 15.9697626557481 \wedge x \leq 18.0641577581413$$
$$x \geq 20.1585528605345 \wedge x \leq 22.2529479629277$$
$$x \geq 24.3473430653209 \wedge x \leq 25.9181393921158$$
$$x \geq 28.012534494509 \wedge x \leq 30.1069295969022$$
$$x \geq 32.2013246992954 \wedge x \leq 34.2957198016886$$
$$x \geq 37.9609112308767 \wedge x \leq 40.0553063332699$$
$$x \geq 42.1497014356631 \wedge x \leq 44.2440965380563$$
$$x \geq 46.3384916404494 \wedge x \leq 47.9092879672443$$
$$x \geq 50.0036830696375 \wedge x \leq 52.0980781720307$$
$$x \geq 54.1924732744239 \wedge x \leq 56.2868683768171$$
$$x \geq 59.9520598060052 \wedge x \leq 62.0464549083984$$
$$x \geq 64.1408500107916 \wedge x \leq 66.2352451131848$$
$$x \geq 68.329640215578 \wedge x \leq 69.9004365423729$$
$$x \geq 71.9948316447661 \wedge x \leq 74.0892267471593$$
$$x \geq 76.1836218495525 \wedge x \leq 78.2780169519457$$
$$x \geq 81.4196096055355 \wedge x \leq 81.9432083811338$$
$$x \geq 84.037603483527 \wedge x \leq 86.1319985859202$$
$$x \geq 88.2263936883134 \wedge x \leq 90.3207887907066$$
$$x \geq 91.8915851175014 \wedge x \leq 93.9859802198946$$
$$x \geq 96.0803753222878 \wedge x \leq 98.174770424681$$
$$x \geq 100.269165527074 \wedge x \leq 136.397481043357$$
$$x \geq 1738.08613559855$$
$$\left|{\sin{\left(3 x \right)}}\right| \leq \frac{\sqrt{2}}{2}$$
Para resolver esta desigualdad primero hay que resolver la ecuación correspondiente:
$$\left|{\sin{\left(3 x \right)}}\right| = \frac{\sqrt{2}}{2}$$
Resolvemos:
Tenemos la ecuación
$$\left|{\sin{\left(3 x \right)}}\right| = \frac{\sqrt{2}}{2}$$
cambiamos
$$\left|{\sin{\left(3 x \right)}}\right| - \frac{\sqrt{2}}{2} = 0$$
$$\left|{\sin{\left(3 x \right)}}\right| - \frac{\sqrt{2}}{2} = 0$$
Sustituimos
$$w = \left|{\sin{\left(3 x \right)}}\right|$$
Abrimos los paréntesis en el miembro izquierdo de la ecuación
w - sqrt2/2 = 0
Dividamos ambos miembros de la ecuación en (w - sqrt(2)/2)/w
w = 0 / ((w - sqrt(2)/2)/w)
Obtenemos la respuesta: w = sqrt(2)/2
hacemos cambio inverso
$$\left|{\sin{\left(3 x \right)}}\right| = w$$
sustituimos w:
$$x_{1} = 56.2868683768171$$
$$x_{2} = -87.7027949127151$$
$$x_{3} = -71.9948316447661$$
$$x_{4} = 18.0641577581413$$
$$x_{5} = -6.02138591938044$$
$$x_{6} = 25.9181393921158$$
$$x_{7} = -65.7116463375865$$
$$x_{8} = 40.0553063332699$$
$$x_{9} = -47.9092879672443$$
$$x_{10} = -19.1113553093379$$
$$x_{11} = 44.2440965380563$$
$$x_{12} = -15.9697626557481$$
$$x_{13} = -43.720497762458$$
$$x_{14} = -62.0464549083984$$
$$x_{15} = -53.6688744988256$$
$$x_{16} = -3.92699081698724$$
$$x_{17} = 88.2263936883134$$
$$x_{18} = -84.037603483527$$
$$x_{19} = -31.6777259236971$$
$$x_{20} = -81.9432083811338$$
$$x_{21} = 74.0892267471593$$
$$x_{22} = -28.012534494509$$
$$x_{23} = 91.8915851175014$$
$$x_{24} = -13.8753675533549$$
$$x_{25} = -172.525796559639$$
$$x_{26} = 0.261799387799149$$
$$x_{27} = 32.2013246992954$$
$$x_{28} = 100.269165527074$$
$$x_{29} = -93.9859802198946$$
$$x_{30} = 3.92699081698724$$
$$x_{31} = 2.35619449019234$$
$$x_{32} = -75.6600230739542$$
$$x_{33} = 76.1836218495525$$
$$x_{34} = -59.9520598060052$$
$$x_{35} = -37.9609112308767$$
$$x_{36} = -9.68657734856853$$
$$x_{37} = 46.3384916404494$$
$$x_{38} = 64.1408500107916$$
$$x_{39} = 78.2780169519457$$
$$x_{40} = 47.9092879672443$$
$$x_{41} = 1738.08613559855$$
$$x_{42} = -91.8915851175014$$
$$x_{43} = 42.1497014356631$$
$$x_{44} = -50.0036830696375$$
$$x_{45} = 96.0803753222878$$
$$x_{46} = -33.7721210260903$$
$$x_{47} = 71.9948316447661$$
$$x_{48} = 86.1319985859202$$
$$x_{49} = -74.0892267471593$$
$$x_{50} = 66.2352451131848$$
$$x_{51} = 28.012534494509$$
$$x_{52} = 12.30457122656$$
$$x_{53} = 50.0036830696375$$
$$x_{54} = 84.037603483527$$
$$x_{55} = 136.397481043357$$
$$x_{56} = 22.2529479629277$$
$$x_{57} = -21.7293491873294$$
$$x_{58} = -52.0980781720307$$
$$x_{59} = 37.9609112308767$$
$$x_{60} = 68.329640215578$$
$$x_{61} = 6.02138591938044$$
$$x_{62} = 59.9520598060052$$
$$x_{63} = -23.8237442897226$$
$$x_{64} = -11.7809724509617$$
$$x_{65} = -79.8488132787406$$
$$x_{66} = -67.8060414399797$$
$$x_{67} = 8.11578102177363$$
$$x_{68} = 62.0464549083984$$
$$x_{69} = -25.9181393921158$$
$$x_{70} = 20.1585528605345$$
$$x_{71} = -96.0803753222878$$
$$x_{72} = -40.0553063332699$$
$$x_{73} = 54.1924732744239$$
$$x_{74} = 81.9432083811338$$
$$x_{75} = 93.9859802198946$$
$$x_{76} = 34.2957198016886$$
$$x_{77} = 90.3207887907066$$
$$x_{78} = -57.857664703612$$
$$x_{79} = -55.7632696012188$$
$$x_{80} = 52.0980781720307$$
$$x_{81} = 81.4196096055355$$
$$x_{82} = 30.1069295969022$$
$$x_{83} = -69.3768377667746$$
$$x_{84} = -89.7971900151083$$
$$x_{85} = -45.8148928648512$$
$$x_{86} = -99.7455667514759$$
$$x_{87} = -18.0641577581413$$
$$x_{88} = -1.83259571459405$$
$$x_{89} = 15.9697626557481$$
$$x_{90} = -77.7544181763474$$
$$x_{91} = 10.2101761241668$$
$$x_{92} = 98.174770424681$$
$$x_{93} = -69.9004365423729$$
$$x_{94} = 69.9004365423729$$
$$x_{95} = -35.8665161284835$$
$$x_{96} = 24.3473430653209$$
$$x_{1} = 56.2868683768171$$
$$x_{2} = -87.7027949127151$$
$$x_{3} = -71.9948316447661$$
$$x_{4} = 18.0641577581413$$
$$x_{5} = -6.02138591938044$$
$$x_{6} = 25.9181393921158$$
$$x_{7} = -65.7116463375865$$
$$x_{8} = 40.0553063332699$$
$$x_{9} = -47.9092879672443$$
$$x_{10} = -19.1113553093379$$
$$x_{11} = 44.2440965380563$$
$$x_{12} = -15.9697626557481$$
$$x_{13} = -43.720497762458$$
$$x_{14} = -62.0464549083984$$
$$x_{15} = -53.6688744988256$$
$$x_{16} = -3.92699081698724$$
$$x_{17} = 88.2263936883134$$
$$x_{18} = -84.037603483527$$
$$x_{19} = -31.6777259236971$$
$$x_{20} = -81.9432083811338$$
$$x_{21} = 74.0892267471593$$
$$x_{22} = -28.012534494509$$
$$x_{23} = 91.8915851175014$$
$$x_{24} = -13.8753675533549$$
$$x_{25} = -172.525796559639$$
$$x_{26} = 0.261799387799149$$
$$x_{27} = 32.2013246992954$$
$$x_{28} = 100.269165527074$$
$$x_{29} = -93.9859802198946$$
$$x_{30} = 3.92699081698724$$
$$x_{31} = 2.35619449019234$$
$$x_{32} = -75.6600230739542$$
$$x_{33} = 76.1836218495525$$
$$x_{34} = -59.9520598060052$$
$$x_{35} = -37.9609112308767$$
$$x_{36} = -9.68657734856853$$
$$x_{37} = 46.3384916404494$$
$$x_{38} = 64.1408500107916$$
$$x_{39} = 78.2780169519457$$
$$x_{40} = 47.9092879672443$$
$$x_{41} = 1738.08613559855$$
$$x_{42} = -91.8915851175014$$
$$x_{43} = 42.1497014356631$$
$$x_{44} = -50.0036830696375$$
$$x_{45} = 96.0803753222878$$
$$x_{46} = -33.7721210260903$$
$$x_{47} = 71.9948316447661$$
$$x_{48} = 86.1319985859202$$
$$x_{49} = -74.0892267471593$$
$$x_{50} = 66.2352451131848$$
$$x_{51} = 28.012534494509$$
$$x_{52} = 12.30457122656$$
$$x_{53} = 50.0036830696375$$
$$x_{54} = 84.037603483527$$
$$x_{55} = 136.397481043357$$
$$x_{56} = 22.2529479629277$$
$$x_{57} = -21.7293491873294$$
$$x_{58} = -52.0980781720307$$
$$x_{59} = 37.9609112308767$$
$$x_{60} = 68.329640215578$$
$$x_{61} = 6.02138591938044$$
$$x_{62} = 59.9520598060052$$
$$x_{63} = -23.8237442897226$$
$$x_{64} = -11.7809724509617$$
$$x_{65} = -79.8488132787406$$
$$x_{66} = -67.8060414399797$$
$$x_{67} = 8.11578102177363$$
$$x_{68} = 62.0464549083984$$
$$x_{69} = -25.9181393921158$$
$$x_{70} = 20.1585528605345$$
$$x_{71} = -96.0803753222878$$
$$x_{72} = -40.0553063332699$$
$$x_{73} = 54.1924732744239$$
$$x_{74} = 81.9432083811338$$
$$x_{75} = 93.9859802198946$$
$$x_{76} = 34.2957198016886$$
$$x_{77} = 90.3207887907066$$
$$x_{78} = -57.857664703612$$
$$x_{79} = -55.7632696012188$$
$$x_{80} = 52.0980781720307$$
$$x_{81} = 81.4196096055355$$
$$x_{82} = 30.1069295969022$$
$$x_{83} = -69.3768377667746$$
$$x_{84} = -89.7971900151083$$
$$x_{85} = -45.8148928648512$$
$$x_{86} = -99.7455667514759$$
$$x_{87} = -18.0641577581413$$
$$x_{88} = -1.83259571459405$$
$$x_{89} = 15.9697626557481$$
$$x_{90} = -77.7544181763474$$
$$x_{91} = 10.2101761241668$$
$$x_{92} = 98.174770424681$$
$$x_{93} = -69.9004365423729$$
$$x_{94} = 69.9004365423729$$
$$x_{95} = -35.8665161284835$$
$$x_{96} = 24.3473430653209$$
Las raíces dadas
$$x_{25} = -172.525796559639$$
$$x_{86} = -99.7455667514759$$
$$x_{71} = -96.0803753222878$$
$$x_{29} = -93.9859802198946$$
$$x_{42} = -91.8915851175014$$
$$x_{84} = -89.7971900151083$$
$$x_{2} = -87.7027949127151$$
$$x_{18} = -84.037603483527$$
$$x_{20} = -81.9432083811338$$
$$x_{65} = -79.8488132787406$$
$$x_{90} = -77.7544181763474$$
$$x_{32} = -75.6600230739542$$
$$x_{49} = -74.0892267471593$$
$$x_{3} = -71.9948316447661$$
$$x_{93} = -69.9004365423729$$
$$x_{83} = -69.3768377667746$$
$$x_{66} = -67.8060414399797$$
$$x_{7} = -65.7116463375865$$
$$x_{14} = -62.0464549083984$$
$$x_{34} = -59.9520598060052$$
$$x_{78} = -57.857664703612$$
$$x_{79} = -55.7632696012188$$
$$x_{15} = -53.6688744988256$$
$$x_{58} = -52.0980781720307$$
$$x_{44} = -50.0036830696375$$
$$x_{9} = -47.9092879672443$$
$$x_{85} = -45.8148928648512$$
$$x_{13} = -43.720497762458$$
$$x_{72} = -40.0553063332699$$
$$x_{35} = -37.9609112308767$$
$$x_{95} = -35.8665161284835$$
$$x_{46} = -33.7721210260903$$
$$x_{19} = -31.6777259236971$$
$$x_{22} = -28.012534494509$$
$$x_{69} = -25.9181393921158$$
$$x_{63} = -23.8237442897226$$
$$x_{57} = -21.7293491873294$$
$$x_{10} = -19.1113553093379$$
$$x_{87} = -18.0641577581413$$
$$x_{12} = -15.9697626557481$$
$$x_{24} = -13.8753675533549$$
$$x_{64} = -11.7809724509617$$
$$x_{36} = -9.68657734856853$$
$$x_{5} = -6.02138591938044$$
$$x_{16} = -3.92699081698724$$
$$x_{88} = -1.83259571459405$$
$$x_{26} = 0.261799387799149$$
$$x_{31} = 2.35619449019234$$
$$x_{30} = 3.92699081698724$$
$$x_{61} = 6.02138591938044$$
$$x_{67} = 8.11578102177363$$
$$x_{91} = 10.2101761241668$$
$$x_{52} = 12.30457122656$$
$$x_{89} = 15.9697626557481$$
$$x_{4} = 18.0641577581413$$
$$x_{70} = 20.1585528605345$$
$$x_{56} = 22.2529479629277$$
$$x_{96} = 24.3473430653209$$
$$x_{6} = 25.9181393921158$$
$$x_{51} = 28.012534494509$$
$$x_{82} = 30.1069295969022$$
$$x_{27} = 32.2013246992954$$
$$x_{76} = 34.2957198016886$$
$$x_{59} = 37.9609112308767$$
$$x_{8} = 40.0553063332699$$
$$x_{43} = 42.1497014356631$$
$$x_{11} = 44.2440965380563$$
$$x_{37} = 46.3384916404494$$
$$x_{40} = 47.9092879672443$$
$$x_{53} = 50.0036830696375$$
$$x_{80} = 52.0980781720307$$
$$x_{73} = 54.1924732744239$$
$$x_{1} = 56.2868683768171$$
$$x_{62} = 59.9520598060052$$
$$x_{68} = 62.0464549083984$$
$$x_{38} = 64.1408500107916$$
$$x_{50} = 66.2352451131848$$
$$x_{60} = 68.329640215578$$
$$x_{94} = 69.9004365423729$$
$$x_{47} = 71.9948316447661$$
$$x_{21} = 74.0892267471593$$
$$x_{33} = 76.1836218495525$$
$$x_{39} = 78.2780169519457$$
$$x_{81} = 81.4196096055355$$
$$x_{74} = 81.9432083811338$$
$$x_{54} = 84.037603483527$$
$$x_{48} = 86.1319985859202$$
$$x_{17} = 88.2263936883134$$
$$x_{77} = 90.3207887907066$$
$$x_{23} = 91.8915851175014$$
$$x_{75} = 93.9859802198946$$
$$x_{45} = 96.0803753222878$$
$$x_{92} = 98.174770424681$$
$$x_{28} = 100.269165527074$$
$$x_{55} = 136.397481043357$$
$$x_{41} = 1738.08613559855$$
son puntos de cambio del signo de desigualdad en las soluciones.
Primero definámonos con el signo hasta el punto extremo izquierdo:
$$x_{0} \leq x_{25}$$
Consideremos, por ejemplo, el punto
$$x_{0} = x_{25} - \frac{1}{10}$$
=
$$-172.525796559639 + - \frac{1}{10}$$
=
$$-172.625796559639$$
lo sustituimos en la expresión
$$\left|{\sin{\left(3 x \right)}}\right| \leq \frac{\sqrt{2}}{2}$$
$$\left|{\sin{\left(\left(-172.625796559639\right) 3 \right)}}\right| \leq \frac{\sqrt{2}}{2}$$
___
\/ 2
0.466560567667758 <= -----
2
significa que una de las soluciones de nuestra ecuación será con:
$$x \leq -172.525796559639$$
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x25 x86 x71 x29 x42 x84 x2 x18 x20 x65 x90 x32 x49 x3 x93 x83 x66 x7 x14 x34 x78 x79 x15 x58 x44 x9 x85 x13 x72 x35 x95 x46 x19 x22 x69 x63 x57 x10 x87 x12 x24 x64 x36 x5 x16 x88 x26 x31 x30 x61 x67 x91 x52 x89 x4 x70 x56 x96 x6 x51 x82 x27 x76 x59 x8 x43 x11 x37 x40 x53 x80 x73 x1 x62 x68 x38 x50 x60 x94 x47 x21 x33 x39 x81 x74 x54 x48 x17 x77 x23 x75 x45 x92 x28 x55 x41Recibiremos otras soluciones de la desigualdad pasando al polo siguiente etc.
etc.
Respuesta:
$$x \leq -172.525796559639$$
$$x \geq -99.7455667514759 \wedge x \leq -96.0803753222878$$
$$x \geq -93.9859802198946 \wedge x \leq -91.8915851175014$$
$$x \geq -89.7971900151083 \wedge x \leq -87.7027949127151$$
$$x \geq -84.037603483527 \wedge x \leq -81.9432083811338$$
$$x \geq -79.8488132787406 \wedge x \leq -77.7544181763474$$
$$x \geq -75.6600230739542 \wedge x \leq -74.0892267471593$$
$$x \geq -71.9948316447661 \wedge x \leq -69.9004365423729$$
$$x \geq -69.3768377667746 \wedge x \leq -67.8060414399797$$
$$x \geq -65.7116463375865 \wedge x \leq -62.0464549083984$$
$$x \geq -59.9520598060052 \wedge x \leq -57.857664703612$$
$$x \geq -55.7632696012188 \wedge x \leq -53.6688744988256$$
$$x \geq -52.0980781720307 \wedge x \leq -50.0036830696375$$
$$x \geq -47.9092879672443 \wedge x \leq -45.8148928648512$$
$$x \geq -43.720497762458 \wedge x \leq -40.0553063332699$$
$$x \geq -37.9609112308767 \wedge x \leq -35.8665161284835$$
$$x \geq -33.7721210260903 \wedge x \leq -31.6777259236971$$
$$x \geq -28.012534494509 \wedge x \leq -25.9181393921158$$
$$x \geq -23.8237442897226 \wedge x \leq -21.7293491873294$$
$$x \geq -19.1113553093379 \wedge x \leq -18.0641577581413$$
$$x \geq -15.9697626557481 \wedge x \leq -13.8753675533549$$
$$x \geq -11.7809724509617 \wedge x \leq -9.68657734856853$$
$$x \geq -6.02138591938044 \wedge x \leq -3.92699081698724$$
$$x \geq -1.83259571459405 \wedge x \leq 0.261799387799149$$
$$x \geq 2.35619449019234 \wedge x \leq 3.92699081698724$$
$$x \geq 6.02138591938044 \wedge x \leq 8.11578102177363$$
$$x \geq 10.2101761241668 \wedge x \leq 12.30457122656$$
$$x \geq 15.9697626557481 \wedge x \leq 18.0641577581413$$
$$x \geq 20.1585528605345 \wedge x \leq 22.2529479629277$$
$$x \geq 24.3473430653209 \wedge x \leq 25.9181393921158$$
$$x \geq 28.012534494509 \wedge x \leq 30.1069295969022$$
$$x \geq 32.2013246992954 \wedge x \leq 34.2957198016886$$
$$x \geq 37.9609112308767 \wedge x \leq 40.0553063332699$$
$$x \geq 42.1497014356631 \wedge x \leq 44.2440965380563$$
$$x \geq 46.3384916404494 \wedge x \leq 47.9092879672443$$
$$x \geq 50.0036830696375 \wedge x \leq 52.0980781720307$$
$$x \geq 54.1924732744239 \wedge x \leq 56.2868683768171$$
$$x \geq 59.9520598060052 \wedge x \leq 62.0464549083984$$
$$x \geq 64.1408500107916 \wedge x \leq 66.2352451131848$$
$$x \geq 68.329640215578 \wedge x \leq 69.9004365423729$$
$$x \geq 71.9948316447661 \wedge x \leq 74.0892267471593$$
$$x \geq 76.1836218495525 \wedge x \leq 78.2780169519457$$
$$x \geq 81.4196096055355 \wedge x \leq 81.9432083811338$$
$$x \geq 84.037603483527 \wedge x \leq 86.1319985859202$$
$$x \geq 88.2263936883134 \wedge x \leq 90.3207887907066$$
$$x \geq 91.8915851175014 \wedge x \leq 93.9859802198946$$
$$x \geq 96.0803753222878 \wedge x \leq 98.174770424681$$
$$x \geq 100.269165527074 \wedge x \leq 136.397481043357$$
$$x \geq 1738.08613559855$$
Solución de la desigualdad en el gráfico