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
-
2-7x>0
-
(x-5)^2/(x-1)>0
-
-16/(x+2)^2-5>=0
- Integral de d{x}:
- |sin(x)|
-
Expresiones idénticas
- |sin(x)|<sqrt(dos)/ dos
- módulo de seno de (x)| menos raíz cuadrada de (2) dividir por 2
- módulo de seno de (x)| menos raíz cuadrada de (dos) dividir por dos
- |sin(x)|<√(2)/2
- |sinx|<sqrt2/2
- |sin(x)|<sqrt(2) dividir por 2
-
Expresiones semejantes
- |sinx|<1
- sin3(x)<sqrt2/2
- |sinx|<|cosx|
- |sinx|<sqrt(2)/2
-
Expresiones con funciones
- Seno sin
- sinx>-1
- sin(x)>1/2
- sin(x)>=1
- sin2x<1/2
- sin(x)<=sqrt(2)/2
- Raíz cuadrada sqrt
- sqrt(x+4)=>x-2
- sqrt(x+1)>sqrt(3-x)
- sqrt(x+4)>x+4
- sqrt(3+2*x-x^2)<=x
- sqrt(x+2)+log(x+3)/log(5)>=0
- Módulo |
- |x-1|<3
- |2x+3|<x+7
- ||2x-1|-2|>3
- |x-5|<=|x+3|
- |x+4|*(x-1)<0
|sin(x)|
Solución
Ha introducido
[src]
___
\/ 2
|sin(x)| < -----
2
$$\left|{\sin{\left(x \right)}}\right| < \frac{\sqrt{2}}{2}$$
Abs(sin(x)) < sqrt(2)/2
Solución detallada
Se da la desigualdad:
$$\left|{\sin{\left(x \right)}}\right| < \frac{\sqrt{2}}{2}$$
Para resolver esta desigualdad primero hay que resolver la ecuación correspondiente:
$$\left|{\sin{\left(x \right)}}\right| = \frac{\sqrt{2}}{2}$$
Resolvemos:
Tenemos la ecuación
$$\left|{\sin{\left(x \right)}}\right| = \frac{\sqrt{2}}{2}$$
cambiamos
$$\left|{\sin{\left(x \right)}}\right| - 1 - \frac{\sqrt{2}}{2} = 0$$
$$\left|{\sin{\left(x \right)}}\right| - 1 - \frac{\sqrt{2}}{2} = 0$$
Sustituimos
$$w = \left|{\sin{\left(x \right)}}\right|$$
Abrimos los paréntesis en el miembro izquierdo de la ecuación
-1 + w - sqrt2/2 = 0
Transportamos los términos libres (sin w)
del miembro izquierdo al derecho, obtenemos:
$$w - \frac{\sqrt{2}}{2} = 1$$
Dividamos ambos miembros de la ecuación en (w - sqrt(2)/2)/w
w = 1 / ((w - sqrt(2)/2)/w)
Obtenemos la respuesta: w = 1 + sqrt(2)/2
hacemos cambio inverso
$$\left|{\sin{\left(x \right)}}\right| = w$$
sustituimos w:
$$x_{1} = -55.7632696012188$$
$$x_{2} = -98.174770424681$$
$$x_{3} = 66.7588438887831$$
$$x_{4} = -11.7809724509617$$
$$x_{5} = 25.9181393921158$$
$$x_{6} = 93.4623814442964$$
$$x_{7} = -40.0553063332699$$
$$x_{8} = 18.0641577581413$$
$$x_{9} = -13.3517687777566$$
$$x_{10} = 55.7632696012188$$
$$x_{11} = -16.4933614313464$$
$$x_{12} = -49.4800842940392$$
$$x_{13} = 96.6039740978861$$
$$x_{14} = -41.6261026600648$$
$$x_{15} = -22.776546738526$$
$$x_{16} = 52.621676947629$$
$$x_{17} = 76.1836218495525$$
$$x_{18} = 69.9004365423729$$
$$x_{19} = 30.6305283725005$$
$$x_{20} = 85.6083998103219$$
$$x_{21} = -19.6349540849362$$
$$x_{22} = -21.2057504117311$$
$$x_{23} = -3.92699081698724$$
$$x_{24} = 38.484510006475$$
$$x_{25} = 32.2013246992954$$
$$x_{26} = 27.4889357189107$$
$$x_{27} = 118.595122673015$$
$$x_{28} = 84.037603483527$$
$$x_{29} = -91.8915851175014$$
$$x_{30} = 98.174770424681$$
$$x_{31} = 44.7676953136546$$
$$x_{32} = -85.6083998103219$$
$$x_{33} = -87.1791961371168$$
$$x_{34} = 41.6261026600648$$
$$x_{35} = 91.8915851175014$$
$$x_{36} = -2.35619449019234$$
$$x_{37} = 16.4933614313464$$
$$x_{38} = 46.3384916404494$$
$$x_{39} = -241.117236163017$$
$$x_{40} = 71.4712328691678$$
$$x_{41} = 63.6172512351933$$
$$x_{42} = 54.1924732744239$$
$$x_{43} = -43.1968989868597$$
$$x_{44} = 245.829625143401$$
$$x_{45} = -84.037603483527$$
$$x_{46} = 0.785398163397448$$
$$x_{47} = -93.4623814442964$$
$$x_{48} = 47.9092879672443$$
$$x_{49} = 24.3473430653209$$
$$x_{50} = -5.49778714378214$$
$$x_{51} = 68.329640215578$$
$$x_{52} = -65.1880475619882$$
$$x_{53} = 62.0464549083984$$
$$x_{54} = -68.329640215578$$
$$x_{55} = -38.484510006475$$
$$x_{56} = 74.6128255227576$$
$$x_{57} = 5.49778714378214$$
$$x_{58} = -47.9092879672443$$
$$x_{59} = 40.0553063332699$$
$$x_{60} = -57.3340659280137$$
$$x_{61} = 49.4800842940392$$
$$x_{62} = -25.9181393921158$$
$$x_{63} = -76.1836218495525$$
$$x_{64} = -200.276531666349$$
$$x_{65} = 88.7499924639117$$
$$x_{66} = 99.7455667514759$$
$$x_{67} = -62.0464549083984$$
$$x_{68} = -82.4668071567321$$
$$x_{69} = -71.4712328691678$$
$$x_{70} = 60.4756585816035$$
$$x_{71} = -33.7721210260903$$
$$x_{72} = 77.7544181763474$$
$$x_{73} = -46.3384916404494$$
$$x_{74} = -63.6172512351933$$
$$x_{75} = -90.3207887907066$$
$$x_{76} = -35.3429173528852$$
$$x_{77} = 8.63937979737193$$
$$x_{78} = -10.2101761241668$$
$$x_{79} = -18.0641577581413$$
$$x_{80} = -27.4889357189107$$
$$x_{81} = -54.1924732744239$$
$$x_{82} = 22.776546738526$$
$$x_{83} = 3.92699081698724$$
$$x_{84} = -79.3252145031423$$
$$x_{85} = -99.7455667514759$$
$$x_{86} = 82.4668071567321$$
$$x_{87} = 33.7721210260903$$
$$x_{88} = 11.7809724509617$$
$$x_{89} = 10.2101761241668$$
$$x_{90} = 19.6349540849362$$
$$x_{91} = -77.7544181763474$$
$$x_{92} = -69.9004365423729$$
$$x_{93} = 2.35619449019234$$
$$x_{94} = -24.3473430653209$$
$$x_{95} = -60.4756585816035$$
$$x_{96} = 90.3207887907066$$
$$x_{97} = -32.2013246992954$$
$$x_{1} = -55.7632696012188$$
$$x_{2} = -98.174770424681$$
$$x_{3} = 66.7588438887831$$
$$x_{4} = -11.7809724509617$$
$$x_{5} = 25.9181393921158$$
$$x_{6} = 93.4623814442964$$
$$x_{7} = -40.0553063332699$$
$$x_{8} = 18.0641577581413$$
$$x_{9} = -13.3517687777566$$
$$x_{10} = 55.7632696012188$$
$$x_{11} = -16.4933614313464$$
$$x_{12} = -49.4800842940392$$
$$x_{13} = 96.6039740978861$$
$$x_{14} = -41.6261026600648$$
$$x_{15} = -22.776546738526$$
$$x_{16} = 52.621676947629$$
$$x_{17} = 76.1836218495525$$
$$x_{18} = 69.9004365423729$$
$$x_{19} = 30.6305283725005$$
$$x_{20} = 85.6083998103219$$
$$x_{21} = -19.6349540849362$$
$$x_{22} = -21.2057504117311$$
$$x_{23} = -3.92699081698724$$
$$x_{24} = 38.484510006475$$
$$x_{25} = 32.2013246992954$$
$$x_{26} = 27.4889357189107$$
$$x_{27} = 118.595122673015$$
$$x_{28} = 84.037603483527$$
$$x_{29} = -91.8915851175014$$
$$x_{30} = 98.174770424681$$
$$x_{31} = 44.7676953136546$$
$$x_{32} = -85.6083998103219$$
$$x_{33} = -87.1791961371168$$
$$x_{34} = 41.6261026600648$$
$$x_{35} = 91.8915851175014$$
$$x_{36} = -2.35619449019234$$
$$x_{37} = 16.4933614313464$$
$$x_{38} = 46.3384916404494$$
$$x_{39} = -241.117236163017$$
$$x_{40} = 71.4712328691678$$
$$x_{41} = 63.6172512351933$$
$$x_{42} = 54.1924732744239$$
$$x_{43} = -43.1968989868597$$
$$x_{44} = 245.829625143401$$
$$x_{45} = -84.037603483527$$
$$x_{46} = 0.785398163397448$$
$$x_{47} = -93.4623814442964$$
$$x_{48} = 47.9092879672443$$
$$x_{49} = 24.3473430653209$$
$$x_{50} = -5.49778714378214$$
$$x_{51} = 68.329640215578$$
$$x_{52} = -65.1880475619882$$
$$x_{53} = 62.0464549083984$$
$$x_{54} = -68.329640215578$$
$$x_{55} = -38.484510006475$$
$$x_{56} = 74.6128255227576$$
$$x_{57} = 5.49778714378214$$
$$x_{58} = -47.9092879672443$$
$$x_{59} = 40.0553063332699$$
$$x_{60} = -57.3340659280137$$
$$x_{61} = 49.4800842940392$$
$$x_{62} = -25.9181393921158$$
$$x_{63} = -76.1836218495525$$
$$x_{64} = -200.276531666349$$
$$x_{65} = 88.7499924639117$$
$$x_{66} = 99.7455667514759$$
$$x_{67} = -62.0464549083984$$
$$x_{68} = -82.4668071567321$$
$$x_{69} = -71.4712328691678$$
$$x_{70} = 60.4756585816035$$
$$x_{71} = -33.7721210260903$$
$$x_{72} = 77.7544181763474$$
$$x_{73} = -46.3384916404494$$
$$x_{74} = -63.6172512351933$$
$$x_{75} = -90.3207887907066$$
$$x_{76} = -35.3429173528852$$
$$x_{77} = 8.63937979737193$$
$$x_{78} = -10.2101761241668$$
$$x_{79} = -18.0641577581413$$
$$x_{80} = -27.4889357189107$$
$$x_{81} = -54.1924732744239$$
$$x_{82} = 22.776546738526$$
$$x_{83} = 3.92699081698724$$
$$x_{84} = -79.3252145031423$$
$$x_{85} = -99.7455667514759$$
$$x_{86} = 82.4668071567321$$
$$x_{87} = 33.7721210260903$$
$$x_{88} = 11.7809724509617$$
$$x_{89} = 10.2101761241668$$
$$x_{90} = 19.6349540849362$$
$$x_{91} = -77.7544181763474$$
$$x_{92} = -69.9004365423729$$
$$x_{93} = 2.35619449019234$$
$$x_{94} = -24.3473430653209$$
$$x_{95} = -60.4756585816035$$
$$x_{96} = 90.3207887907066$$
$$x_{97} = -32.2013246992954$$
Las raíces dadas
$$x_{39} = -241.117236163017$$
$$x_{64} = -200.276531666349$$
$$x_{85} = -99.7455667514759$$
$$x_{2} = -98.174770424681$$
$$x_{47} = -93.4623814442964$$
$$x_{29} = -91.8915851175014$$
$$x_{75} = -90.3207887907066$$
$$x_{33} = -87.1791961371168$$
$$x_{32} = -85.6083998103219$$
$$x_{45} = -84.037603483527$$
$$x_{68} = -82.4668071567321$$
$$x_{84} = -79.3252145031423$$
$$x_{91} = -77.7544181763474$$
$$x_{63} = -76.1836218495525$$
$$x_{69} = -71.4712328691678$$
$$x_{92} = -69.9004365423729$$
$$x_{54} = -68.329640215578$$
$$x_{52} = -65.1880475619882$$
$$x_{74} = -63.6172512351933$$
$$x_{67} = -62.0464549083984$$
$$x_{95} = -60.4756585816035$$
$$x_{60} = -57.3340659280137$$
$$x_{1} = -55.7632696012188$$
$$x_{81} = -54.1924732744239$$
$$x_{12} = -49.4800842940392$$
$$x_{58} = -47.9092879672443$$
$$x_{73} = -46.3384916404494$$
$$x_{43} = -43.1968989868597$$
$$x_{14} = -41.6261026600648$$
$$x_{7} = -40.0553063332699$$
$$x_{55} = -38.484510006475$$
$$x_{76} = -35.3429173528852$$
$$x_{71} = -33.7721210260903$$
$$x_{97} = -32.2013246992954$$
$$x_{80} = -27.4889357189107$$
$$x_{62} = -25.9181393921158$$
$$x_{94} = -24.3473430653209$$
$$x_{15} = -22.776546738526$$
$$x_{22} = -21.2057504117311$$
$$x_{21} = -19.6349540849362$$
$$x_{79} = -18.0641577581413$$
$$x_{11} = -16.4933614313464$$
$$x_{9} = -13.3517687777566$$
$$x_{4} = -11.7809724509617$$
$$x_{78} = -10.2101761241668$$
$$x_{50} = -5.49778714378214$$
$$x_{23} = -3.92699081698724$$
$$x_{36} = -2.35619449019234$$
$$x_{46} = 0.785398163397448$$
$$x_{93} = 2.35619449019234$$
$$x_{83} = 3.92699081698724$$
$$x_{57} = 5.49778714378214$$
$$x_{77} = 8.63937979737193$$
$$x_{89} = 10.2101761241668$$
$$x_{88} = 11.7809724509617$$
$$x_{37} = 16.4933614313464$$
$$x_{8} = 18.0641577581413$$
$$x_{90} = 19.6349540849362$$
$$x_{82} = 22.776546738526$$
$$x_{49} = 24.3473430653209$$
$$x_{5} = 25.9181393921158$$
$$x_{26} = 27.4889357189107$$
$$x_{19} = 30.6305283725005$$
$$x_{25} = 32.2013246992954$$
$$x_{87} = 33.7721210260903$$
$$x_{24} = 38.484510006475$$
$$x_{59} = 40.0553063332699$$
$$x_{34} = 41.6261026600648$$
$$x_{31} = 44.7676953136546$$
$$x_{38} = 46.3384916404494$$
$$x_{48} = 47.9092879672443$$
$$x_{61} = 49.4800842940392$$
$$x_{16} = 52.621676947629$$
$$x_{42} = 54.1924732744239$$
$$x_{10} = 55.7632696012188$$
$$x_{70} = 60.4756585816035$$
$$x_{53} = 62.0464549083984$$
$$x_{41} = 63.6172512351933$$
$$x_{3} = 66.7588438887831$$
$$x_{51} = 68.329640215578$$
$$x_{18} = 69.9004365423729$$
$$x_{40} = 71.4712328691678$$
$$x_{56} = 74.6128255227576$$
$$x_{17} = 76.1836218495525$$
$$x_{72} = 77.7544181763474$$
$$x_{86} = 82.4668071567321$$
$$x_{28} = 84.037603483527$$
$$x_{20} = 85.6083998103219$$
$$x_{65} = 88.7499924639117$$
$$x_{96} = 90.3207887907066$$
$$x_{35} = 91.8915851175014$$
$$x_{6} = 93.4623814442964$$
$$x_{13} = 96.6039740978861$$
$$x_{30} = 98.174770424681$$
$$x_{66} = 99.7455667514759$$
$$x_{27} = 118.595122673015$$
$$x_{44} = 245.829625143401$$
son puntos de cambio del signo de desigualdad en las soluciones.
Primero definámonos con el signo hasta el punto extremo izquierdo:
$$x_{0} < x_{39}$$
Consideremos, por ejemplo, el punto
$$x_{0} = x_{39} - \frac{1}{10}$$
=
$$-241.117236163017 + - \frac{1}{10}$$
=
$$-241.217236163017$$
lo sustituimos en la expresión
$$\left|{\sin{\left(x \right)}}\right| < \frac{\sqrt{2}}{2}$$
$$\left|{\sin{\left(-241.217236163017 \right)}}\right| < \frac{\sqrt{2}}{2}$$
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\/ 2
0.632981306676965 < -----
2
significa que una de las soluciones de nuestra ecuación será con:
$$x < -241.117236163017$$
_____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____
\ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \
-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------
x39 x64 x85 x2 x47 x29 x75 x33 x32 x45 x68 x84 x91 x63 x69 x92 x54 x52 x74 x67 x95 x60 x1 x81 x12 x58 x73 x43 x14 x7 x55 x76 x71 x97 x80 x62 x94 x15 x22 x21 x79 x11 x9 x4 x78 x50 x23 x36 x46 x93 x83 x57 x77 x89 x88 x37 x8 x90 x82 x49 x5 x26 x19 x25 x87 x24 x59 x34 x31 x38 x48 x61 x16 x42 x10 x70 x53 x41 x3 x51 x18 x40 x56 x17 x72 x86 x28 x20 x65 x96 x35 x6 x13 x30 x66 x27 x44
Recibiremos otras soluciones de la desigualdad pasando al polo siguiente etc.
etc.
Respuesta:
$$x < -241.117236163017$$
$$x > -200.276531666349 \wedge x < -99.7455667514759$$
$$x > -98.174770424681 \wedge x < -93.4623814442964$$
$$x > -91.8915851175014 \wedge x < -90.3207887907066$$
$$x > -87.1791961371168 \wedge x < -85.6083998103219$$
$$x > -84.037603483527 \wedge x < -82.4668071567321$$
$$x > -79.3252145031423 \wedge x < -77.7544181763474$$
$$x > -76.1836218495525 \wedge x < -71.4712328691678$$
$$x > -69.9004365423729 \wedge x < -68.329640215578$$
$$x > -65.1880475619882 \wedge x < -63.6172512351933$$
$$x > -62.0464549083984 \wedge x < -60.4756585816035$$
$$x > -57.3340659280137 \wedge x < -55.7632696012188$$
$$x > -54.1924732744239 \wedge x < -49.4800842940392$$
$$x > -47.9092879672443 \wedge x < -46.3384916404494$$
$$x > -43.1968989868597 \wedge x < -41.6261026600648$$
$$x > -40.0553063332699 \wedge x < -38.484510006475$$
$$x > -35.3429173528852 \wedge x < -33.7721210260903$$
$$x > -32.2013246992954 \wedge x < -27.4889357189107$$
$$x > -25.9181393921158 \wedge x < -24.3473430653209$$
$$x > -22.776546738526 \wedge x < -21.2057504117311$$
$$x > -19.6349540849362 \wedge x < -18.0641577581413$$
$$x > -16.4933614313464 \wedge x < -13.3517687777566$$
$$x > -11.7809724509617 \wedge x < -10.2101761241668$$
$$x > -5.49778714378214 \wedge x < -3.92699081698724$$
$$x > -2.35619449019234 \wedge x < 0.785398163397448$$
$$x > 2.35619449019234 \wedge x < 3.92699081698724$$
$$x > 5.49778714378214 \wedge x < 8.63937979737193$$
$$x > 10.2101761241668 \wedge x < 11.7809724509617$$
$$x > 16.4933614313464 \wedge x < 18.0641577581413$$
$$x > 19.6349540849362 \wedge x < 22.776546738526$$
$$x > 24.3473430653209 \wedge x < 25.9181393921158$$
$$x > 27.4889357189107 \wedge x < 30.6305283725005$$
$$x > 32.2013246992954 \wedge x < 33.7721210260903$$
$$x > 38.484510006475 \wedge x < 40.0553063332699$$
$$x > 41.6261026600648 \wedge x < 44.7676953136546$$
$$x > 46.3384916404494 \wedge x < 47.9092879672443$$
$$x > 49.4800842940392 \wedge x < 52.621676947629$$
$$x > 54.1924732744239 \wedge x < 55.7632696012188$$
$$x > 60.4756585816035 \wedge x < 62.0464549083984$$
$$x > 63.6172512351933 \wedge x < 66.7588438887831$$
$$x > 68.329640215578 \wedge x < 69.9004365423729$$
$$x > 71.4712328691678 \wedge x < 74.6128255227576$$
$$x > 76.1836218495525 \wedge x < 77.7544181763474$$
$$x > 82.4668071567321 \wedge x < 84.037603483527$$
$$x > 85.6083998103219 \wedge x < 88.7499924639117$$
$$x > 90.3207887907066 \wedge x < 91.8915851175014$$
$$x > 93.4623814442964 \wedge x < 96.6039740978861$$
$$x > 98.174770424681 \wedge x < 99.7455667514759$$
$$x > 118.595122673015 \wedge x < 245.829625143401$$
Solución de la desigualdad en el gráfico
Gráfico
Solución
Ha introducido
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|sin(x)| < -----
2
$$\left|{\sin{\left(x \right)}}\right| < \frac{\sqrt{2}}{2}$$
Abs(sin(x)) < sqrt(2)/2
Solución detallada
Se da la desigualdad:
$$\left|{\sin{\left(x \right)}}\right| < \frac{\sqrt{2}}{2}$$
Para resolver esta desigualdad primero hay que resolver la ecuación correspondiente:
$$\left|{\sin{\left(x \right)}}\right| = \frac{\sqrt{2}}{2}$$
Resolvemos:
Tenemos la ecuación
$$\left|{\sin{\left(x \right)}}\right| = \frac{\sqrt{2}}{2}$$
cambiamos
$$\left|{\sin{\left(x \right)}}\right| - 1 - \frac{\sqrt{2}}{2} = 0$$
$$\left|{\sin{\left(x \right)}}\right| - 1 - \frac{\sqrt{2}}{2} = 0$$
Sustituimos
$$w = \left|{\sin{\left(x \right)}}\right|$$
Abrimos los paréntesis en el miembro izquierdo de la ecuación
Transportamos los términos libres (sin w)
del miembro izquierdo al derecho, obtenemos:
$$w - \frac{\sqrt{2}}{2} = 1$$
Dividamos ambos miembros de la ecuación en (w - sqrt(2)/2)/w
Obtenemos la respuesta: w = 1 + sqrt(2)/2
hacemos cambio inverso
$$\left|{\sin{\left(x \right)}}\right| = w$$
sustituimos w:
$$x_{1} = -55.7632696012188$$
$$x_{2} = -98.174770424681$$
$$x_{3} = 66.7588438887831$$
$$x_{4} = -11.7809724509617$$
$$x_{5} = 25.9181393921158$$
$$x_{6} = 93.4623814442964$$
$$x_{7} = -40.0553063332699$$
$$x_{8} = 18.0641577581413$$
$$x_{9} = -13.3517687777566$$
$$x_{10} = 55.7632696012188$$
$$x_{11} = -16.4933614313464$$
$$x_{12} = -49.4800842940392$$
$$x_{13} = 96.6039740978861$$
$$x_{14} = -41.6261026600648$$
$$x_{15} = -22.776546738526$$
$$x_{16} = 52.621676947629$$
$$x_{17} = 76.1836218495525$$
$$x_{18} = 69.9004365423729$$
$$x_{19} = 30.6305283725005$$
$$x_{20} = 85.6083998103219$$
$$x_{21} = -19.6349540849362$$
$$x_{22} = -21.2057504117311$$
$$x_{23} = -3.92699081698724$$
$$x_{24} = 38.484510006475$$
$$x_{25} = 32.2013246992954$$
$$x_{26} = 27.4889357189107$$
$$x_{27} = 118.595122673015$$
$$x_{28} = 84.037603483527$$
$$x_{29} = -91.8915851175014$$
$$x_{30} = 98.174770424681$$
$$x_{31} = 44.7676953136546$$
$$x_{32} = -85.6083998103219$$
$$x_{33} = -87.1791961371168$$
$$x_{34} = 41.6261026600648$$
$$x_{35} = 91.8915851175014$$
$$x_{36} = -2.35619449019234$$
$$x_{37} = 16.4933614313464$$
$$x_{38} = 46.3384916404494$$
$$x_{39} = -241.117236163017$$
$$x_{40} = 71.4712328691678$$
$$x_{41} = 63.6172512351933$$
$$x_{42} = 54.1924732744239$$
$$x_{43} = -43.1968989868597$$
$$x_{44} = 245.829625143401$$
$$x_{45} = -84.037603483527$$
$$x_{46} = 0.785398163397448$$
$$x_{47} = -93.4623814442964$$
$$x_{48} = 47.9092879672443$$
$$x_{49} = 24.3473430653209$$
$$x_{50} = -5.49778714378214$$
$$x_{51} = 68.329640215578$$
$$x_{52} = -65.1880475619882$$
$$x_{53} = 62.0464549083984$$
$$x_{54} = -68.329640215578$$
$$x_{55} = -38.484510006475$$
$$x_{56} = 74.6128255227576$$
$$x_{57} = 5.49778714378214$$
$$x_{58} = -47.9092879672443$$
$$x_{59} = 40.0553063332699$$
$$x_{60} = -57.3340659280137$$
$$x_{61} = 49.4800842940392$$
$$x_{62} = -25.9181393921158$$
$$x_{63} = -76.1836218495525$$
$$x_{64} = -200.276531666349$$
$$x_{65} = 88.7499924639117$$
$$x_{66} = 99.7455667514759$$
$$x_{67} = -62.0464549083984$$
$$x_{68} = -82.4668071567321$$
$$x_{69} = -71.4712328691678$$
$$x_{70} = 60.4756585816035$$
$$x_{71} = -33.7721210260903$$
$$x_{72} = 77.7544181763474$$
$$x_{73} = -46.3384916404494$$
$$x_{74} = -63.6172512351933$$
$$x_{75} = -90.3207887907066$$
$$x_{76} = -35.3429173528852$$
$$x_{77} = 8.63937979737193$$
$$x_{78} = -10.2101761241668$$
$$x_{79} = -18.0641577581413$$
$$x_{80} = -27.4889357189107$$
$$x_{81} = -54.1924732744239$$
$$x_{82} = 22.776546738526$$
$$x_{83} = 3.92699081698724$$
$$x_{84} = -79.3252145031423$$
$$x_{85} = -99.7455667514759$$
$$x_{86} = 82.4668071567321$$
$$x_{87} = 33.7721210260903$$
$$x_{88} = 11.7809724509617$$
$$x_{89} = 10.2101761241668$$
$$x_{90} = 19.6349540849362$$
$$x_{91} = -77.7544181763474$$
$$x_{92} = -69.9004365423729$$
$$x_{93} = 2.35619449019234$$
$$x_{94} = -24.3473430653209$$
$$x_{95} = -60.4756585816035$$
$$x_{96} = 90.3207887907066$$
$$x_{97} = -32.2013246992954$$
$$x_{1} = -55.7632696012188$$
$$x_{2} = -98.174770424681$$
$$x_{3} = 66.7588438887831$$
$$x_{4} = -11.7809724509617$$
$$x_{5} = 25.9181393921158$$
$$x_{6} = 93.4623814442964$$
$$x_{7} = -40.0553063332699$$
$$x_{8} = 18.0641577581413$$
$$x_{9} = -13.3517687777566$$
$$x_{10} = 55.7632696012188$$
$$x_{11} = -16.4933614313464$$
$$x_{12} = -49.4800842940392$$
$$x_{13} = 96.6039740978861$$
$$x_{14} = -41.6261026600648$$
$$x_{15} = -22.776546738526$$
$$x_{16} = 52.621676947629$$
$$x_{17} = 76.1836218495525$$
$$x_{18} = 69.9004365423729$$
$$x_{19} = 30.6305283725005$$
$$x_{20} = 85.6083998103219$$
$$x_{21} = -19.6349540849362$$
$$x_{22} = -21.2057504117311$$
$$x_{23} = -3.92699081698724$$
$$x_{24} = 38.484510006475$$
$$x_{25} = 32.2013246992954$$
$$x_{26} = 27.4889357189107$$
$$x_{27} = 118.595122673015$$
$$x_{28} = 84.037603483527$$
$$x_{29} = -91.8915851175014$$
$$x_{30} = 98.174770424681$$
$$x_{31} = 44.7676953136546$$
$$x_{32} = -85.6083998103219$$
$$x_{33} = -87.1791961371168$$
$$x_{34} = 41.6261026600648$$
$$x_{35} = 91.8915851175014$$
$$x_{36} = -2.35619449019234$$
$$x_{37} = 16.4933614313464$$
$$x_{38} = 46.3384916404494$$
$$x_{39} = -241.117236163017$$
$$x_{40} = 71.4712328691678$$
$$x_{41} = 63.6172512351933$$
$$x_{42} = 54.1924732744239$$
$$x_{43} = -43.1968989868597$$
$$x_{44} = 245.829625143401$$
$$x_{45} = -84.037603483527$$
$$x_{46} = 0.785398163397448$$
$$x_{47} = -93.4623814442964$$
$$x_{48} = 47.9092879672443$$
$$x_{49} = 24.3473430653209$$
$$x_{50} = -5.49778714378214$$
$$x_{51} = 68.329640215578$$
$$x_{52} = -65.1880475619882$$
$$x_{53} = 62.0464549083984$$
$$x_{54} = -68.329640215578$$
$$x_{55} = -38.484510006475$$
$$x_{56} = 74.6128255227576$$
$$x_{57} = 5.49778714378214$$
$$x_{58} = -47.9092879672443$$
$$x_{59} = 40.0553063332699$$
$$x_{60} = -57.3340659280137$$
$$x_{61} = 49.4800842940392$$
$$x_{62} = -25.9181393921158$$
$$x_{63} = -76.1836218495525$$
$$x_{64} = -200.276531666349$$
$$x_{65} = 88.7499924639117$$
$$x_{66} = 99.7455667514759$$
$$x_{67} = -62.0464549083984$$
$$x_{68} = -82.4668071567321$$
$$x_{69} = -71.4712328691678$$
$$x_{70} = 60.4756585816035$$
$$x_{71} = -33.7721210260903$$
$$x_{72} = 77.7544181763474$$
$$x_{73} = -46.3384916404494$$
$$x_{74} = -63.6172512351933$$
$$x_{75} = -90.3207887907066$$
$$x_{76} = -35.3429173528852$$
$$x_{77} = 8.63937979737193$$
$$x_{78} = -10.2101761241668$$
$$x_{79} = -18.0641577581413$$
$$x_{80} = -27.4889357189107$$
$$x_{81} = -54.1924732744239$$
$$x_{82} = 22.776546738526$$
$$x_{83} = 3.92699081698724$$
$$x_{84} = -79.3252145031423$$
$$x_{85} = -99.7455667514759$$
$$x_{86} = 82.4668071567321$$
$$x_{87} = 33.7721210260903$$
$$x_{88} = 11.7809724509617$$
$$x_{89} = 10.2101761241668$$
$$x_{90} = 19.6349540849362$$
$$x_{91} = -77.7544181763474$$
$$x_{92} = -69.9004365423729$$
$$x_{93} = 2.35619449019234$$
$$x_{94} = -24.3473430653209$$
$$x_{95} = -60.4756585816035$$
$$x_{96} = 90.3207887907066$$
$$x_{97} = -32.2013246992954$$
Las raíces dadas
$$x_{39} = -241.117236163017$$
$$x_{64} = -200.276531666349$$
$$x_{85} = -99.7455667514759$$
$$x_{2} = -98.174770424681$$
$$x_{47} = -93.4623814442964$$
$$x_{29} = -91.8915851175014$$
$$x_{75} = -90.3207887907066$$
$$x_{33} = -87.1791961371168$$
$$x_{32} = -85.6083998103219$$
$$x_{45} = -84.037603483527$$
$$x_{68} = -82.4668071567321$$
$$x_{84} = -79.3252145031423$$
$$x_{91} = -77.7544181763474$$
$$x_{63} = -76.1836218495525$$
$$x_{69} = -71.4712328691678$$
$$x_{92} = -69.9004365423729$$
$$x_{54} = -68.329640215578$$
$$x_{52} = -65.1880475619882$$
$$x_{74} = -63.6172512351933$$
$$x_{67} = -62.0464549083984$$
$$x_{95} = -60.4756585816035$$
$$x_{60} = -57.3340659280137$$
$$x_{1} = -55.7632696012188$$
$$x_{81} = -54.1924732744239$$
$$x_{12} = -49.4800842940392$$
$$x_{58} = -47.9092879672443$$
$$x_{73} = -46.3384916404494$$
$$x_{43} = -43.1968989868597$$
$$x_{14} = -41.6261026600648$$
$$x_{7} = -40.0553063332699$$
$$x_{55} = -38.484510006475$$
$$x_{76} = -35.3429173528852$$
$$x_{71} = -33.7721210260903$$
$$x_{97} = -32.2013246992954$$
$$x_{80} = -27.4889357189107$$
$$x_{62} = -25.9181393921158$$
$$x_{94} = -24.3473430653209$$
$$x_{15} = -22.776546738526$$
$$x_{22} = -21.2057504117311$$
$$x_{21} = -19.6349540849362$$
$$x_{79} = -18.0641577581413$$
$$x_{11} = -16.4933614313464$$
$$x_{9} = -13.3517687777566$$
$$x_{4} = -11.7809724509617$$
$$x_{78} = -10.2101761241668$$
$$x_{50} = -5.49778714378214$$
$$x_{23} = -3.92699081698724$$
$$x_{36} = -2.35619449019234$$
$$x_{46} = 0.785398163397448$$
$$x_{93} = 2.35619449019234$$
$$x_{83} = 3.92699081698724$$
$$x_{57} = 5.49778714378214$$
$$x_{77} = 8.63937979737193$$
$$x_{89} = 10.2101761241668$$
$$x_{88} = 11.7809724509617$$
$$x_{37} = 16.4933614313464$$
$$x_{8} = 18.0641577581413$$
$$x_{90} = 19.6349540849362$$
$$x_{82} = 22.776546738526$$
$$x_{49} = 24.3473430653209$$
$$x_{5} = 25.9181393921158$$
$$x_{26} = 27.4889357189107$$
$$x_{19} = 30.6305283725005$$
$$x_{25} = 32.2013246992954$$
$$x_{87} = 33.7721210260903$$
$$x_{24} = 38.484510006475$$
$$x_{59} = 40.0553063332699$$
$$x_{34} = 41.6261026600648$$
$$x_{31} = 44.7676953136546$$
$$x_{38} = 46.3384916404494$$
$$x_{48} = 47.9092879672443$$
$$x_{61} = 49.4800842940392$$
$$x_{16} = 52.621676947629$$
$$x_{42} = 54.1924732744239$$
$$x_{10} = 55.7632696012188$$
$$x_{70} = 60.4756585816035$$
$$x_{53} = 62.0464549083984$$
$$x_{41} = 63.6172512351933$$
$$x_{3} = 66.7588438887831$$
$$x_{51} = 68.329640215578$$
$$x_{18} = 69.9004365423729$$
$$x_{40} = 71.4712328691678$$
$$x_{56} = 74.6128255227576$$
$$x_{17} = 76.1836218495525$$
$$x_{72} = 77.7544181763474$$
$$x_{86} = 82.4668071567321$$
$$x_{28} = 84.037603483527$$
$$x_{20} = 85.6083998103219$$
$$x_{65} = 88.7499924639117$$
$$x_{96} = 90.3207887907066$$
$$x_{35} = 91.8915851175014$$
$$x_{6} = 93.4623814442964$$
$$x_{13} = 96.6039740978861$$
$$x_{30} = 98.174770424681$$
$$x_{66} = 99.7455667514759$$
$$x_{27} = 118.595122673015$$
$$x_{44} = 245.829625143401$$
son puntos de cambio del signo de desigualdad en las soluciones.
Primero definámonos con el signo hasta el punto extremo izquierdo:
$$x_{0} < x_{39}$$
Consideremos, por ejemplo, el punto
$$x_{0} = x_{39} - \frac{1}{10}$$
=
$$-241.117236163017 + - \frac{1}{10}$$
=
$$-241.217236163017$$
lo sustituimos en la expresión
$$\left|{\sin{\left(x \right)}}\right| < \frac{\sqrt{2}}{2}$$
$$\left|{\sin{\left(-241.217236163017 \right)}}\right| < \frac{\sqrt{2}}{2}$$
significa que una de las soluciones de nuestra ecuación será con:
$$x < -241.117236163017$$
Recibiremos otras soluciones de la desigualdad pasando al polo siguiente etc.
etc.
Respuesta:
$$x < -241.117236163017$$
$$x > -200.276531666349 \wedge x < -99.7455667514759$$
$$x > -98.174770424681 \wedge x < -93.4623814442964$$
$$x > -91.8915851175014 \wedge x < -90.3207887907066$$
$$x > -87.1791961371168 \wedge x < -85.6083998103219$$
$$x > -84.037603483527 \wedge x < -82.4668071567321$$
$$x > -79.3252145031423 \wedge x < -77.7544181763474$$
$$x > -76.1836218495525 \wedge x < -71.4712328691678$$
$$x > -69.9004365423729 \wedge x < -68.329640215578$$
$$x > -65.1880475619882 \wedge x < -63.6172512351933$$
$$x > -62.0464549083984 \wedge x < -60.4756585816035$$
$$x > -57.3340659280137 \wedge x < -55.7632696012188$$
$$x > -54.1924732744239 \wedge x < -49.4800842940392$$
$$x > -47.9092879672443 \wedge x < -46.3384916404494$$
$$x > -43.1968989868597 \wedge x < -41.6261026600648$$
$$x > -40.0553063332699 \wedge x < -38.484510006475$$
$$x > -35.3429173528852 \wedge x < -33.7721210260903$$
$$x > -32.2013246992954 \wedge x < -27.4889357189107$$
$$x > -25.9181393921158 \wedge x < -24.3473430653209$$
$$x > -22.776546738526 \wedge x < -21.2057504117311$$
$$x > -19.6349540849362 \wedge x < -18.0641577581413$$
$$x > -16.4933614313464 \wedge x < -13.3517687777566$$
$$x > -11.7809724509617 \wedge x < -10.2101761241668$$
$$x > -5.49778714378214 \wedge x < -3.92699081698724$$
$$x > -2.35619449019234 \wedge x < 0.785398163397448$$
$$x > 2.35619449019234 \wedge x < 3.92699081698724$$
$$x > 5.49778714378214 \wedge x < 8.63937979737193$$
$$x > 10.2101761241668 \wedge x < 11.7809724509617$$
$$x > 16.4933614313464 \wedge x < 18.0641577581413$$
$$x > 19.6349540849362 \wedge x < 22.776546738526$$
$$x > 24.3473430653209 \wedge x < 25.9181393921158$$
$$x > 27.4889357189107 \wedge x < 30.6305283725005$$
$$x > 32.2013246992954 \wedge x < 33.7721210260903$$
$$x > 38.484510006475 \wedge x < 40.0553063332699$$
$$x > 41.6261026600648 \wedge x < 44.7676953136546$$
$$x > 46.3384916404494 \wedge x < 47.9092879672443$$
$$x > 49.4800842940392 \wedge x < 52.621676947629$$
$$x > 54.1924732744239 \wedge x < 55.7632696012188$$
$$x > 60.4756585816035 \wedge x < 62.0464549083984$$
$$x > 63.6172512351933 \wedge x < 66.7588438887831$$
$$x > 68.329640215578 \wedge x < 69.9004365423729$$
$$x > 71.4712328691678 \wedge x < 74.6128255227576$$
$$x > 76.1836218495525 \wedge x < 77.7544181763474$$
$$x > 82.4668071567321 \wedge x < 84.037603483527$$
$$x > 85.6083998103219 \wedge x < 88.7499924639117$$
$$x > 90.3207887907066 \wedge x < 91.8915851175014$$
$$x > 93.4623814442964 \wedge x < 96.6039740978861$$
$$x > 98.174770424681 \wedge x < 99.7455667514759$$
$$x > 118.595122673015 \wedge x < 245.829625143401$$
$$\left|{\sin{\left(x \right)}}\right| < \frac{\sqrt{2}}{2}$$
Para resolver esta desigualdad primero hay que resolver la ecuación correspondiente:
$$\left|{\sin{\left(x \right)}}\right| = \frac{\sqrt{2}}{2}$$
Resolvemos:
Tenemos la ecuación
$$\left|{\sin{\left(x \right)}}\right| = \frac{\sqrt{2}}{2}$$
cambiamos
$$\left|{\sin{\left(x \right)}}\right| - 1 - \frac{\sqrt{2}}{2} = 0$$
$$\left|{\sin{\left(x \right)}}\right| - 1 - \frac{\sqrt{2}}{2} = 0$$
Sustituimos
$$w = \left|{\sin{\left(x \right)}}\right|$$
Abrimos los paréntesis en el miembro izquierdo de la ecuación
-1 + w - sqrt2/2 = 0
Transportamos los términos libres (sin w)
del miembro izquierdo al derecho, obtenemos:
$$w - \frac{\sqrt{2}}{2} = 1$$
Dividamos ambos miembros de la ecuación en (w - sqrt(2)/2)/w
w = 1 / ((w - sqrt(2)/2)/w)
Obtenemos la respuesta: w = 1 + sqrt(2)/2
hacemos cambio inverso
$$\left|{\sin{\left(x \right)}}\right| = w$$
sustituimos w:
$$x_{1} = -55.7632696012188$$
$$x_{2} = -98.174770424681$$
$$x_{3} = 66.7588438887831$$
$$x_{4} = -11.7809724509617$$
$$x_{5} = 25.9181393921158$$
$$x_{6} = 93.4623814442964$$
$$x_{7} = -40.0553063332699$$
$$x_{8} = 18.0641577581413$$
$$x_{9} = -13.3517687777566$$
$$x_{10} = 55.7632696012188$$
$$x_{11} = -16.4933614313464$$
$$x_{12} = -49.4800842940392$$
$$x_{13} = 96.6039740978861$$
$$x_{14} = -41.6261026600648$$
$$x_{15} = -22.776546738526$$
$$x_{16} = 52.621676947629$$
$$x_{17} = 76.1836218495525$$
$$x_{18} = 69.9004365423729$$
$$x_{19} = 30.6305283725005$$
$$x_{20} = 85.6083998103219$$
$$x_{21} = -19.6349540849362$$
$$x_{22} = -21.2057504117311$$
$$x_{23} = -3.92699081698724$$
$$x_{24} = 38.484510006475$$
$$x_{25} = 32.2013246992954$$
$$x_{26} = 27.4889357189107$$
$$x_{27} = 118.595122673015$$
$$x_{28} = 84.037603483527$$
$$x_{29} = -91.8915851175014$$
$$x_{30} = 98.174770424681$$
$$x_{31} = 44.7676953136546$$
$$x_{32} = -85.6083998103219$$
$$x_{33} = -87.1791961371168$$
$$x_{34} = 41.6261026600648$$
$$x_{35} = 91.8915851175014$$
$$x_{36} = -2.35619449019234$$
$$x_{37} = 16.4933614313464$$
$$x_{38} = 46.3384916404494$$
$$x_{39} = -241.117236163017$$
$$x_{40} = 71.4712328691678$$
$$x_{41} = 63.6172512351933$$
$$x_{42} = 54.1924732744239$$
$$x_{43} = -43.1968989868597$$
$$x_{44} = 245.829625143401$$
$$x_{45} = -84.037603483527$$
$$x_{46} = 0.785398163397448$$
$$x_{47} = -93.4623814442964$$
$$x_{48} = 47.9092879672443$$
$$x_{49} = 24.3473430653209$$
$$x_{50} = -5.49778714378214$$
$$x_{51} = 68.329640215578$$
$$x_{52} = -65.1880475619882$$
$$x_{53} = 62.0464549083984$$
$$x_{54} = -68.329640215578$$
$$x_{55} = -38.484510006475$$
$$x_{56} = 74.6128255227576$$
$$x_{57} = 5.49778714378214$$
$$x_{58} = -47.9092879672443$$
$$x_{59} = 40.0553063332699$$
$$x_{60} = -57.3340659280137$$
$$x_{61} = 49.4800842940392$$
$$x_{62} = -25.9181393921158$$
$$x_{63} = -76.1836218495525$$
$$x_{64} = -200.276531666349$$
$$x_{65} = 88.7499924639117$$
$$x_{66} = 99.7455667514759$$
$$x_{67} = -62.0464549083984$$
$$x_{68} = -82.4668071567321$$
$$x_{69} = -71.4712328691678$$
$$x_{70} = 60.4756585816035$$
$$x_{71} = -33.7721210260903$$
$$x_{72} = 77.7544181763474$$
$$x_{73} = -46.3384916404494$$
$$x_{74} = -63.6172512351933$$
$$x_{75} = -90.3207887907066$$
$$x_{76} = -35.3429173528852$$
$$x_{77} = 8.63937979737193$$
$$x_{78} = -10.2101761241668$$
$$x_{79} = -18.0641577581413$$
$$x_{80} = -27.4889357189107$$
$$x_{81} = -54.1924732744239$$
$$x_{82} = 22.776546738526$$
$$x_{83} = 3.92699081698724$$
$$x_{84} = -79.3252145031423$$
$$x_{85} = -99.7455667514759$$
$$x_{86} = 82.4668071567321$$
$$x_{87} = 33.7721210260903$$
$$x_{88} = 11.7809724509617$$
$$x_{89} = 10.2101761241668$$
$$x_{90} = 19.6349540849362$$
$$x_{91} = -77.7544181763474$$
$$x_{92} = -69.9004365423729$$
$$x_{93} = 2.35619449019234$$
$$x_{94} = -24.3473430653209$$
$$x_{95} = -60.4756585816035$$
$$x_{96} = 90.3207887907066$$
$$x_{97} = -32.2013246992954$$
$$x_{1} = -55.7632696012188$$
$$x_{2} = -98.174770424681$$
$$x_{3} = 66.7588438887831$$
$$x_{4} = -11.7809724509617$$
$$x_{5} = 25.9181393921158$$
$$x_{6} = 93.4623814442964$$
$$x_{7} = -40.0553063332699$$
$$x_{8} = 18.0641577581413$$
$$x_{9} = -13.3517687777566$$
$$x_{10} = 55.7632696012188$$
$$x_{11} = -16.4933614313464$$
$$x_{12} = -49.4800842940392$$
$$x_{13} = 96.6039740978861$$
$$x_{14} = -41.6261026600648$$
$$x_{15} = -22.776546738526$$
$$x_{16} = 52.621676947629$$
$$x_{17} = 76.1836218495525$$
$$x_{18} = 69.9004365423729$$
$$x_{19} = 30.6305283725005$$
$$x_{20} = 85.6083998103219$$
$$x_{21} = -19.6349540849362$$
$$x_{22} = -21.2057504117311$$
$$x_{23} = -3.92699081698724$$
$$x_{24} = 38.484510006475$$
$$x_{25} = 32.2013246992954$$
$$x_{26} = 27.4889357189107$$
$$x_{27} = 118.595122673015$$
$$x_{28} = 84.037603483527$$
$$x_{29} = -91.8915851175014$$
$$x_{30} = 98.174770424681$$
$$x_{31} = 44.7676953136546$$
$$x_{32} = -85.6083998103219$$
$$x_{33} = -87.1791961371168$$
$$x_{34} = 41.6261026600648$$
$$x_{35} = 91.8915851175014$$
$$x_{36} = -2.35619449019234$$
$$x_{37} = 16.4933614313464$$
$$x_{38} = 46.3384916404494$$
$$x_{39} = -241.117236163017$$
$$x_{40} = 71.4712328691678$$
$$x_{41} = 63.6172512351933$$
$$x_{42} = 54.1924732744239$$
$$x_{43} = -43.1968989868597$$
$$x_{44} = 245.829625143401$$
$$x_{45} = -84.037603483527$$
$$x_{46} = 0.785398163397448$$
$$x_{47} = -93.4623814442964$$
$$x_{48} = 47.9092879672443$$
$$x_{49} = 24.3473430653209$$
$$x_{50} = -5.49778714378214$$
$$x_{51} = 68.329640215578$$
$$x_{52} = -65.1880475619882$$
$$x_{53} = 62.0464549083984$$
$$x_{54} = -68.329640215578$$
$$x_{55} = -38.484510006475$$
$$x_{56} = 74.6128255227576$$
$$x_{57} = 5.49778714378214$$
$$x_{58} = -47.9092879672443$$
$$x_{59} = 40.0553063332699$$
$$x_{60} = -57.3340659280137$$
$$x_{61} = 49.4800842940392$$
$$x_{62} = -25.9181393921158$$
$$x_{63} = -76.1836218495525$$
$$x_{64} = -200.276531666349$$
$$x_{65} = 88.7499924639117$$
$$x_{66} = 99.7455667514759$$
$$x_{67} = -62.0464549083984$$
$$x_{68} = -82.4668071567321$$
$$x_{69} = -71.4712328691678$$
$$x_{70} = 60.4756585816035$$
$$x_{71} = -33.7721210260903$$
$$x_{72} = 77.7544181763474$$
$$x_{73} = -46.3384916404494$$
$$x_{74} = -63.6172512351933$$
$$x_{75} = -90.3207887907066$$
$$x_{76} = -35.3429173528852$$
$$x_{77} = 8.63937979737193$$
$$x_{78} = -10.2101761241668$$
$$x_{79} = -18.0641577581413$$
$$x_{80} = -27.4889357189107$$
$$x_{81} = -54.1924732744239$$
$$x_{82} = 22.776546738526$$
$$x_{83} = 3.92699081698724$$
$$x_{84} = -79.3252145031423$$
$$x_{85} = -99.7455667514759$$
$$x_{86} = 82.4668071567321$$
$$x_{87} = 33.7721210260903$$
$$x_{88} = 11.7809724509617$$
$$x_{89} = 10.2101761241668$$
$$x_{90} = 19.6349540849362$$
$$x_{91} = -77.7544181763474$$
$$x_{92} = -69.9004365423729$$
$$x_{93} = 2.35619449019234$$
$$x_{94} = -24.3473430653209$$
$$x_{95} = -60.4756585816035$$
$$x_{96} = 90.3207887907066$$
$$x_{97} = -32.2013246992954$$
Las raíces dadas
$$x_{39} = -241.117236163017$$
$$x_{64} = -200.276531666349$$
$$x_{85} = -99.7455667514759$$
$$x_{2} = -98.174770424681$$
$$x_{47} = -93.4623814442964$$
$$x_{29} = -91.8915851175014$$
$$x_{75} = -90.3207887907066$$
$$x_{33} = -87.1791961371168$$
$$x_{32} = -85.6083998103219$$
$$x_{45} = -84.037603483527$$
$$x_{68} = -82.4668071567321$$
$$x_{84} = -79.3252145031423$$
$$x_{91} = -77.7544181763474$$
$$x_{63} = -76.1836218495525$$
$$x_{69} = -71.4712328691678$$
$$x_{92} = -69.9004365423729$$
$$x_{54} = -68.329640215578$$
$$x_{52} = -65.1880475619882$$
$$x_{74} = -63.6172512351933$$
$$x_{67} = -62.0464549083984$$
$$x_{95} = -60.4756585816035$$
$$x_{60} = -57.3340659280137$$
$$x_{1} = -55.7632696012188$$
$$x_{81} = -54.1924732744239$$
$$x_{12} = -49.4800842940392$$
$$x_{58} = -47.9092879672443$$
$$x_{73} = -46.3384916404494$$
$$x_{43} = -43.1968989868597$$
$$x_{14} = -41.6261026600648$$
$$x_{7} = -40.0553063332699$$
$$x_{55} = -38.484510006475$$
$$x_{76} = -35.3429173528852$$
$$x_{71} = -33.7721210260903$$
$$x_{97} = -32.2013246992954$$
$$x_{80} = -27.4889357189107$$
$$x_{62} = -25.9181393921158$$
$$x_{94} = -24.3473430653209$$
$$x_{15} = -22.776546738526$$
$$x_{22} = -21.2057504117311$$
$$x_{21} = -19.6349540849362$$
$$x_{79} = -18.0641577581413$$
$$x_{11} = -16.4933614313464$$
$$x_{9} = -13.3517687777566$$
$$x_{4} = -11.7809724509617$$
$$x_{78} = -10.2101761241668$$
$$x_{50} = -5.49778714378214$$
$$x_{23} = -3.92699081698724$$
$$x_{36} = -2.35619449019234$$
$$x_{46} = 0.785398163397448$$
$$x_{93} = 2.35619449019234$$
$$x_{83} = 3.92699081698724$$
$$x_{57} = 5.49778714378214$$
$$x_{77} = 8.63937979737193$$
$$x_{89} = 10.2101761241668$$
$$x_{88} = 11.7809724509617$$
$$x_{37} = 16.4933614313464$$
$$x_{8} = 18.0641577581413$$
$$x_{90} = 19.6349540849362$$
$$x_{82} = 22.776546738526$$
$$x_{49} = 24.3473430653209$$
$$x_{5} = 25.9181393921158$$
$$x_{26} = 27.4889357189107$$
$$x_{19} = 30.6305283725005$$
$$x_{25} = 32.2013246992954$$
$$x_{87} = 33.7721210260903$$
$$x_{24} = 38.484510006475$$
$$x_{59} = 40.0553063332699$$
$$x_{34} = 41.6261026600648$$
$$x_{31} = 44.7676953136546$$
$$x_{38} = 46.3384916404494$$
$$x_{48} = 47.9092879672443$$
$$x_{61} = 49.4800842940392$$
$$x_{16} = 52.621676947629$$
$$x_{42} = 54.1924732744239$$
$$x_{10} = 55.7632696012188$$
$$x_{70} = 60.4756585816035$$
$$x_{53} = 62.0464549083984$$
$$x_{41} = 63.6172512351933$$
$$x_{3} = 66.7588438887831$$
$$x_{51} = 68.329640215578$$
$$x_{18} = 69.9004365423729$$
$$x_{40} = 71.4712328691678$$
$$x_{56} = 74.6128255227576$$
$$x_{17} = 76.1836218495525$$
$$x_{72} = 77.7544181763474$$
$$x_{86} = 82.4668071567321$$
$$x_{28} = 84.037603483527$$
$$x_{20} = 85.6083998103219$$
$$x_{65} = 88.7499924639117$$
$$x_{96} = 90.3207887907066$$
$$x_{35} = 91.8915851175014$$
$$x_{6} = 93.4623814442964$$
$$x_{13} = 96.6039740978861$$
$$x_{30} = 98.174770424681$$
$$x_{66} = 99.7455667514759$$
$$x_{27} = 118.595122673015$$
$$x_{44} = 245.829625143401$$
son puntos de cambio del signo de desigualdad en las soluciones.
Primero definámonos con el signo hasta el punto extremo izquierdo:
$$x_{0} < x_{39}$$
Consideremos, por ejemplo, el punto
$$x_{0} = x_{39} - \frac{1}{10}$$
=
$$-241.117236163017 + - \frac{1}{10}$$
=
$$-241.217236163017$$
lo sustituimos en la expresión
$$\left|{\sin{\left(x \right)}}\right| < \frac{\sqrt{2}}{2}$$
$$\left|{\sin{\left(-241.217236163017 \right)}}\right| < \frac{\sqrt{2}}{2}$$
___
\/ 2
0.632981306676965 < -----
2
significa que una de las soluciones de nuestra ecuación será con:
$$x < -241.117236163017$$
_____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____
\ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \
-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------
x39 x64 x85 x2 x47 x29 x75 x33 x32 x45 x68 x84 x91 x63 x69 x92 x54 x52 x74 x67 x95 x60 x1 x81 x12 x58 x73 x43 x14 x7 x55 x76 x71 x97 x80 x62 x94 x15 x22 x21 x79 x11 x9 x4 x78 x50 x23 x36 x46 x93 x83 x57 x77 x89 x88 x37 x8 x90 x82 x49 x5 x26 x19 x25 x87 x24 x59 x34 x31 x38 x48 x61 x16 x42 x10 x70 x53 x41 x3 x51 x18 x40 x56 x17 x72 x86 x28 x20 x65 x96 x35 x6 x13 x30 x66 x27 x44Recibiremos otras soluciones de la desigualdad pasando al polo siguiente etc.
etc.
Respuesta:
$$x < -241.117236163017$$
$$x > -200.276531666349 \wedge x < -99.7455667514759$$
$$x > -98.174770424681 \wedge x < -93.4623814442964$$
$$x > -91.8915851175014 \wedge x < -90.3207887907066$$
$$x > -87.1791961371168 \wedge x < -85.6083998103219$$
$$x > -84.037603483527 \wedge x < -82.4668071567321$$
$$x > -79.3252145031423 \wedge x < -77.7544181763474$$
$$x > -76.1836218495525 \wedge x < -71.4712328691678$$
$$x > -69.9004365423729 \wedge x < -68.329640215578$$
$$x > -65.1880475619882 \wedge x < -63.6172512351933$$
$$x > -62.0464549083984 \wedge x < -60.4756585816035$$
$$x > -57.3340659280137 \wedge x < -55.7632696012188$$
$$x > -54.1924732744239 \wedge x < -49.4800842940392$$
$$x > -47.9092879672443 \wedge x < -46.3384916404494$$
$$x > -43.1968989868597 \wedge x < -41.6261026600648$$
$$x > -40.0553063332699 \wedge x < -38.484510006475$$
$$x > -35.3429173528852 \wedge x < -33.7721210260903$$
$$x > -32.2013246992954 \wedge x < -27.4889357189107$$
$$x > -25.9181393921158 \wedge x < -24.3473430653209$$
$$x > -22.776546738526 \wedge x < -21.2057504117311$$
$$x > -19.6349540849362 \wedge x < -18.0641577581413$$
$$x > -16.4933614313464 \wedge x < -13.3517687777566$$
$$x > -11.7809724509617 \wedge x < -10.2101761241668$$
$$x > -5.49778714378214 \wedge x < -3.92699081698724$$
$$x > -2.35619449019234 \wedge x < 0.785398163397448$$
$$x > 2.35619449019234 \wedge x < 3.92699081698724$$
$$x > 5.49778714378214 \wedge x < 8.63937979737193$$
$$x > 10.2101761241668 \wedge x < 11.7809724509617$$
$$x > 16.4933614313464 \wedge x < 18.0641577581413$$
$$x > 19.6349540849362 \wedge x < 22.776546738526$$
$$x > 24.3473430653209 \wedge x < 25.9181393921158$$
$$x > 27.4889357189107 \wedge x < 30.6305283725005$$
$$x > 32.2013246992954 \wedge x < 33.7721210260903$$
$$x > 38.484510006475 \wedge x < 40.0553063332699$$
$$x > 41.6261026600648 \wedge x < 44.7676953136546$$
$$x > 46.3384916404494 \wedge x < 47.9092879672443$$
$$x > 49.4800842940392 \wedge x < 52.621676947629$$
$$x > 54.1924732744239 \wedge x < 55.7632696012188$$
$$x > 60.4756585816035 \wedge x < 62.0464549083984$$
$$x > 63.6172512351933 \wedge x < 66.7588438887831$$
$$x > 68.329640215578 \wedge x < 69.9004365423729$$
$$x > 71.4712328691678 \wedge x < 74.6128255227576$$
$$x > 76.1836218495525 \wedge x < 77.7544181763474$$
$$x > 82.4668071567321 \wedge x < 84.037603483527$$
$$x > 85.6083998103219 \wedge x < 88.7499924639117$$
$$x > 90.3207887907066 \wedge x < 91.8915851175014$$
$$x > 93.4623814442964 \wedge x < 96.6039740978861$$
$$x > 98.174770424681 \wedge x < 99.7455667514759$$
$$x > 118.595122673015 \wedge x < 245.829625143401$$
Solución de la desigualdad en el gráfico
Gráfico