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|sin(x)|≥(sqrt(2))*1/2 desigualdades

En la desigualdad la incógnita

Solución

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|sin(x)| >= -----
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$$\left|{\sin{\left(x \right)}}\right| \geq \frac{\sqrt{2}}{2}$$
Abs(sin(x)) >= sqrt(2)/2
Solución detallada
Se da la desigualdad:
$$\left|{\sin{\left(x \right)}}\right| \geq \frac{\sqrt{2}}{2}$$
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:
$$x_{1} = -41.6261026600648$$
$$x_{2} = 18.0641577581413$$
$$x_{3} = 25.9181393921158$$
$$x_{4} = 41.6261026600648$$
$$x_{5} = 40.0553063332699$$
$$x_{6} = 63.6172512351933$$
$$x_{7} = -47.9092879672443$$
$$x_{8} = 88.7499924639117$$
$$x_{9} = -22.776546738526$$
$$x_{10} = 96.6039740978861$$
$$x_{11} = 82.4668071567321$$
$$x_{12} = -13.3517687777566$$
$$x_{13} = -62.0464549083984$$
$$x_{14} = 44.7676953136546$$
$$x_{15} = -200.276531666349$$
$$x_{16} = 0.785398163397448$$
$$x_{17} = -24.3473430653209$$
$$x_{18} = -16.4933614313464$$
$$x_{19} = 19.6349540849362$$
$$x_{20} = -3.92699081698724$$
$$x_{21} = 52.621676947629$$
$$x_{22} = -84.037603483527$$
$$x_{23} = -49.4800842940392$$
$$x_{24} = -63.6172512351933$$
$$x_{25} = 55.7632696012188$$
$$x_{26} = 91.8915851175014$$
$$x_{27} = -87.1791961371168$$
$$x_{28} = 93.4623814442964$$
$$x_{29} = 60.4756585816035$$
$$x_{30} = -10.2101761241668$$
$$x_{31} = 71.4712328691678$$
$$x_{32} = 32.2013246992954$$
$$x_{33} = -85.6083998103219$$
$$x_{34} = 33.7721210260903$$
$$x_{35} = -57.3340659280137$$
$$x_{36} = -32.2013246992954$$
$$x_{37} = 30.6305283725005$$
$$x_{38} = -82.4668071567321$$
$$x_{39} = 3.92699081698724$$
$$x_{40} = 2.35619449019234$$
$$x_{41} = 16.4933614313464$$
$$x_{42} = 76.1836218495525$$
$$x_{43} = -2.35619449019234$$
$$x_{44} = -21.2057504117311$$
$$x_{45} = -90.3207887907066$$
$$x_{46} = -46.3384916404494$$
$$x_{47} = -98.174770424681$$
$$x_{48} = 46.3384916404494$$
$$x_{49} = 5.49778714378214$$
$$x_{50} = 47.9092879672443$$
$$x_{51} = -43.1968989868597$$
$$x_{52} = 99.7455667514759$$
$$x_{53} = -91.8915851175014$$
$$x_{54} = -33.7721210260903$$
$$x_{55} = -19.6349540849362$$
$$x_{56} = -60.4756585816035$$
$$x_{57} = 27.4889357189107$$
$$x_{58} = -71.4712328691678$$
$$x_{59} = 84.037603483527$$
$$x_{60} = -76.1836218495525$$
$$x_{61} = -38.484510006475$$
$$x_{62} = 22.776546738526$$
$$x_{63} = 68.329640215578$$
$$x_{64} = 66.7588438887831$$
$$x_{65} = -241.117236163017$$
$$x_{66} = -11.7809724509617$$
$$x_{67} = -27.4889357189107$$
$$x_{68} = 62.0464549083984$$
$$x_{69} = -25.9181393921158$$
$$x_{70} = -40.0553063332699$$
$$x_{71} = -65.1880475619882$$
$$x_{72} = -5.49778714378214$$
$$x_{73} = 245.829625143401$$
$$x_{74} = 54.1924732744239$$
$$x_{75} = -35.3429173528852$$
$$x_{76} = 90.3207887907066$$
$$x_{77} = -55.7632696012188$$
$$x_{78} = 11.7809724509617$$
$$x_{79} = 8.63937979737193$$
$$x_{80} = 74.6128255227576$$
$$x_{81} = -54.1924732744239$$
$$x_{82} = -99.7455667514759$$
$$x_{83} = -18.0641577581413$$
$$x_{84} = 38.484510006475$$
$$x_{85} = -77.7544181763474$$
$$x_{86} = -79.3252145031423$$
$$x_{87} = 10.2101761241668$$
$$x_{88} = 85.6083998103219$$
$$x_{89} = 98.174770424681$$
$$x_{90} = -68.329640215578$$
$$x_{91} = 118.595122673015$$
$$x_{92} = -69.9004365423729$$
$$x_{93} = 49.4800842940392$$
$$x_{94} = 69.9004365423729$$
$$x_{95} = 24.3473430653209$$
$$x_{96} = -93.4623814442964$$
$$x_{97} = 77.7544181763474$$
$$x_{1} = -41.6261026600648$$
$$x_{2} = 18.0641577581413$$
$$x_{3} = 25.9181393921158$$
$$x_{4} = 41.6261026600648$$
$$x_{5} = 40.0553063332699$$
$$x_{6} = 63.6172512351933$$
$$x_{7} = -47.9092879672443$$
$$x_{8} = 88.7499924639117$$
$$x_{9} = -22.776546738526$$
$$x_{10} = 96.6039740978861$$
$$x_{11} = 82.4668071567321$$
$$x_{12} = -13.3517687777566$$
$$x_{13} = -62.0464549083984$$
$$x_{14} = 44.7676953136546$$
$$x_{15} = -200.276531666349$$
$$x_{16} = 0.785398163397448$$
$$x_{17} = -24.3473430653209$$
$$x_{18} = -16.4933614313464$$
$$x_{19} = 19.6349540849362$$
$$x_{20} = -3.92699081698724$$
$$x_{21} = 52.621676947629$$
$$x_{22} = -84.037603483527$$
$$x_{23} = -49.4800842940392$$
$$x_{24} = -63.6172512351933$$
$$x_{25} = 55.7632696012188$$
$$x_{26} = 91.8915851175014$$
$$x_{27} = -87.1791961371168$$
$$x_{28} = 93.4623814442964$$
$$x_{29} = 60.4756585816035$$
$$x_{30} = -10.2101761241668$$
$$x_{31} = 71.4712328691678$$
$$x_{32} = 32.2013246992954$$
$$x_{33} = -85.6083998103219$$
$$x_{34} = 33.7721210260903$$
$$x_{35} = -57.3340659280137$$
$$x_{36} = -32.2013246992954$$
$$x_{37} = 30.6305283725005$$
$$x_{38} = -82.4668071567321$$
$$x_{39} = 3.92699081698724$$
$$x_{40} = 2.35619449019234$$
$$x_{41} = 16.4933614313464$$
$$x_{42} = 76.1836218495525$$
$$x_{43} = -2.35619449019234$$
$$x_{44} = -21.2057504117311$$
$$x_{45} = -90.3207887907066$$
$$x_{46} = -46.3384916404494$$
$$x_{47} = -98.174770424681$$
$$x_{48} = 46.3384916404494$$
$$x_{49} = 5.49778714378214$$
$$x_{50} = 47.9092879672443$$
$$x_{51} = -43.1968989868597$$
$$x_{52} = 99.7455667514759$$
$$x_{53} = -91.8915851175014$$
$$x_{54} = -33.7721210260903$$
$$x_{55} = -19.6349540849362$$
$$x_{56} = -60.4756585816035$$
$$x_{57} = 27.4889357189107$$
$$x_{58} = -71.4712328691678$$
$$x_{59} = 84.037603483527$$
$$x_{60} = -76.1836218495525$$
$$x_{61} = -38.484510006475$$
$$x_{62} = 22.776546738526$$
$$x_{63} = 68.329640215578$$
$$x_{64} = 66.7588438887831$$
$$x_{65} = -241.117236163017$$
$$x_{66} = -11.7809724509617$$
$$x_{67} = -27.4889357189107$$
$$x_{68} = 62.0464549083984$$
$$x_{69} = -25.9181393921158$$
$$x_{70} = -40.0553063332699$$
$$x_{71} = -65.1880475619882$$
$$x_{72} = -5.49778714378214$$
$$x_{73} = 245.829625143401$$
$$x_{74} = 54.1924732744239$$
$$x_{75} = -35.3429173528852$$
$$x_{76} = 90.3207887907066$$
$$x_{77} = -55.7632696012188$$
$$x_{78} = 11.7809724509617$$
$$x_{79} = 8.63937979737193$$
$$x_{80} = 74.6128255227576$$
$$x_{81} = -54.1924732744239$$
$$x_{82} = -99.7455667514759$$
$$x_{83} = -18.0641577581413$$
$$x_{84} = 38.484510006475$$
$$x_{85} = -77.7544181763474$$
$$x_{86} = -79.3252145031423$$
$$x_{87} = 10.2101761241668$$
$$x_{88} = 85.6083998103219$$
$$x_{89} = 98.174770424681$$
$$x_{90} = -68.329640215578$$
$$x_{91} = 118.595122673015$$
$$x_{92} = -69.9004365423729$$
$$x_{93} = 49.4800842940392$$
$$x_{94} = 69.9004365423729$$
$$x_{95} = 24.3473430653209$$
$$x_{96} = -93.4623814442964$$
$$x_{97} = 77.7544181763474$$
Las raíces dadas
$$x_{65} = -241.117236163017$$
$$x_{15} = -200.276531666349$$
$$x_{82} = -99.7455667514759$$
$$x_{47} = -98.174770424681$$
$$x_{96} = -93.4623814442964$$
$$x_{53} = -91.8915851175014$$
$$x_{45} = -90.3207887907066$$
$$x_{27} = -87.1791961371168$$
$$x_{33} = -85.6083998103219$$
$$x_{22} = -84.037603483527$$
$$x_{38} = -82.4668071567321$$
$$x_{86} = -79.3252145031423$$
$$x_{85} = -77.7544181763474$$
$$x_{60} = -76.1836218495525$$
$$x_{58} = -71.4712328691678$$
$$x_{92} = -69.9004365423729$$
$$x_{90} = -68.329640215578$$
$$x_{71} = -65.1880475619882$$
$$x_{24} = -63.6172512351933$$
$$x_{13} = -62.0464549083984$$
$$x_{56} = -60.4756585816035$$
$$x_{35} = -57.3340659280137$$
$$x_{77} = -55.7632696012188$$
$$x_{81} = -54.1924732744239$$
$$x_{23} = -49.4800842940392$$
$$x_{7} = -47.9092879672443$$
$$x_{46} = -46.3384916404494$$
$$x_{51} = -43.1968989868597$$
$$x_{1} = -41.6261026600648$$
$$x_{70} = -40.0553063332699$$
$$x_{61} = -38.484510006475$$
$$x_{75} = -35.3429173528852$$
$$x_{54} = -33.7721210260903$$
$$x_{36} = -32.2013246992954$$
$$x_{67} = -27.4889357189107$$
$$x_{69} = -25.9181393921158$$
$$x_{17} = -24.3473430653209$$
$$x_{9} = -22.776546738526$$
$$x_{44} = -21.2057504117311$$
$$x_{55} = -19.6349540849362$$
$$x_{83} = -18.0641577581413$$
$$x_{18} = -16.4933614313464$$
$$x_{12} = -13.3517687777566$$
$$x_{66} = -11.7809724509617$$
$$x_{30} = -10.2101761241668$$
$$x_{72} = -5.49778714378214$$
$$x_{20} = -3.92699081698724$$
$$x_{43} = -2.35619449019234$$
$$x_{16} = 0.785398163397448$$
$$x_{40} = 2.35619449019234$$
$$x_{39} = 3.92699081698724$$
$$x_{49} = 5.49778714378214$$
$$x_{79} = 8.63937979737193$$
$$x_{87} = 10.2101761241668$$
$$x_{78} = 11.7809724509617$$
$$x_{41} = 16.4933614313464$$
$$x_{2} = 18.0641577581413$$
$$x_{19} = 19.6349540849362$$
$$x_{62} = 22.776546738526$$
$$x_{95} = 24.3473430653209$$
$$x_{3} = 25.9181393921158$$
$$x_{57} = 27.4889357189107$$
$$x_{37} = 30.6305283725005$$
$$x_{32} = 32.2013246992954$$
$$x_{34} = 33.7721210260903$$
$$x_{84} = 38.484510006475$$
$$x_{5} = 40.0553063332699$$
$$x_{4} = 41.6261026600648$$
$$x_{14} = 44.7676953136546$$
$$x_{48} = 46.3384916404494$$
$$x_{50} = 47.9092879672443$$
$$x_{93} = 49.4800842940392$$
$$x_{21} = 52.621676947629$$
$$x_{74} = 54.1924732744239$$
$$x_{25} = 55.7632696012188$$
$$x_{29} = 60.4756585816035$$
$$x_{68} = 62.0464549083984$$
$$x_{6} = 63.6172512351933$$
$$x_{64} = 66.7588438887831$$
$$x_{63} = 68.329640215578$$
$$x_{94} = 69.9004365423729$$
$$x_{31} = 71.4712328691678$$
$$x_{80} = 74.6128255227576$$
$$x_{42} = 76.1836218495525$$
$$x_{97} = 77.7544181763474$$
$$x_{11} = 82.4668071567321$$
$$x_{59} = 84.037603483527$$
$$x_{88} = 85.6083998103219$$
$$x_{8} = 88.7499924639117$$
$$x_{76} = 90.3207887907066$$
$$x_{26} = 91.8915851175014$$
$$x_{28} = 93.4623814442964$$
$$x_{10} = 96.6039740978861$$
$$x_{89} = 98.174770424681$$
$$x_{52} = 99.7455667514759$$
$$x_{91} = 118.595122673015$$
$$x_{73} = 245.829625143401$$
son puntos de cambio del signo de desigualdad en las soluciones.
Primero definámonos con el signo hasta el punto extremo izquierdo:
$$x_{0} \leq x_{65}$$
Consideremos, por ejemplo, el punto
$$x_{0} = x_{65} - \frac{1}{10}$$
=
$$-241.117236163017 + - \frac{1}{10}$$
=
$$-241.217236163017$$
lo sustituimos en la expresión
$$\left|{\sin{\left(x \right)}}\right| \geq \frac{\sqrt{2}}{2}$$
$$\left|{\sin{\left(-241.217236163017 \right)}}\right| \geq \frac{\sqrt{2}}{2}$$
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0.632981306676965 >= -----
                       2  
                     

pero
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0.632981306676965 < -----
                      2  
                    

Entonces
$$x \leq -241.117236163017$$
no se cumple
significa que una de las soluciones de nuestra ecuación será con:
$$x \geq -241.117236163017 \wedge x \leq -200.276531666349$$
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       x65      x15      x82      x47      x96      x53      x45      x27      x33      x22      x38      x86      x85      x60      x58      x92      x90      x71      x24      x13      x56      x35      x77      x81      x23      x7      x46      x51      x1      x70      x61      x75      x54      x36      x67      x69      x17      x9      x44      x55      x83      x18      x12      x66      x30      x72      x20      x43      x16      x40      x39      x49      x79      x87      x78      x41      x2      x19      x62      x95      x3      x57      x37      x32      x34      x84      x5      x4      x14      x48      x50      x93      x21      x74      x25      x29      x68      x6      x64      x63      x94      x31      x80      x42      x97      x11      x59      x88      x8      x76      x26      x28      x10      x89      x52      x91      x73

Recibiremos otras soluciones de la desigualdad pasando al polo siguiente etc.
etc.
Respuesta:
$$x \geq -241.117236163017 \wedge x \leq -200.276531666349$$
$$x \geq -99.7455667514759 \wedge x \leq -98.174770424681$$
$$x \geq -93.4623814442964 \wedge x \leq -91.8915851175014$$
$$x \geq -90.3207887907066 \wedge x \leq -87.1791961371168$$
$$x \geq -85.6083998103219 \wedge x \leq -84.037603483527$$
$$x \geq -82.4668071567321 \wedge x \leq -79.3252145031423$$
$$x \geq -77.7544181763474 \wedge x \leq -76.1836218495525$$
$$x \geq -71.4712328691678 \wedge x \leq -69.9004365423729$$
$$x \geq -68.329640215578 \wedge x \leq -65.1880475619882$$
$$x \geq -63.6172512351933 \wedge x \leq -62.0464549083984$$
$$x \geq -60.4756585816035 \wedge x \leq -57.3340659280137$$
$$x \geq -55.7632696012188 \wedge x \leq -54.1924732744239$$
$$x \geq -49.4800842940392 \wedge x \leq -47.9092879672443$$
$$x \geq -46.3384916404494 \wedge x \leq -43.1968989868597$$
$$x \geq -41.6261026600648 \wedge x \leq -40.0553063332699$$
$$x \geq -38.484510006475 \wedge x \leq -35.3429173528852$$
$$x \geq -33.7721210260903 \wedge x \leq -32.2013246992954$$
$$x \geq -27.4889357189107 \wedge x \leq -25.9181393921158$$
$$x \geq -24.3473430653209 \wedge x \leq -22.776546738526$$
$$x \geq -21.2057504117311 \wedge x \leq -19.6349540849362$$
$$x \geq -18.0641577581413 \wedge x \leq -16.4933614313464$$
$$x \geq -13.3517687777566 \wedge x \leq -11.7809724509617$$
$$x \geq -10.2101761241668 \wedge x \leq -5.49778714378214$$
$$x \geq -3.92699081698724 \wedge x \leq -2.35619449019234$$
$$x \geq 0.785398163397448 \wedge x \leq 2.35619449019234$$
$$x \geq 3.92699081698724 \wedge x \leq 5.49778714378214$$
$$x \geq 8.63937979737193 \wedge x \leq 10.2101761241668$$
$$x \geq 11.7809724509617 \wedge x \leq 16.4933614313464$$
$$x \geq 18.0641577581413 \wedge x \leq 19.6349540849362$$
$$x \geq 22.776546738526 \wedge x \leq 24.3473430653209$$
$$x \geq 25.9181393921158 \wedge x \leq 27.4889357189107$$
$$x \geq 30.6305283725005 \wedge x \leq 32.2013246992954$$
$$x \geq 33.7721210260903 \wedge x \leq 38.484510006475$$
$$x \geq 40.0553063332699 \wedge x \leq 41.6261026600648$$
$$x \geq 44.7676953136546 \wedge x \leq 46.3384916404494$$
$$x \geq 47.9092879672443 \wedge x \leq 49.4800842940392$$
$$x \geq 52.621676947629 \wedge x \leq 54.1924732744239$$
$$x \geq 55.7632696012188 \wedge x \leq 60.4756585816035$$
$$x \geq 62.0464549083984 \wedge x \leq 63.6172512351933$$
$$x \geq 66.7588438887831 \wedge x \leq 68.329640215578$$
$$x \geq 69.9004365423729 \wedge x \leq 71.4712328691678$$
$$x \geq 74.6128255227576 \wedge x \leq 76.1836218495525$$
$$x \geq 77.7544181763474 \wedge x \leq 82.4668071567321$$
$$x \geq 84.037603483527 \wedge x \leq 85.6083998103219$$
$$x \geq 88.7499924639117 \wedge x \leq 90.3207887907066$$
$$x \geq 91.8915851175014 \wedge x \leq 93.4623814442964$$
$$x \geq 96.6039740978861 \wedge x \leq 98.174770424681$$
$$x \geq 99.7455667514759 \wedge x \leq 118.595122673015$$
$$x \geq 245.829625143401$$
Solución de la desigualdad en el gráfico