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