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