Se da la desigualdad:
∣ cos ( x ) ∣ ≥ 2 2 \left|{\cos{\left(x \right)}}\right| \geq \frac{\sqrt{2}}{2} ∣ cos ( x ) ∣ ≥ 2 2 Para resolver esta desigualdad primero hay que resolver la ecuación correspondiente:
∣ cos ( x ) ∣ = 2 2 \left|{\cos{\left(x \right)}}\right| = \frac{\sqrt{2}}{2} ∣ cos ( x ) ∣ = 2 2 Resolvemos:
Tenemos la ecuación
∣ cos ( x ) ∣ = 2 2 \left|{\cos{\left(x \right)}}\right| = \frac{\sqrt{2}}{2} ∣ cos ( x ) ∣ = 2 2 cambiamos
∣ cos ( x ) ∣ − 1 − 2 2 = 0 \left|{\cos{\left(x \right)}}\right| - 1 - \frac{\sqrt{2}}{2} = 0 ∣ cos ( x ) ∣ − 1 − 2 2 = 0 ∣ cos ( x ) ∣ − 1 − 2 2 = 0 \left|{\cos{\left(x \right)}}\right| - 1 - \frac{\sqrt{2}}{2} = 0 ∣ cos ( x ) ∣ − 1 − 2 2 = 0 Sustituimos
w = ∣ cos ( x ) ∣ w = \left|{\cos{\left(x \right)}}\right| w = ∣ cos ( 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
∣ cos ( x ) ∣ = w \left|{\cos{\left(x \right)}}\right| = w ∣ cos ( x ) ∣ = w sustituimos w:
x 1 = − 74.6128255227576 x_{1} = -74.6128255227576 x 1 = − 74.6128255227576 x 2 = 55.7632696012188 x_{2} = 55.7632696012188 x 2 = 55.7632696012188 x 3 = − 33.7721210260903 x_{3} = -33.7721210260903 x 3 = − 33.7721210260903 x 4 = 3.92699081698724 x_{4} = 3.92699081698724 x 4 = 3.92699081698724 x 5 = − 96.6039740978861 x_{5} = -96.6039740978861 x 5 = − 96.6039740978861 x 6 = − 16.4933614313464 x_{6} = -16.4933614313464 x 6 = − 16.4933614313464 x 7 = − 47.9092879672443 x_{7} = -47.9092879672443 x 7 = − 47.9092879672443 x 8 = 71.4712328691678 x_{8} = 71.4712328691678 x 8 = 71.4712328691678 x 9 = 46.3384916404494 x_{9} = 46.3384916404494 x 9 = 46.3384916404494 x 10 = 16.4933614313464 x_{10} = 16.4933614313464 x 10 = 16.4933614313464 x 11 = − 76.1836218495525 x_{11} = -76.1836218495525 x 11 = − 76.1836218495525 x 12 = 90.3207887907066 x_{12} = 90.3207887907066 x 12 = 90.3207887907066 x 13 = 60.4756585816035 x_{13} = 60.4756585816035 x 13 = 60.4756585816035 x 14 = − 99.7455667514759 x_{14} = -99.7455667514759 x 14 = − 99.7455667514759 x 15 = − 98.174770424681 x_{15} = -98.174770424681 x 15 = − 98.174770424681 x 16 = − 10.2101761241668 x_{16} = -10.2101761241668 x 16 = − 10.2101761241668 x 17 = − 85.6083998103219 x_{17} = -85.6083998103219 x 17 = − 85.6083998103219 x 18 = − 2.35619449019234 x_{18} = -2.35619449019234 x 18 = − 2.35619449019234 x 19 = − 55.7632696012188 x_{19} = -55.7632696012188 x 19 = − 55.7632696012188 x 20 = − 14.9225651045515 x_{20} = -14.9225651045515 x 20 = − 14.9225651045515 x 21 = 63.6172512351933 x_{21} = 63.6172512351933 x 21 = 63.6172512351933 x 22 = 32.2013246992954 x_{22} = 32.2013246992954 x 22 = 32.2013246992954 x 23 = 18.0641577581413 x_{23} = 18.0641577581413 x 23 = 18.0641577581413 x 24 = − 82.4668071567321 x_{24} = -82.4668071567321 x 24 = − 82.4668071567321 x 25 = − 91.8915851175014 x_{25} = -91.8915851175014 x 25 = − 91.8915851175014 x 26 = 77.7544181763474 x_{26} = 77.7544181763474 x 26 = 77.7544181763474 x 27 = − 90.3207887907066 x_{27} = -90.3207887907066 x 27 = − 90.3207887907066 x 28 = − 60.4756585816035 x_{28} = -60.4756585816035 x 28 = − 60.4756585816035 x 29 = − 13.3517687777566 x_{29} = -13.3517687777566 x 29 = − 13.3517687777566 x 30 = 91.8915851175014 x_{30} = 91.8915851175014 x 30 = 91.8915851175014 x 31 = − 3.92699081698724 x_{31} = -3.92699081698724 x 31 = − 3.92699081698724 x 32 = − 71.4712328691678 x_{32} = -71.4712328691678 x 32 = − 71.4712328691678 x 33 = − 36.9137136796801 x_{33} = -36.9137136796801 x 33 = − 36.9137136796801 x 34 = 40.0553063332699 x_{34} = 40.0553063332699 x 34 = 40.0553063332699 x 35 = − 25.9181393921158 x_{35} = -25.9181393921158 x 35 = − 25.9181393921158 x 36 = 33.7721210260903 x_{36} = 33.7721210260903 x 36 = 33.7721210260903 x 37 = 2.35619449019234 x_{37} = 2.35619449019234 x 37 = 2.35619449019234 x 38 = 47.9092879672443 x_{38} = 47.9092879672443 x 38 = 47.9092879672443 x 39 = 99.7455667514759 x_{39} = 99.7455667514759 x 39 = 99.7455667514759 x 40 = 96.6039740978861 x_{40} = 96.6039740978861 x 40 = 96.6039740978861 x 41 = − 11.7809724509617 x_{41} = -11.7809724509617 x 41 = − 11.7809724509617 x 42 = − 62.0464549083984 x_{42} = -62.0464549083984 x 42 = − 62.0464549083984 x 43 = 589.834020711484 x_{43} = 589.834020711484 x 43 = 589.834020711484 x 44 = − 18.0641577581413 x_{44} = -18.0641577581413 x 44 = − 18.0641577581413 x 45 = 82.4668071567321 x_{45} = 82.4668071567321 x 45 = 82.4668071567321 x 46 = 54.1924732744239 x_{46} = 54.1924732744239 x 46 = 54.1924732744239 x 47 = 5.49778714378214 x_{47} = 5.49778714378214 x 47 = 5.49778714378214 x 48 = − 49.4800842940392 x_{48} = -49.4800842940392 x 48 = − 49.4800842940392 x 49 = 84.037603483527 x_{49} = 84.037603483527 x 49 = 84.037603483527 x 50 = 88.7499924639117 x_{50} = 88.7499924639117 x 50 = 88.7499924639117 x 51 = − 77.7544181763474 x_{51} = -77.7544181763474 x 51 = − 77.7544181763474 x 52 = − 46.3384916404494 x_{52} = -46.3384916404494 x 52 = − 46.3384916404494 x 53 = 24.3473430653209 x_{53} = 24.3473430653209 x 53 = 24.3473430653209 x 54 = − 38.484510006475 x_{54} = -38.484510006475 x 54 = − 38.484510006475 x 55 = 19.6349540849362 x_{55} = 19.6349540849362 x 55 = 19.6349540849362 x 56 = 85.6083998103219 x_{56} = 85.6083998103219 x 56 = 85.6083998103219 x 57 = 62.0464549083984 x_{57} = 62.0464549083984 x 57 = 62.0464549083984 x 58 = 5350.91768722682 x_{58} = 5350.91768722682 x 58 = 5350.91768722682 x 59 = − 57.3340659280137 x_{59} = -57.3340659280137 x 59 = − 57.3340659280137 x 60 = 76.1836218495525 x_{60} = 76.1836218495525 x 60 = 76.1836218495525 x 61 = 69.9004365423729 x_{61} = 69.9004365423729 x 61 = 69.9004365423729 x 62 = 8.63937979737193 x_{62} = 8.63937979737193 x 62 = 8.63937979737193 x 63 = − 69.9004365423729 x_{63} = -69.9004365423729 x 63 = − 69.9004365423729 x 64 = 68.329640215578 x_{64} = 68.329640215578 x 64 = 68.329640215578 x 65 = − 63.6172512351933 x_{65} = -63.6172512351933 x 65 = − 63.6172512351933 x 66 = 98.174770424681 x_{66} = 98.174770424681 x 66 = 98.174770424681 x 67 = 41.6261026600648 x_{67} = 41.6261026600648 x 67 = 41.6261026600648 x 68 = − 19.6349540849362 x_{68} = -19.6349540849362 x 68 = − 19.6349540849362 x 69 = − 24.3473430653209 x_{69} = -24.3473430653209 x 69 = − 24.3473430653209 x 70 = − 93.4623814442964 x_{70} = -93.4623814442964 x 70 = − 93.4623814442964 x 71 = − 41.6261026600648 x_{71} = -41.6261026600648 x 71 = − 41.6261026600648 x 72 = − 27.4889357189107 x_{72} = -27.4889357189107 x 72 = − 27.4889357189107 x 73 = 291.382718620453 x_{73} = 291.382718620453 x 73 = 291.382718620453 x 74 = 30.6305283725005 x_{74} = 30.6305283725005 x 74 = 30.6305283725005 x 75 = − 84.037603483527 x_{75} = -84.037603483527 x 75 = − 84.037603483527 x 76 = 336.935812097505 x_{76} = 336.935812097505 x 76 = 336.935812097505 x 77 = 10.2101761241668 x_{77} = 10.2101761241668 x 77 = 10.2101761241668 x 78 = 25.9181393921158 x_{78} = 25.9181393921158 x 78 = 25.9181393921158 x 79 = 74.6128255227576 x_{79} = 74.6128255227576 x 79 = 74.6128255227576 x 80 = − 68.329640215578 x_{80} = -68.329640215578 x 80 = − 68.329640215578 x 81 = − 79.3252145031423 x_{81} = -79.3252145031423 x 81 = − 79.3252145031423 x 82 = − 40.0553063332699 x_{82} = -40.0553063332699 x 82 = − 40.0553063332699 x 83 = 52.621676947629 x_{83} = 52.621676947629 x 83 = 52.621676947629 x 84 = 11.7809724509617 x_{84} = 11.7809724509617 x 84 = 11.7809724509617 x 85 = − 54.1924732744239 x_{85} = -54.1924732744239 x 85 = − 54.1924732744239 x 86 = − 32.2013246992954 x_{86} = -32.2013246992954 x 86 = − 32.2013246992954 x 87 = 38.484510006475 x_{87} = 38.484510006475 x 87 = 38.484510006475 x 88 = − 5.49778714378214 x_{88} = -5.49778714378214 x 88 = − 5.49778714378214 x 89 = − 35.3429173528852 x_{89} = -35.3429173528852 x 89 = − 35.3429173528852 x 1 = − 74.6128255227576 x_{1} = -74.6128255227576 x 1 = − 74.6128255227576 x 2 = 55.7632696012188 x_{2} = 55.7632696012188 x 2 = 55.7632696012188 x 3 = − 33.7721210260903 x_{3} = -33.7721210260903 x 3 = − 33.7721210260903 x 4 = 3.92699081698724 x_{4} = 3.92699081698724 x 4 = 3.92699081698724 x 5 = − 96.6039740978861 x_{5} = -96.6039740978861 x 5 = − 96.6039740978861 x 6 = − 16.4933614313464 x_{6} = -16.4933614313464 x 6 = − 16.4933614313464 x 7 = − 47.9092879672443 x_{7} = -47.9092879672443 x 7 = − 47.9092879672443 x 8 = 71.4712328691678 x_{8} = 71.4712328691678 x 8 = 71.4712328691678 x 9 = 46.3384916404494 x_{9} = 46.3384916404494 x 9 = 46.3384916404494 x 10 = 16.4933614313464 x_{10} = 16.4933614313464 x 10 = 16.4933614313464 x 11 = − 76.1836218495525 x_{11} = -76.1836218495525 x 11 = − 76.1836218495525 x 12 = 90.3207887907066 x_{12} = 90.3207887907066 x 12 = 90.3207887907066 x 13 = 60.4756585816035 x_{13} = 60.4756585816035 x 13 = 60.4756585816035 x 14 = − 99.7455667514759 x_{14} = -99.7455667514759 x 14 = − 99.7455667514759 x 15 = − 98.174770424681 x_{15} = -98.174770424681 x 15 = − 98.174770424681 x 16 = − 10.2101761241668 x_{16} = -10.2101761241668 x 16 = − 10.2101761241668 x 17 = − 85.6083998103219 x_{17} = -85.6083998103219 x 17 = − 85.6083998103219 x 18 = − 2.35619449019234 x_{18} = -2.35619449019234 x 18 = − 2.35619449019234 x 19 = − 55.7632696012188 x_{19} = -55.7632696012188 x 19 = − 55.7632696012188 x 20 = − 14.9225651045515 x_{20} = -14.9225651045515 x 20 = − 14.9225651045515 x 21 = 63.6172512351933 x_{21} = 63.6172512351933 x 21 = 63.6172512351933 x 22 = 32.2013246992954 x_{22} = 32.2013246992954 x 22 = 32.2013246992954 x 23 = 18.0641577581413 x_{23} = 18.0641577581413 x 23 = 18.0641577581413 x 24 = − 82.4668071567321 x_{24} = -82.4668071567321 x 24 = − 82.4668071567321 x 25 = − 91.8915851175014 x_{25} = -91.8915851175014 x 25 = − 91.8915851175014 x 26 = 77.7544181763474 x_{26} = 77.7544181763474 x 26 = 77.7544181763474 x 27 = − 90.3207887907066 x_{27} = -90.3207887907066 x 27 = − 90.3207887907066 x 28 = − 60.4756585816035 x_{28} = -60.4756585816035 x 28 = − 60.4756585816035 x 29 = − 13.3517687777566 x_{29} = -13.3517687777566 x 29 = − 13.3517687777566 x 30 = 91.8915851175014 x_{30} = 91.8915851175014 x 30 = 91.8915851175014 x 31 = − 3.92699081698724 x_{31} = -3.92699081698724 x 31 = − 3.92699081698724 x 32 = − 71.4712328691678 x_{32} = -71.4712328691678 x 32 = − 71.4712328691678 x 33 = − 36.9137136796801 x_{33} = -36.9137136796801 x 33 = − 36.9137136796801 x 34 = 40.0553063332699 x_{34} = 40.0553063332699 x 34 = 40.0553063332699 x 35 = − 25.9181393921158 x_{35} = -25.9181393921158 x 35 = − 25.9181393921158 x 36 = 33.7721210260903 x_{36} = 33.7721210260903 x 36 = 33.7721210260903 x 37 = 2.35619449019234 x_{37} = 2.35619449019234 x 37 = 2.35619449019234 x 38 = 47.9092879672443 x_{38} = 47.9092879672443 x 38 = 47.9092879672443 x 39 = 99.7455667514759 x_{39} = 99.7455667514759 x 39 = 99.7455667514759 x 40 = 96.6039740978861 x_{40} = 96.6039740978861 x 40 = 96.6039740978861 x 41 = − 11.7809724509617 x_{41} = -11.7809724509617 x 41 = − 11.7809724509617 x 42 = − 62.0464549083984 x_{42} = -62.0464549083984 x 42 = − 62.0464549083984 x 43 = 589.834020711484 x_{43} = 589.834020711484 x 43 = 589.834020711484 x 44 = − 18.0641577581413 x_{44} = -18.0641577581413 x 44 = − 18.0641577581413 x 45 = 82.4668071567321 x_{45} = 82.4668071567321 x 45 = 82.4668071567321 x 46 = 54.1924732744239 x_{46} = 54.1924732744239 x 46 = 54.1924732744239 x 47 = 5.49778714378214 x_{47} = 5.49778714378214 x 47 = 5.49778714378214 x 48 = − 49.4800842940392 x_{48} = -49.4800842940392 x 48 = − 49.4800842940392 x 49 = 84.037603483527 x_{49} = 84.037603483527 x 49 = 84.037603483527 x 50 = 88.7499924639117 x_{50} = 88.7499924639117 x 50 = 88.7499924639117 x 51 = − 77.7544181763474 x_{51} = -77.7544181763474 x 51 = − 77.7544181763474 x 52 = − 46.3384916404494 x_{52} = -46.3384916404494 x 52 = − 46.3384916404494 x 53 = 24.3473430653209 x_{53} = 24.3473430653209 x 53 = 24.3473430653209 x 54 = − 38.484510006475 x_{54} = -38.484510006475 x 54 = − 38.484510006475 x 55 = 19.6349540849362 x_{55} = 19.6349540849362 x 55 = 19.6349540849362 x 56 = 85.6083998103219 x_{56} = 85.6083998103219 x 56 = 85.6083998103219 x 57 = 62.0464549083984 x_{57} = 62.0464549083984 x 57 = 62.0464549083984 x 58 = 5350.91768722682 x_{58} = 5350.91768722682 x 58 = 5350.91768722682 x 59 = − 57.3340659280137 x_{59} = -57.3340659280137 x 59 = − 57.3340659280137 x 60 = 76.1836218495525 x_{60} = 76.1836218495525 x 60 = 76.1836218495525 x 61 = 69.9004365423729 x_{61} = 69.9004365423729 x 61 = 69.9004365423729 x 62 = 8.63937979737193 x_{62} = 8.63937979737193 x 62 = 8.63937979737193 x 63 = − 69.9004365423729 x_{63} = -69.9004365423729 x 63 = − 69.9004365423729 x 64 = 68.329640215578 x_{64} = 68.329640215578 x 64 = 68.329640215578 x 65 = − 63.6172512351933 x_{65} = -63.6172512351933 x 65 = − 63.6172512351933 x 66 = 98.174770424681 x_{66} = 98.174770424681 x 66 = 98.174770424681 x 67 = 41.6261026600648 x_{67} = 41.6261026600648 x 67 = 41.6261026600648 x 68 = − 19.6349540849362 x_{68} = -19.6349540849362 x 68 = − 19.6349540849362 x 69 = − 24.3473430653209 x_{69} = -24.3473430653209 x 69 = − 24.3473430653209 x 70 = − 93.4623814442964 x_{70} = -93.4623814442964 x 70 = − 93.4623814442964 x 71 = − 41.6261026600648 x_{71} = -41.6261026600648 x 71 = − 41.6261026600648 x 72 = − 27.4889357189107 x_{72} = -27.4889357189107 x 72 = − 27.4889357189107 x 73 = 291.382718620453 x_{73} = 291.382718620453 x 73 = 291.382718620453 x 74 = 30.6305283725005 x_{74} = 30.6305283725005 x 74 = 30.6305283725005 x 75 = − 84.037603483527 x_{75} = -84.037603483527 x 75 = − 84.037603483527 x 76 = 336.935812097505 x_{76} = 336.935812097505 x 76 = 336.935812097505 x 77 = 10.2101761241668 x_{77} = 10.2101761241668 x 77 = 10.2101761241668 x 78 = 25.9181393921158 x_{78} = 25.9181393921158 x 78 = 25.9181393921158 x 79 = 74.6128255227576 x_{79} = 74.6128255227576 x 79 = 74.6128255227576 x 80 = − 68.329640215578 x_{80} = -68.329640215578 x 80 = − 68.329640215578 x 81 = − 79.3252145031423 x_{81} = -79.3252145031423 x 81 = − 79.3252145031423 x 82 = − 40.0553063332699 x_{82} = -40.0553063332699 x 82 = − 40.0553063332699 x 83 = 52.621676947629 x_{83} = 52.621676947629 x 83 = 52.621676947629 x 84 = 11.7809724509617 x_{84} = 11.7809724509617 x 84 = 11.7809724509617 x 85 = − 54.1924732744239 x_{85} = -54.1924732744239 x 85 = − 54.1924732744239 x 86 = − 32.2013246992954 x_{86} = -32.2013246992954 x 86 = − 32.2013246992954 x 87 = 38.484510006475 x_{87} = 38.484510006475 x 87 = 38.484510006475 x 88 = − 5.49778714378214 x_{88} = -5.49778714378214 x 88 = − 5.49778714378214 x 89 = − 35.3429173528852 x_{89} = -35.3429173528852 x 89 = − 35.3429173528852 Las raíces dadas
x 14 = − 99.7455667514759 x_{14} = -99.7455667514759 x 14 = − 99.7455667514759 x 15 = − 98.174770424681 x_{15} = -98.174770424681 x 15 = − 98.174770424681 x 5 = − 96.6039740978861 x_{5} = -96.6039740978861 x 5 = − 96.6039740978861 x 70 = − 93.4623814442964 x_{70} = -93.4623814442964 x 70 = − 93.4623814442964 x 25 = − 91.8915851175014 x_{25} = -91.8915851175014 x 25 = − 91.8915851175014 x 27 = − 90.3207887907066 x_{27} = -90.3207887907066 x 27 = − 90.3207887907066 x 17 = − 85.6083998103219 x_{17} = -85.6083998103219 x 17 = − 85.6083998103219 x 75 = − 84.037603483527 x_{75} = -84.037603483527 x 75 = − 84.037603483527 x 24 = − 82.4668071567321 x_{24} = -82.4668071567321 x 24 = − 82.4668071567321 x 81 = − 79.3252145031423 x_{81} = -79.3252145031423 x 81 = − 79.3252145031423 x 51 = − 77.7544181763474 x_{51} = -77.7544181763474 x 51 = − 77.7544181763474 x 11 = − 76.1836218495525 x_{11} = -76.1836218495525 x 11 = − 76.1836218495525 x 1 = − 74.6128255227576 x_{1} = -74.6128255227576 x 1 = − 74.6128255227576 x 32 = − 71.4712328691678 x_{32} = -71.4712328691678 x 32 = − 71.4712328691678 x 63 = − 69.9004365423729 x_{63} = -69.9004365423729 x 63 = − 69.9004365423729 x 80 = − 68.329640215578 x_{80} = -68.329640215578 x 80 = − 68.329640215578 x 65 = − 63.6172512351933 x_{65} = -63.6172512351933 x 65 = − 63.6172512351933 x 42 = − 62.0464549083984 x_{42} = -62.0464549083984 x 42 = − 62.0464549083984 x 28 = − 60.4756585816035 x_{28} = -60.4756585816035 x 28 = − 60.4756585816035 x 59 = − 57.3340659280137 x_{59} = -57.3340659280137 x 59 = − 57.3340659280137 x 19 = − 55.7632696012188 x_{19} = -55.7632696012188 x 19 = − 55.7632696012188 x 85 = − 54.1924732744239 x_{85} = -54.1924732744239 x 85 = − 54.1924732744239 x 48 = − 49.4800842940392 x_{48} = -49.4800842940392 x 48 = − 49.4800842940392 x 7 = − 47.9092879672443 x_{7} = -47.9092879672443 x 7 = − 47.9092879672443 x 52 = − 46.3384916404494 x_{52} = -46.3384916404494 x 52 = − 46.3384916404494 x 71 = − 41.6261026600648 x_{71} = -41.6261026600648 x 71 = − 41.6261026600648 x 82 = − 40.0553063332699 x_{82} = -40.0553063332699 x 82 = − 40.0553063332699 x 54 = − 38.484510006475 x_{54} = -38.484510006475 x 54 = − 38.484510006475 x 33 = − 36.9137136796801 x_{33} = -36.9137136796801 x 33 = − 36.9137136796801 x 89 = − 35.3429173528852 x_{89} = -35.3429173528852 x 89 = − 35.3429173528852 x 3 = − 33.7721210260903 x_{3} = -33.7721210260903 x 3 = − 33.7721210260903 x 86 = − 32.2013246992954 x_{86} = -32.2013246992954 x 86 = − 32.2013246992954 x 72 = − 27.4889357189107 x_{72} = -27.4889357189107 x 72 = − 27.4889357189107 x 35 = − 25.9181393921158 x_{35} = -25.9181393921158 x 35 = − 25.9181393921158 x 69 = − 24.3473430653209 x_{69} = -24.3473430653209 x 69 = − 24.3473430653209 x 68 = − 19.6349540849362 x_{68} = -19.6349540849362 x 68 = − 19.6349540849362 x 44 = − 18.0641577581413 x_{44} = -18.0641577581413 x 44 = − 18.0641577581413 x 6 = − 16.4933614313464 x_{6} = -16.4933614313464 x 6 = − 16.4933614313464 x 20 = − 14.9225651045515 x_{20} = -14.9225651045515 x 20 = − 14.9225651045515 x 29 = − 13.3517687777566 x_{29} = -13.3517687777566 x 29 = − 13.3517687777566 x 41 = − 11.7809724509617 x_{41} = -11.7809724509617 x 41 = − 11.7809724509617 x 16 = − 10.2101761241668 x_{16} = -10.2101761241668 x 16 = − 10.2101761241668 x 88 = − 5.49778714378214 x_{88} = -5.49778714378214 x 88 = − 5.49778714378214 x 31 = − 3.92699081698724 x_{31} = -3.92699081698724 x 31 = − 3.92699081698724 x 18 = − 2.35619449019234 x_{18} = -2.35619449019234 x 18 = − 2.35619449019234 x 37 = 2.35619449019234 x_{37} = 2.35619449019234 x 37 = 2.35619449019234 x 4 = 3.92699081698724 x_{4} = 3.92699081698724 x 4 = 3.92699081698724 x 47 = 5.49778714378214 x_{47} = 5.49778714378214 x 47 = 5.49778714378214 x 62 = 8.63937979737193 x_{62} = 8.63937979737193 x 62 = 8.63937979737193 x 77 = 10.2101761241668 x_{77} = 10.2101761241668 x 77 = 10.2101761241668 x 84 = 11.7809724509617 x_{84} = 11.7809724509617 x 84 = 11.7809724509617 x 10 = 16.4933614313464 x_{10} = 16.4933614313464 x 10 = 16.4933614313464 x 23 = 18.0641577581413 x_{23} = 18.0641577581413 x 23 = 18.0641577581413 x 55 = 19.6349540849362 x_{55} = 19.6349540849362 x 55 = 19.6349540849362 x 53 = 24.3473430653209 x_{53} = 24.3473430653209 x 53 = 24.3473430653209 x 78 = 25.9181393921158 x_{78} = 25.9181393921158 x 78 = 25.9181393921158 x 74 = 30.6305283725005 x_{74} = 30.6305283725005 x 74 = 30.6305283725005 x 22 = 32.2013246992954 x_{22} = 32.2013246992954 x 22 = 32.2013246992954 x 36 = 33.7721210260903 x_{36} = 33.7721210260903 x 36 = 33.7721210260903 x 87 = 38.484510006475 x_{87} = 38.484510006475 x 87 = 38.484510006475 x 34 = 40.0553063332699 x_{34} = 40.0553063332699 x 34 = 40.0553063332699 x 67 = 41.6261026600648 x_{67} = 41.6261026600648 x 67 = 41.6261026600648 x 9 = 46.3384916404494 x_{9} = 46.3384916404494 x 9 = 46.3384916404494 x 38 = 47.9092879672443 x_{38} = 47.9092879672443 x 38 = 47.9092879672443 x 83 = 52.621676947629 x_{83} = 52.621676947629 x 83 = 52.621676947629 x 46 = 54.1924732744239 x_{46} = 54.1924732744239 x 46 = 54.1924732744239 x 2 = 55.7632696012188 x_{2} = 55.7632696012188 x 2 = 55.7632696012188 x 13 = 60.4756585816035 x_{13} = 60.4756585816035 x 13 = 60.4756585816035 x 57 = 62.0464549083984 x_{57} = 62.0464549083984 x 57 = 62.0464549083984 x 21 = 63.6172512351933 x_{21} = 63.6172512351933 x 21 = 63.6172512351933 x 64 = 68.329640215578 x_{64} = 68.329640215578 x 64 = 68.329640215578 x 61 = 69.9004365423729 x_{61} = 69.9004365423729 x 61 = 69.9004365423729 x 8 = 71.4712328691678 x_{8} = 71.4712328691678 x 8 = 71.4712328691678 x 79 = 74.6128255227576 x_{79} = 74.6128255227576 x 79 = 74.6128255227576 x 60 = 76.1836218495525 x_{60} = 76.1836218495525 x 60 = 76.1836218495525 x 26 = 77.7544181763474 x_{26} = 77.7544181763474 x 26 = 77.7544181763474 x 45 = 82.4668071567321 x_{45} = 82.4668071567321 x 45 = 82.4668071567321 x 49 = 84.037603483527 x_{49} = 84.037603483527 x 49 = 84.037603483527 x 56 = 85.6083998103219 x_{56} = 85.6083998103219 x 56 = 85.6083998103219 x 50 = 88.7499924639117 x_{50} = 88.7499924639117 x 50 = 88.7499924639117 x 12 = 90.3207887907066 x_{12} = 90.3207887907066 x 12 = 90.3207887907066 x 30 = 91.8915851175014 x_{30} = 91.8915851175014 x 30 = 91.8915851175014 x 40 = 96.6039740978861 x_{40} = 96.6039740978861 x 40 = 96.6039740978861 x 66 = 98.174770424681 x_{66} = 98.174770424681 x 66 = 98.174770424681 x 39 = 99.7455667514759 x_{39} = 99.7455667514759 x 39 = 99.7455667514759 x 73 = 291.382718620453 x_{73} = 291.382718620453 x 73 = 291.382718620453 x 76 = 336.935812097505 x_{76} = 336.935812097505 x 76 = 336.935812097505 x 43 = 589.834020711484 x_{43} = 589.834020711484 x 43 = 589.834020711484 x 58 = 5350.91768722682 x_{58} = 5350.91768722682 x 58 = 5350.91768722682 son puntos de cambio del signo de desigualdad en las soluciones.
Primero definámonos con el signo hasta el punto extremo izquierdo:
x 0 ≤ x 14 x_{0} \leq x_{14} x 0 ≤ x 14 Consideremos, por ejemplo, el punto
x 0 = x 14 − 1 10 x_{0} = x_{14} - \frac{1}{10} x 0 = x 14 − 10 1 =
− 99.7455667514759 + − 1 10 -99.7455667514759 + - \frac{1}{10} − 99.7455667514759 + − 10 1 =
− 99.8455667514759 -99.8455667514759 − 99.8455667514759 lo sustituimos en la expresión
∣ cos ( x ) ∣ ≥ 2 2 \left|{\cos{\left(x \right)}}\right| \geq \frac{\sqrt{2}}{2} ∣ cos ( x ) ∣ ≥ 2 2 ∣ cos ( − 99.8455667514759 ) ∣ ≥ 2 2 \left|{\cos{\left(-99.8455667514759 \right)}}\right| \geq \frac{\sqrt{2}}{2} ∣ cos ( − 99.8455667514759 ) ∣ ≥ 2 2 ___
\/ 2
0.774167078476942 >= -----
2
significa que una de las soluciones de nuestra ecuación será con:
x ≤ − 99.7455667514759 x \leq -99.7455667514759 x ≤ − 99.7455667514759 _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____
\ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \
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x14 x15 x5 x70 x25 x27 x17 x75 x24 x81 x51 x11 x1 x32 x63 x80 x65 x42 x28 x59 x19 x85 x48 x7 x52 x71 x82 x54 x33 x89 x3 x86 x72 x35 x69 x68 x44 x6 x20 x29 x41 x16 x88 x31 x18 x37 x4 x47 x62 x77 x84 x10 x23 x55 x53 x78 x74 x22 x36 x87 x34 x67 x9 x38 x83 x46 x2 x13 x57 x21 x64 x61 x8 x79 x60 x26 x45 x49 x56 x50 x12 x30 x40 x66 x39 x73 x76 x43 x58 Recibiremos otras soluciones de la desigualdad pasando al polo siguiente etc.
etc.
Respuesta:
x ≤ − 99.7455667514759 x \leq -99.7455667514759 x ≤ − 99.7455667514759 x ≥ − 98.174770424681 ∧ x ≤ − 96.6039740978861 x \geq -98.174770424681 \wedge x \leq -96.6039740978861 x ≥ − 98.174770424681 ∧ x ≤ − 96.6039740978861 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 ≤ − 85.6083998103219 x \geq -90.3207887907066 \wedge x \leq -85.6083998103219 x ≥ − 90.3207887907066 ∧ x ≤ − 85.6083998103219 x ≥ − 84.037603483527 ∧ x ≤ − 82.4668071567321 x \geq -84.037603483527 \wedge x \leq -82.4668071567321 x ≥ − 84.037603483527 ∧ x ≤ − 82.4668071567321 x ≥ − 79.3252145031423 ∧ x ≤ − 77.7544181763474 x \geq -79.3252145031423 \wedge x \leq -77.7544181763474 x ≥ − 79.3252145031423 ∧ x ≤ − 77.7544181763474 x ≥ − 76.1836218495525 ∧ x ≤ − 74.6128255227576 x \geq -76.1836218495525 \wedge x \leq -74.6128255227576 x ≥ − 76.1836218495525 ∧ x ≤ − 74.6128255227576 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 ≤ − 63.6172512351933 x \geq -68.329640215578 \wedge x \leq -63.6172512351933 x ≥ − 68.329640215578 ∧ x ≤ − 63.6172512351933 x ≥ − 62.0464549083984 ∧ x ≤ − 60.4756585816035 x \geq -62.0464549083984 \wedge x \leq -60.4756585816035 x ≥ − 62.0464549083984 ∧ x ≤ − 60.4756585816035 x ≥ − 57.3340659280137 ∧ x ≤ − 55.7632696012188 x \geq -57.3340659280137 \wedge x \leq -55.7632696012188 x ≥ − 57.3340659280137 ∧ x ≤ − 55.7632696012188 x ≥ − 54.1924732744239 ∧ x ≤ − 49.4800842940392 x \geq -54.1924732744239 \wedge x \leq -49.4800842940392 x ≥ − 54.1924732744239 ∧ x ≤ − 49.4800842940392 x ≥ − 47.9092879672443 ∧ x ≤ − 46.3384916404494 x \geq -47.9092879672443 \wedge x \leq -46.3384916404494 x ≥ − 47.9092879672443 ∧ x ≤ − 46.3384916404494 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 ≤ − 36.9137136796801 x \geq -38.484510006475 \wedge x \leq -36.9137136796801 x ≥ − 38.484510006475 ∧ x ≤ − 36.9137136796801 x ≥ − 35.3429173528852 ∧ x ≤ − 33.7721210260903 x \geq -35.3429173528852 \wedge x \leq -33.7721210260903 x ≥ − 35.3429173528852 ∧ x ≤ − 33.7721210260903 x ≥ − 32.2013246992954 ∧ x ≤ − 27.4889357189107 x \geq -32.2013246992954 \wedge x \leq -27.4889357189107 x ≥ − 32.2013246992954 ∧ x ≤ − 27.4889357189107 x ≥ − 25.9181393921158 ∧ x ≤ − 24.3473430653209 x \geq -25.9181393921158 \wedge x \leq -24.3473430653209 x ≥ − 25.9181393921158 ∧ x ≤ − 24.3473430653209 x ≥ − 19.6349540849362 ∧ x ≤ − 18.0641577581413 x \geq -19.6349540849362 \wedge x \leq -18.0641577581413 x ≥ − 19.6349540849362 ∧ x ≤ − 18.0641577581413 x ≥ − 16.4933614313464 ∧ x ≤ − 14.9225651045515 x \geq -16.4933614313464 \wedge x \leq -14.9225651045515 x ≥ − 16.4933614313464 ∧ x ≤ − 14.9225651045515 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 ≥ 2.35619449019234 ∧ x ≤ 3.92699081698724 x \geq 2.35619449019234 \wedge x \leq 3.92699081698724 x ≥ 2.35619449019234 ∧ x ≤ 3.92699081698724 x ≥ 5.49778714378214 ∧ x ≤ 8.63937979737193 x \geq 5.49778714378214 \wedge x \leq 8.63937979737193 x ≥ 5.49778714378214 ∧ x ≤ 8.63937979737193 x ≥ 10.2101761241668 ∧ x ≤ 11.7809724509617 x \geq 10.2101761241668 \wedge x \leq 11.7809724509617 x ≥ 10.2101761241668 ∧ x ≤ 11.7809724509617 x ≥ 16.4933614313464 ∧ x ≤ 18.0641577581413 x \geq 16.4933614313464 \wedge x \leq 18.0641577581413 x ≥ 16.4933614313464 ∧ x ≤ 18.0641577581413 x ≥ 19.6349540849362 ∧ x ≤ 24.3473430653209 x \geq 19.6349540849362 \wedge x \leq 24.3473430653209 x ≥ 19.6349540849362 ∧ x ≤ 24.3473430653209 x ≥ 25.9181393921158 ∧ x ≤ 30.6305283725005 x \geq 25.9181393921158 \wedge x \leq 30.6305283725005 x ≥ 25.9181393921158 ∧ x ≤ 30.6305283725005 x ≥ 32.2013246992954 ∧ x ≤ 33.7721210260903 x \geq 32.2013246992954 \wedge x \leq 33.7721210260903 x ≥ 32.2013246992954 ∧ x ≤ 33.7721210260903 x ≥ 38.484510006475 ∧ x ≤ 40.0553063332699 x \geq 38.484510006475 \wedge x \leq 40.0553063332699 x ≥ 38.484510006475 ∧ x ≤ 40.0553063332699 x ≥ 41.6261026600648 ∧ x ≤ 46.3384916404494 x \geq 41.6261026600648 \wedge x \leq 46.3384916404494 x ≥ 41.6261026600648 ∧ x ≤ 46.3384916404494 x ≥ 47.9092879672443 ∧ x ≤ 52.621676947629 x \geq 47.9092879672443 \wedge x \leq 52.621676947629 x ≥ 47.9092879672443 ∧ x ≤ 52.621676947629 x ≥ 54.1924732744239 ∧ x ≤ 55.7632696012188 x \geq 54.1924732744239 \wedge x \leq 55.7632696012188 x ≥ 54.1924732744239 ∧ x ≤ 55.7632696012188 x ≥ 60.4756585816035 ∧ x ≤ 62.0464549083984 x \geq 60.4756585816035 \wedge x \leq 62.0464549083984 x ≥ 60.4756585816035 ∧ x ≤ 62.0464549083984 x ≥ 63.6172512351933 ∧ x ≤ 68.329640215578 x \geq 63.6172512351933 \wedge x \leq 68.329640215578 x ≥ 63.6172512351933 ∧ 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 ≤ 96.6039740978861 x \geq 91.8915851175014 \wedge x \leq 96.6039740978861 x ≥ 91.8915851175014 ∧ x ≤ 96.6039740978861 x ≥ 98.174770424681 ∧ x ≤ 99.7455667514759 x \geq 98.174770424681 \wedge x \leq 99.7455667514759 x ≥ 98.174770424681 ∧ x ≤ 99.7455667514759 x ≥ 291.382718620453 ∧ x ≤ 336.935812097505 x \geq 291.382718620453 \wedge x \leq 336.935812097505 x ≥ 291.382718620453 ∧ x ≤ 336.935812097505 x ≥ 589.834020711484 ∧ x ≤ 5350.91768722682 x \geq 589.834020711484 \wedge x \leq 5350.91768722682 x ≥ 589.834020711484 ∧ x ≤ 5350.91768722682 
-x^2+3x-2<0
x^2+4x-5<0
x+1>0
x^2-6x-27>0