Se da la desigualdad:
∣ sin ( x ) ∣ ≤ 3 2 \left|{\sin{\left(x \right)}}\right| \leq \frac{\sqrt{3}}{2} ∣ sin ( x ) ∣ ≤ 2 3 Para resolver esta desigualdad primero hay que resolver la ecuación correspondiente:
∣ sin ( x ) ∣ = 3 2 \left|{\sin{\left(x \right)}}\right| = \frac{\sqrt{3}}{2} ∣ sin ( x ) ∣ = 2 3 Resolvemos:
Tenemos la ecuación
∣ sin ( x ) ∣ = 3 2 \left|{\sin{\left(x \right)}}\right| = \frac{\sqrt{3}}{2} ∣ sin ( x ) ∣ = 2 3 cambiamos
∣ sin ( x ) ∣ − 1 − 3 2 = 0 \left|{\sin{\left(x \right)}}\right| - 1 - \frac{\sqrt{3}}{2} = 0 ∣ sin ( x ) ∣ − 1 − 2 3 = 0 ∣ sin ( x ) ∣ − 1 − 3 2 = 0 \left|{\sin{\left(x \right)}}\right| - 1 - \frac{\sqrt{3}}{2} = 0 ∣ sin ( x ) ∣ − 1 − 2 3 = 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 - sqrt3/2 = 0 Transportamos los términos libres (sin w)
del miembro izquierdo al derecho, obtenemos:
w − 3 2 = 1 w - \frac{\sqrt{3}}{2} = 1 w − 2 3 = 1 Dividamos ambos miembros de la ecuación en (w - sqrt(3)/2)/w
w = 1 / ((w - sqrt(3)/2)/w) Obtenemos la respuesta: w = 1 + sqrt(3)/2
hacemos cambio inverso
∣ sin ( x ) ∣ = w \left|{\sin{\left(x \right)}}\right| = w ∣ sin ( x ) ∣ = w sustituimos w:
x 1 = − 99.4837673636768 x_{1} = -99.4837673636768 x 1 = − 99.4837673636768 x 2 = − 60.7374579694027 x_{2} = -60.7374579694027 x 2 = − 60.7374579694027 x 3 = − 5.23598775598299 x_{3} = -5.23598775598299 x 3 = − 5.23598775598299 x 4 = 2.0943951023932 x_{4} = 2.0943951023932 x 4 = 2.0943951023932 x 5 = 90.0589894029074 x_{5} = 90.0589894029074 x 5 = 90.0589894029074 x 6 = − 93.2005820564972 x_{6} = -93.2005820564972 x 6 = − 93.2005820564972 x 7 = − 17.8023583703422 x_{7} = -17.8023583703422 x 7 = − 17.8023583703422 x 8 = 19.8967534727354 x_{8} = 19.8967534727354 x 8 = 19.8967534727354 x 9 = − 39.7935069454707 x_{9} = -39.7935069454707 x 9 = − 39.7935069454707 x 10 = 61.7846555205993 x_{10} = 61.7846555205993 x 10 = 61.7846555205993 x 11 = − 48.1710873550435 x_{11} = -48.1710873550435 x 11 = − 48.1710873550435 x 12 = − 10.471975511966 x_{12} = -10.471975511966 x 12 = − 10.471975511966 x 13 = − 70.162235930172 x_{13} = -70.162235930172 x 13 = − 70.162235930172 x 14 = − 79.5870138909414 x_{14} = -79.5870138909414 x 14 = − 79.5870138909414 x 15 = − 32.4631240870945 x_{15} = -32.4631240870945 x 15 = − 32.4631240870945 x 16 = − 71.2094334813686 x_{16} = -71.2094334813686 x 16 = − 71.2094334813686 x 17 = − 86.9173967493176 x_{17} = -86.9173967493176 x 17 = − 86.9173967493176 x 18 = 83.7758040957278 x_{18} = 83.7758040957278 x 18 = 83.7758040957278 x 19 = − 41.8879020478639 x_{19} = -41.8879020478639 x 19 = − 41.8879020478639 x 20 = 30.3687289847013 x_{20} = 30.3687289847013 x 20 = 30.3687289847013 x 21 = − 92.1533845053006 x_{21} = -92.1533845053006 x 21 = − 92.1533845053006 x 22 = 89.0117918517108 x_{22} = 89.0117918517108 x 22 = 89.0117918517108 x 23 = 99.4837673636768 x_{23} = 99.4837673636768 x 23 = 99.4837673636768 x 24 = 86.9173967493176 x_{24} = 86.9173967493176 x 24 = 86.9173967493176 x 25 = 76.4454212373516 x_{25} = 76.4454212373516 x 25 = 76.4454212373516 x 26 = − 19.8967534727354 x_{26} = -19.8967534727354 x 26 = − 19.8967534727354 x 27 = 4.18879020478639 x_{27} = 4.18879020478639 x 27 = 4.18879020478639 x 28 = 85.870199198121 x_{28} = 85.870199198121 x 28 = 85.870199198121 x 29 = 8.37758040957278 x_{29} = 8.37758040957278 x 29 = 8.37758040957278 x 30 = − 77.4926187885482 x_{30} = -77.4926187885482 x 30 = − 77.4926187885482 x 31 = 60.7374579694027 x_{31} = 60.7374579694027 x 31 = 60.7374579694027 x 32 = − 13.6135681655558 x_{32} = -13.6135681655558 x 32 = − 13.6135681655558 x 33 = − 49.2182849062401 x_{33} = -49.2182849062401 x 33 = − 49.2182849062401 x 34 = 5.23598775598299 x_{34} = 5.23598775598299 x 34 = 5.23598775598299 x 35 = 68.0678408277789 x_{35} = 68.0678408277789 x 35 = 68.0678408277789 x 36 = 11.5191730631626 x_{36} = 11.5191730631626 x 36 = 11.5191730631626 x 37 = 77.4926187885482 x_{37} = 77.4926187885482 x 37 = 77.4926187885482 x 38 = − 90.0589894029074 x_{38} = -90.0589894029074 x 38 = − 90.0589894029074 x 39 = − 85.870199198121 x_{39} = -85.870199198121 x 39 = − 85.870199198121 x 40 = − 57.5958653158129 x_{40} = -57.5958653158129 x 40 = − 57.5958653158129 x 41 = − 63.8790506229925 x_{41} = -63.8790506229925 x 41 = − 63.8790506229925 x 42 = 33.5103216382911 x_{42} = 33.5103216382911 x 42 = 33.5103216382911 x 43 = − 24.0855436775217 x_{43} = -24.0855436775217 x 43 = − 24.0855436775217 x 44 = 1.0471975511966 x_{44} = 1.0471975511966 x 44 = 1.0471975511966 x 45 = 45.0294947014537 x_{45} = 45.0294947014537 x 45 = 45.0294947014537 x 46 = − 16.7551608191456 x_{46} = -16.7551608191456 x 46 = − 16.7551608191456 x 47 = 74.3510261349584 x_{47} = 74.3510261349584 x 47 = 74.3510261349584 x 48 = − 35.6047167406843 x_{48} = -35.6047167406843 x 48 = − 35.6047167406843 x 49 = − 83.7758040957278 x_{49} = -83.7758040957278 x 49 = − 83.7758040957278 x 50 = 96.342174710087 x_{50} = 96.342174710087 x 50 = 96.342174710087 x 51 = − 98.4365698124802 x_{51} = -98.4365698124802 x 51 = − 98.4365698124802 x 52 = − 42.9350995990605 x_{52} = -42.9350995990605 x 52 = − 42.9350995990605 x 53 = 71.2094334813686 x_{53} = 71.2094334813686 x 53 = 71.2094334813686 x 54 = − 33.5103216382911 x_{54} = -33.5103216382911 x 54 = − 33.5103216382911 x 55 = − 61.7846555205993 x_{55} = -61.7846555205993 x 55 = − 61.7846555205993 x 56 = − 26.1799387799149 x_{56} = -26.1799387799149 x 56 = − 26.1799387799149 x 57 = − 36.6519142918809 x_{57} = -36.6519142918809 x 57 = − 36.6519142918809 x 58 = 26.1799387799149 x_{58} = 26.1799387799149 x 58 = 26.1799387799149 x 59 = − 80.634211442138 x_{59} = -80.634211442138 x 59 = − 80.634211442138 x 60 = − 38.7463093942741 x_{60} = -38.7463093942741 x 60 = − 38.7463093942741 x 61 = 67.0206432765823 x_{61} = 67.0206432765823 x 61 = 67.0206432765823 x 62 = − 76.4454212373516 x_{62} = -76.4454212373516 x 62 = − 76.4454212373516 x 63 = 70.162235930172 x_{63} = 70.162235930172 x 63 = 70.162235930172 x 64 = 10.471975511966 x_{64} = 10.471975511966 x 64 = 10.471975511966 x 65 = 16.7551608191456 x_{65} = 16.7551608191456 x 65 = 16.7551608191456 x 66 = 55.5014702134197 x_{66} = 55.5014702134197 x 66 = 55.5014702134197 x 67 = 17.8023583703422 x_{67} = 17.8023583703422 x 67 = 17.8023583703422 x 68 = 48.1710873550435 x_{68} = 48.1710873550435 x 68 = 48.1710873550435 x 69 = 98.4365698124802 x_{69} = 98.4365698124802 x 69 = 98.4365698124802 x 70 = 82.7286065445312 x_{70} = 82.7286065445312 x 70 = 82.7286065445312 x 71 = − 11.5191730631626 x_{71} = -11.5191730631626 x 71 = − 11.5191730631626 x 72 = 92.1533845053006 x_{72} = 92.1533845053006 x 72 = 92.1533845053006 x 73 = − 82.7286065445312 x_{73} = -82.7286065445312 x 73 = − 82.7286065445312 x 74 = 52.3598775598299 x_{74} = 52.3598775598299 x 74 = 52.3598775598299 x 75 = 38.7463093942741 x_{75} = 38.7463093942741 x 75 = 38.7463093942741 x 76 = 27.2271363311115 x_{76} = 27.2271363311115 x 76 = 27.2271363311115 x 77 = 41.8879020478639 x_{77} = 41.8879020478639 x 77 = 41.8879020478639 x 78 = − 4.18879020478639 x_{78} = -4.18879020478639 x 78 = − 4.18879020478639 x 79 = 32.4631240870945 x_{79} = 32.4631240870945 x 79 = 32.4631240870945 x 80 = − 55.5014702134197 x_{80} = -55.5014702134197 x 80 = − 55.5014702134197 x 81 = − 20.943951023932 x_{81} = -20.943951023932 x 81 = − 20.943951023932 x 82 = − 54.4542726622231 x_{82} = -54.4542726622231 x 82 = − 54.4542726622231 x 83 = 39.7935069454707 x_{83} = 39.7935069454707 x 83 = 39.7935069454707 x 84 = 23.0383461263252 x_{84} = 23.0383461263252 x 84 = 23.0383461263252 x 85 = − 2.0943951023932 x_{85} = -2.0943951023932 x 85 = − 2.0943951023932 x 86 = − 68.0678408277789 x_{86} = -68.0678408277789 x 86 = − 68.0678408277789 x 87 = − 64.9262481741891 x_{87} = -64.9262481741891 x 87 = − 64.9262481741891 x 88 = 93.2005820564972 x_{88} = 93.2005820564972 x 88 = 93.2005820564972 x 89 = 24.0855436775217 x_{89} = 24.0855436775217 x 89 = 24.0855436775217 x 90 = 46.0766922526503 x_{90} = 46.0766922526503 x 90 = 46.0766922526503 x 91 = 49.2182849062401 x_{91} = 49.2182849062401 x 91 = 49.2182849062401 x 92 = 54.4542726622231 x_{92} = 54.4542726622231 x 92 = 54.4542726622231 x 93 = 63.8790506229925 x_{93} = 63.8790506229925 x 93 = 63.8790506229925 x 94 = − 46.0766922526503 x_{94} = -46.0766922526503 x 94 = − 46.0766922526503 x 95 = − 27.2271363311115 x_{95} = -27.2271363311115 x 95 = − 27.2271363311115 x 1 = − 99.4837673636768 x_{1} = -99.4837673636768 x 1 = − 99.4837673636768 x 2 = − 60.7374579694027 x_{2} = -60.7374579694027 x 2 = − 60.7374579694027 x 3 = − 5.23598775598299 x_{3} = -5.23598775598299 x 3 = − 5.23598775598299 x 4 = 2.0943951023932 x_{4} = 2.0943951023932 x 4 = 2.0943951023932 x 5 = 90.0589894029074 x_{5} = 90.0589894029074 x 5 = 90.0589894029074 x 6 = − 93.2005820564972 x_{6} = -93.2005820564972 x 6 = − 93.2005820564972 x 7 = − 17.8023583703422 x_{7} = -17.8023583703422 x 7 = − 17.8023583703422 x 8 = 19.8967534727354 x_{8} = 19.8967534727354 x 8 = 19.8967534727354 x 9 = − 39.7935069454707 x_{9} = -39.7935069454707 x 9 = − 39.7935069454707 x 10 = 61.7846555205993 x_{10} = 61.7846555205993 x 10 = 61.7846555205993 x 11 = − 48.1710873550435 x_{11} = -48.1710873550435 x 11 = − 48.1710873550435 x 12 = − 10.471975511966 x_{12} = -10.471975511966 x 12 = − 10.471975511966 x 13 = − 70.162235930172 x_{13} = -70.162235930172 x 13 = − 70.162235930172 x 14 = − 79.5870138909414 x_{14} = -79.5870138909414 x 14 = − 79.5870138909414 x 15 = − 32.4631240870945 x_{15} = -32.4631240870945 x 15 = − 32.4631240870945 x 16 = − 71.2094334813686 x_{16} = -71.2094334813686 x 16 = − 71.2094334813686 x 17 = − 86.9173967493176 x_{17} = -86.9173967493176 x 17 = − 86.9173967493176 x 18 = 83.7758040957278 x_{18} = 83.7758040957278 x 18 = 83.7758040957278 x 19 = − 41.8879020478639 x_{19} = -41.8879020478639 x 19 = − 41.8879020478639 x 20 = 30.3687289847013 x_{20} = 30.3687289847013 x 20 = 30.3687289847013 x 21 = − 92.1533845053006 x_{21} = -92.1533845053006 x 21 = − 92.1533845053006 x 22 = 89.0117918517108 x_{22} = 89.0117918517108 x 22 = 89.0117918517108 x 23 = 99.4837673636768 x_{23} = 99.4837673636768 x 23 = 99.4837673636768 x 24 = 86.9173967493176 x_{24} = 86.9173967493176 x 24 = 86.9173967493176 x 25 = 76.4454212373516 x_{25} = 76.4454212373516 x 25 = 76.4454212373516 x 26 = − 19.8967534727354 x_{26} = -19.8967534727354 x 26 = − 19.8967534727354 x 27 = 4.18879020478639 x_{27} = 4.18879020478639 x 27 = 4.18879020478639 x 28 = 85.870199198121 x_{28} = 85.870199198121 x 28 = 85.870199198121 x 29 = 8.37758040957278 x_{29} = 8.37758040957278 x 29 = 8.37758040957278 x 30 = − 77.4926187885482 x_{30} = -77.4926187885482 x 30 = − 77.4926187885482 x 31 = 60.7374579694027 x_{31} = 60.7374579694027 x 31 = 60.7374579694027 x 32 = − 13.6135681655558 x_{32} = -13.6135681655558 x 32 = − 13.6135681655558 x 33 = − 49.2182849062401 x_{33} = -49.2182849062401 x 33 = − 49.2182849062401 x 34 = 5.23598775598299 x_{34} = 5.23598775598299 x 34 = 5.23598775598299 x 35 = 68.0678408277789 x_{35} = 68.0678408277789 x 35 = 68.0678408277789 x 36 = 11.5191730631626 x_{36} = 11.5191730631626 x 36 = 11.5191730631626 x 37 = 77.4926187885482 x_{37} = 77.4926187885482 x 37 = 77.4926187885482 x 38 = − 90.0589894029074 x_{38} = -90.0589894029074 x 38 = − 90.0589894029074 x 39 = − 85.870199198121 x_{39} = -85.870199198121 x 39 = − 85.870199198121 x 40 = − 57.5958653158129 x_{40} = -57.5958653158129 x 40 = − 57.5958653158129 x 41 = − 63.8790506229925 x_{41} = -63.8790506229925 x 41 = − 63.8790506229925 x 42 = 33.5103216382911 x_{42} = 33.5103216382911 x 42 = 33.5103216382911 x 43 = − 24.0855436775217 x_{43} = -24.0855436775217 x 43 = − 24.0855436775217 x 44 = 1.0471975511966 x_{44} = 1.0471975511966 x 44 = 1.0471975511966 x 45 = 45.0294947014537 x_{45} = 45.0294947014537 x 45 = 45.0294947014537 x 46 = − 16.7551608191456 x_{46} = -16.7551608191456 x 46 = − 16.7551608191456 x 47 = 74.3510261349584 x_{47} = 74.3510261349584 x 47 = 74.3510261349584 x 48 = − 35.6047167406843 x_{48} = -35.6047167406843 x 48 = − 35.6047167406843 x 49 = − 83.7758040957278 x_{49} = -83.7758040957278 x 49 = − 83.7758040957278 x 50 = 96.342174710087 x_{50} = 96.342174710087 x 50 = 96.342174710087 x 51 = − 98.4365698124802 x_{51} = -98.4365698124802 x 51 = − 98.4365698124802 x 52 = − 42.9350995990605 x_{52} = -42.9350995990605 x 52 = − 42.9350995990605 x 53 = 71.2094334813686 x_{53} = 71.2094334813686 x 53 = 71.2094334813686 x 54 = − 33.5103216382911 x_{54} = -33.5103216382911 x 54 = − 33.5103216382911 x 55 = − 61.7846555205993 x_{55} = -61.7846555205993 x 55 = − 61.7846555205993 x 56 = − 26.1799387799149 x_{56} = -26.1799387799149 x 56 = − 26.1799387799149 x 57 = − 36.6519142918809 x_{57} = -36.6519142918809 x 57 = − 36.6519142918809 x 58 = 26.1799387799149 x_{58} = 26.1799387799149 x 58 = 26.1799387799149 x 59 = − 80.634211442138 x_{59} = -80.634211442138 x 59 = − 80.634211442138 x 60 = − 38.7463093942741 x_{60} = -38.7463093942741 x 60 = − 38.7463093942741 x 61 = 67.0206432765823 x_{61} = 67.0206432765823 x 61 = 67.0206432765823 x 62 = − 76.4454212373516 x_{62} = -76.4454212373516 x 62 = − 76.4454212373516 x 63 = 70.162235930172 x_{63} = 70.162235930172 x 63 = 70.162235930172 x 64 = 10.471975511966 x_{64} = 10.471975511966 x 64 = 10.471975511966 x 65 = 16.7551608191456 x_{65} = 16.7551608191456 x 65 = 16.7551608191456 x 66 = 55.5014702134197 x_{66} = 55.5014702134197 x 66 = 55.5014702134197 x 67 = 17.8023583703422 x_{67} = 17.8023583703422 x 67 = 17.8023583703422 x 68 = 48.1710873550435 x_{68} = 48.1710873550435 x 68 = 48.1710873550435 x 69 = 98.4365698124802 x_{69} = 98.4365698124802 x 69 = 98.4365698124802 x 70 = 82.7286065445312 x_{70} = 82.7286065445312 x 70 = 82.7286065445312 x 71 = − 11.5191730631626 x_{71} = -11.5191730631626 x 71 = − 11.5191730631626 x 72 = 92.1533845053006 x_{72} = 92.1533845053006 x 72 = 92.1533845053006 x 73 = − 82.7286065445312 x_{73} = -82.7286065445312 x 73 = − 82.7286065445312 x 74 = 52.3598775598299 x_{74} = 52.3598775598299 x 74 = 52.3598775598299 x 75 = 38.7463093942741 x_{75} = 38.7463093942741 x 75 = 38.7463093942741 x 76 = 27.2271363311115 x_{76} = 27.2271363311115 x 76 = 27.2271363311115 x 77 = 41.8879020478639 x_{77} = 41.8879020478639 x 77 = 41.8879020478639 x 78 = − 4.18879020478639 x_{78} = -4.18879020478639 x 78 = − 4.18879020478639 x 79 = 32.4631240870945 x_{79} = 32.4631240870945 x 79 = 32.4631240870945 x 80 = − 55.5014702134197 x_{80} = -55.5014702134197 x 80 = − 55.5014702134197 x 81 = − 20.943951023932 x_{81} = -20.943951023932 x 81 = − 20.943951023932 x 82 = − 54.4542726622231 x_{82} = -54.4542726622231 x 82 = − 54.4542726622231 x 83 = 39.7935069454707 x_{83} = 39.7935069454707 x 83 = 39.7935069454707 x 84 = 23.0383461263252 x_{84} = 23.0383461263252 x 84 = 23.0383461263252 x 85 = − 2.0943951023932 x_{85} = -2.0943951023932 x 85 = − 2.0943951023932 x 86 = − 68.0678408277789 x_{86} = -68.0678408277789 x 86 = − 68.0678408277789 x 87 = − 64.9262481741891 x_{87} = -64.9262481741891 x 87 = − 64.9262481741891 x 88 = 93.2005820564972 x_{88} = 93.2005820564972 x 88 = 93.2005820564972 x 89 = 24.0855436775217 x_{89} = 24.0855436775217 x 89 = 24.0855436775217 x 90 = 46.0766922526503 x_{90} = 46.0766922526503 x 90 = 46.0766922526503 x 91 = 49.2182849062401 x_{91} = 49.2182849062401 x 91 = 49.2182849062401 x 92 = 54.4542726622231 x_{92} = 54.4542726622231 x 92 = 54.4542726622231 x 93 = 63.8790506229925 x_{93} = 63.8790506229925 x 93 = 63.8790506229925 x 94 = − 46.0766922526503 x_{94} = -46.0766922526503 x 94 = − 46.0766922526503 x 95 = − 27.2271363311115 x_{95} = -27.2271363311115 x 95 = − 27.2271363311115 Las raíces dadas
x 1 = − 99.4837673636768 x_{1} = -99.4837673636768 x 1 = − 99.4837673636768 x 51 = − 98.4365698124802 x_{51} = -98.4365698124802 x 51 = − 98.4365698124802 x 6 = − 93.2005820564972 x_{6} = -93.2005820564972 x 6 = − 93.2005820564972 x 21 = − 92.1533845053006 x_{21} = -92.1533845053006 x 21 = − 92.1533845053006 x 38 = − 90.0589894029074 x_{38} = -90.0589894029074 x 38 = − 90.0589894029074 x 17 = − 86.9173967493176 x_{17} = -86.9173967493176 x 17 = − 86.9173967493176 x 39 = − 85.870199198121 x_{39} = -85.870199198121 x 39 = − 85.870199198121 x 49 = − 83.7758040957278 x_{49} = -83.7758040957278 x 49 = − 83.7758040957278 x 73 = − 82.7286065445312 x_{73} = -82.7286065445312 x 73 = − 82.7286065445312 x 59 = − 80.634211442138 x_{59} = -80.634211442138 x 59 = − 80.634211442138 x 14 = − 79.5870138909414 x_{14} = -79.5870138909414 x 14 = − 79.5870138909414 x 30 = − 77.4926187885482 x_{30} = -77.4926187885482 x 30 = − 77.4926187885482 x 62 = − 76.4454212373516 x_{62} = -76.4454212373516 x 62 = − 76.4454212373516 x 16 = − 71.2094334813686 x_{16} = -71.2094334813686 x 16 = − 71.2094334813686 x 13 = − 70.162235930172 x_{13} = -70.162235930172 x 13 = − 70.162235930172 x 86 = − 68.0678408277789 x_{86} = -68.0678408277789 x 86 = − 68.0678408277789 x 87 = − 64.9262481741891 x_{87} = -64.9262481741891 x 87 = − 64.9262481741891 x 41 = − 63.8790506229925 x_{41} = -63.8790506229925 x 41 = − 63.8790506229925 x 55 = − 61.7846555205993 x_{55} = -61.7846555205993 x 55 = − 61.7846555205993 x 2 = − 60.7374579694027 x_{2} = -60.7374579694027 x 2 = − 60.7374579694027 x 40 = − 57.5958653158129 x_{40} = -57.5958653158129 x 40 = − 57.5958653158129 x 80 = − 55.5014702134197 x_{80} = -55.5014702134197 x 80 = − 55.5014702134197 x 82 = − 54.4542726622231 x_{82} = -54.4542726622231 x 82 = − 54.4542726622231 x 33 = − 49.2182849062401 x_{33} = -49.2182849062401 x 33 = − 49.2182849062401 x 11 = − 48.1710873550435 x_{11} = -48.1710873550435 x 11 = − 48.1710873550435 x 94 = − 46.0766922526503 x_{94} = -46.0766922526503 x 94 = − 46.0766922526503 x 52 = − 42.9350995990605 x_{52} = -42.9350995990605 x 52 = − 42.9350995990605 x 19 = − 41.8879020478639 x_{19} = -41.8879020478639 x 19 = − 41.8879020478639 x 9 = − 39.7935069454707 x_{9} = -39.7935069454707 x 9 = − 39.7935069454707 x 60 = − 38.7463093942741 x_{60} = -38.7463093942741 x 60 = − 38.7463093942741 x 57 = − 36.6519142918809 x_{57} = -36.6519142918809 x 57 = − 36.6519142918809 x 48 = − 35.6047167406843 x_{48} = -35.6047167406843 x 48 = − 35.6047167406843 x 54 = − 33.5103216382911 x_{54} = -33.5103216382911 x 54 = − 33.5103216382911 x 15 = − 32.4631240870945 x_{15} = -32.4631240870945 x 15 = − 32.4631240870945 x 95 = − 27.2271363311115 x_{95} = -27.2271363311115 x 95 = − 27.2271363311115 x 56 = − 26.1799387799149 x_{56} = -26.1799387799149 x 56 = − 26.1799387799149 x 43 = − 24.0855436775217 x_{43} = -24.0855436775217 x 43 = − 24.0855436775217 x 81 = − 20.943951023932 x_{81} = -20.943951023932 x 81 = − 20.943951023932 x 26 = − 19.8967534727354 x_{26} = -19.8967534727354 x 26 = − 19.8967534727354 x 7 = − 17.8023583703422 x_{7} = -17.8023583703422 x 7 = − 17.8023583703422 x 46 = − 16.7551608191456 x_{46} = -16.7551608191456 x 46 = − 16.7551608191456 x 32 = − 13.6135681655558 x_{32} = -13.6135681655558 x 32 = − 13.6135681655558 x 71 = − 11.5191730631626 x_{71} = -11.5191730631626 x 71 = − 11.5191730631626 x 12 = − 10.471975511966 x_{12} = -10.471975511966 x 12 = − 10.471975511966 x 3 = − 5.23598775598299 x_{3} = -5.23598775598299 x 3 = − 5.23598775598299 x 78 = − 4.18879020478639 x_{78} = -4.18879020478639 x 78 = − 4.18879020478639 x 85 = − 2.0943951023932 x_{85} = -2.0943951023932 x 85 = − 2.0943951023932 x 44 = 1.0471975511966 x_{44} = 1.0471975511966 x 44 = 1.0471975511966 x 4 = 2.0943951023932 x_{4} = 2.0943951023932 x 4 = 2.0943951023932 x 27 = 4.18879020478639 x_{27} = 4.18879020478639 x 27 = 4.18879020478639 x 34 = 5.23598775598299 x_{34} = 5.23598775598299 x 34 = 5.23598775598299 x 29 = 8.37758040957278 x_{29} = 8.37758040957278 x 29 = 8.37758040957278 x 64 = 10.471975511966 x_{64} = 10.471975511966 x 64 = 10.471975511966 x 36 = 11.5191730631626 x_{36} = 11.5191730631626 x 36 = 11.5191730631626 x 65 = 16.7551608191456 x_{65} = 16.7551608191456 x 65 = 16.7551608191456 x 67 = 17.8023583703422 x_{67} = 17.8023583703422 x 67 = 17.8023583703422 x 8 = 19.8967534727354 x_{8} = 19.8967534727354 x 8 = 19.8967534727354 x 84 = 23.0383461263252 x_{84} = 23.0383461263252 x 84 = 23.0383461263252 x 89 = 24.0855436775217 x_{89} = 24.0855436775217 x 89 = 24.0855436775217 x 58 = 26.1799387799149 x_{58} = 26.1799387799149 x 58 = 26.1799387799149 x 76 = 27.2271363311115 x_{76} = 27.2271363311115 x 76 = 27.2271363311115 x 20 = 30.3687289847013 x_{20} = 30.3687289847013 x 20 = 30.3687289847013 x 79 = 32.4631240870945 x_{79} = 32.4631240870945 x 79 = 32.4631240870945 x 42 = 33.5103216382911 x_{42} = 33.5103216382911 x 42 = 33.5103216382911 x 75 = 38.7463093942741 x_{75} = 38.7463093942741 x 75 = 38.7463093942741 x 83 = 39.7935069454707 x_{83} = 39.7935069454707 x 83 = 39.7935069454707 x 77 = 41.8879020478639 x_{77} = 41.8879020478639 x 77 = 41.8879020478639 x 45 = 45.0294947014537 x_{45} = 45.0294947014537 x 45 = 45.0294947014537 x 90 = 46.0766922526503 x_{90} = 46.0766922526503 x 90 = 46.0766922526503 x 68 = 48.1710873550435 x_{68} = 48.1710873550435 x 68 = 48.1710873550435 x 91 = 49.2182849062401 x_{91} = 49.2182849062401 x 91 = 49.2182849062401 x 74 = 52.3598775598299 x_{74} = 52.3598775598299 x 74 = 52.3598775598299 x 92 = 54.4542726622231 x_{92} = 54.4542726622231 x 92 = 54.4542726622231 x 66 = 55.5014702134197 x_{66} = 55.5014702134197 x 66 = 55.5014702134197 x 31 = 60.7374579694027 x_{31} = 60.7374579694027 x 31 = 60.7374579694027 x 10 = 61.7846555205993 x_{10} = 61.7846555205993 x 10 = 61.7846555205993 x 93 = 63.8790506229925 x_{93} = 63.8790506229925 x 93 = 63.8790506229925 x 61 = 67.0206432765823 x_{61} = 67.0206432765823 x 61 = 67.0206432765823 x 35 = 68.0678408277789 x_{35} = 68.0678408277789 x 35 = 68.0678408277789 x 63 = 70.162235930172 x_{63} = 70.162235930172 x 63 = 70.162235930172 x 53 = 71.2094334813686 x_{53} = 71.2094334813686 x 53 = 71.2094334813686 x 47 = 74.3510261349584 x_{47} = 74.3510261349584 x 47 = 74.3510261349584 x 25 = 76.4454212373516 x_{25} = 76.4454212373516 x 25 = 76.4454212373516 x 37 = 77.4926187885482 x_{37} = 77.4926187885482 x 37 = 77.4926187885482 x 70 = 82.7286065445312 x_{70} = 82.7286065445312 x 70 = 82.7286065445312 x 18 = 83.7758040957278 x_{18} = 83.7758040957278 x 18 = 83.7758040957278 x 28 = 85.870199198121 x_{28} = 85.870199198121 x 28 = 85.870199198121 x 24 = 86.9173967493176 x_{24} = 86.9173967493176 x 24 = 86.9173967493176 x 22 = 89.0117918517108 x_{22} = 89.0117918517108 x 22 = 89.0117918517108 x 5 = 90.0589894029074 x_{5} = 90.0589894029074 x 5 = 90.0589894029074 x 72 = 92.1533845053006 x_{72} = 92.1533845053006 x 72 = 92.1533845053006 x 88 = 93.2005820564972 x_{88} = 93.2005820564972 x 88 = 93.2005820564972 x 50 = 96.342174710087 x_{50} = 96.342174710087 x 50 = 96.342174710087 x 69 = 98.4365698124802 x_{69} = 98.4365698124802 x 69 = 98.4365698124802 x 23 = 99.4837673636768 x_{23} = 99.4837673636768 x 23 = 99.4837673636768 son puntos de cambio del signo de desigualdad en las soluciones.
Primero definámonos con el signo hasta el punto extremo izquierdo:
x 0 ≤ x 1 x_{0} \leq x_{1} x 0 ≤ x 1 Consideremos, por ejemplo, el punto
x 0 = x 1 − 1 10 x_{0} = x_{1} - \frac{1}{10} x 0 = x 1 − 10 1 =
− 99.4837673636768 + − 1 10 -99.4837673636768 + - \frac{1}{10} − 99.4837673636768 + − 10 1 =
− 99.5837673636768 -99.5837673636768 − 99.5837673636768 lo sustituimos en la expresión
∣ sin ( x ) ∣ ≤ 3 2 \left|{\sin{\left(x \right)}}\right| \leq \frac{\sqrt{3}}{2} ∣ sin ( x ) ∣ ≤ 2 3 ∣ sin ( − 99.5837673636768 ) ∣ ≤ 3 2 \left|{\sin{\left(-99.5837673636768 \right)}}\right| \leq \frac{\sqrt{3}}{2} ∣ sin ( − 99.5837673636768 ) ∣ ≤ 2 3 ___
\/ 3
0.811782175678687 <= -----
2
significa que una de las soluciones de nuestra ecuación será con:
x ≤ − 99.4837673636768 x \leq -99.4837673636768 x ≤ − 99.4837673636768 _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____
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x1 x51 x6 x21 x38 x17 x39 x49 x73 x59 x14 x30 x62 x16 x13 x86 x87 x41 x55 x2 x40 x80 x82 x33 x11 x94 x52 x19 x9 x60 x57 x48 x54 x15 x95 x56 x43 x81 x26 x7 x46 x32 x71 x12 x3 x78 x85 x44 x4 x27 x34 x29 x64 x36 x65 x67 x8 x84 x89 x58 x76 x20 x79 x42 x75 x83 x77 x45 x90 x68 x91 x74 x92 x66 x31 x10 x93 x61 x35 x63 x53 x47 x25 x37 x70 x18 x28 x24 x22 x5 x72 x88 x50 x69 x23 Recibiremos otras soluciones de la desigualdad pasando al polo siguiente etc.
etc.
Respuesta:
x ≤ − 99.4837673636768 x \leq -99.4837673636768 x ≤ − 99.4837673636768 x ≥ − 98.4365698124802 ∧ x ≤ − 93.2005820564972 x \geq -98.4365698124802 \wedge x \leq -93.2005820564972 x ≥ − 98.4365698124802 ∧ x ≤ − 93.2005820564972 x ≥ − 92.1533845053006 ∧ x ≤ − 90.0589894029074 x \geq -92.1533845053006 \wedge x \leq -90.0589894029074 x ≥ − 92.1533845053006 ∧ x ≤ − 90.0589894029074 x ≥ − 86.9173967493176 ∧ x ≤ − 85.870199198121 x \geq -86.9173967493176 \wedge x \leq -85.870199198121 x ≥ − 86.9173967493176 ∧ x ≤ − 85.870199198121 x ≥ − 83.7758040957278 ∧ x ≤ − 82.7286065445312 x \geq -83.7758040957278 \wedge x \leq -82.7286065445312 x ≥ − 83.7758040957278 ∧ x ≤ − 82.7286065445312 x ≥ − 80.634211442138 ∧ x ≤ − 79.5870138909414 x \geq -80.634211442138 \wedge x \leq -79.5870138909414 x ≥ − 80.634211442138 ∧ x ≤ − 79.5870138909414 x ≥ − 77.4926187885482 ∧ x ≤ − 76.4454212373516 x \geq -77.4926187885482 \wedge x \leq -76.4454212373516 x ≥ − 77.4926187885482 ∧ x ≤ − 76.4454212373516 x ≥ − 71.2094334813686 ∧ x ≤ − 70.162235930172 x \geq -71.2094334813686 \wedge x \leq -70.162235930172 x ≥ − 71.2094334813686 ∧ x ≤ − 70.162235930172 x ≥ − 68.0678408277789 ∧ x ≤ − 64.9262481741891 x \geq -68.0678408277789 \wedge x \leq -64.9262481741891 x ≥ − 68.0678408277789 ∧ x ≤ − 64.9262481741891 x ≥ − 63.8790506229925 ∧ x ≤ − 61.7846555205993 x \geq -63.8790506229925 \wedge x \leq -61.7846555205993 x ≥ − 63.8790506229925 ∧ x ≤ − 61.7846555205993 x ≥ − 60.7374579694027 ∧ x ≤ − 57.5958653158129 x \geq -60.7374579694027 \wedge x \leq -57.5958653158129 x ≥ − 60.7374579694027 ∧ x ≤ − 57.5958653158129 x ≥ − 55.5014702134197 ∧ x ≤ − 54.4542726622231 x \geq -55.5014702134197 \wedge x \leq -54.4542726622231 x ≥ − 55.5014702134197 ∧ x ≤ − 54.4542726622231 x ≥ − 49.2182849062401 ∧ x ≤ − 48.1710873550435 x \geq -49.2182849062401 \wedge x \leq -48.1710873550435 x ≥ − 49.2182849062401 ∧ x ≤ − 48.1710873550435 x ≥ − 46.0766922526503 ∧ x ≤ − 42.9350995990605 x \geq -46.0766922526503 \wedge x \leq -42.9350995990605 x ≥ − 46.0766922526503 ∧ x ≤ − 42.9350995990605 x ≥ − 41.8879020478639 ∧ x ≤ − 39.7935069454707 x \geq -41.8879020478639 \wedge x \leq -39.7935069454707 x ≥ − 41.8879020478639 ∧ x ≤ − 39.7935069454707 x ≥ − 38.7463093942741 ∧ x ≤ − 36.6519142918809 x \geq -38.7463093942741 \wedge x \leq -36.6519142918809 x ≥ − 38.7463093942741 ∧ x ≤ − 36.6519142918809 x ≥ − 35.6047167406843 ∧ x ≤ − 33.5103216382911 x \geq -35.6047167406843 \wedge x \leq -33.5103216382911 x ≥ − 35.6047167406843 ∧ x ≤ − 33.5103216382911 x ≥ − 32.4631240870945 ∧ x ≤ − 27.2271363311115 x \geq -32.4631240870945 \wedge x \leq -27.2271363311115 x ≥ − 32.4631240870945 ∧ x ≤ − 27.2271363311115 x ≥ − 26.1799387799149 ∧ x ≤ − 24.0855436775217 x \geq -26.1799387799149 \wedge x \leq -24.0855436775217 x ≥ − 26.1799387799149 ∧ x ≤ − 24.0855436775217 x ≥ − 20.943951023932 ∧ x ≤ − 19.8967534727354 x \geq -20.943951023932 \wedge x \leq -19.8967534727354 x ≥ − 20.943951023932 ∧ x ≤ − 19.8967534727354 x ≥ − 17.8023583703422 ∧ x ≤ − 16.7551608191456 x \geq -17.8023583703422 \wedge x \leq -16.7551608191456 x ≥ − 17.8023583703422 ∧ x ≤ − 16.7551608191456 x ≥ − 13.6135681655558 ∧ x ≤ − 11.5191730631626 x \geq -13.6135681655558 \wedge x \leq -11.5191730631626 x ≥ − 13.6135681655558 ∧ x ≤ − 11.5191730631626 x ≥ − 10.471975511966 ∧ x ≤ − 5.23598775598299 x \geq -10.471975511966 \wedge x \leq -5.23598775598299 x ≥ − 10.471975511966 ∧ x ≤ − 5.23598775598299 x ≥ − 4.18879020478639 ∧ x ≤ − 2.0943951023932 x \geq -4.18879020478639 \wedge x \leq -2.0943951023932 x ≥ − 4.18879020478639 ∧ x ≤ − 2.0943951023932 x ≥ 1.0471975511966 ∧ x ≤ 2.0943951023932 x \geq 1.0471975511966 \wedge x \leq 2.0943951023932 x ≥ 1.0471975511966 ∧ x ≤ 2.0943951023932 x ≥ 4.18879020478639 ∧ x ≤ 5.23598775598299 x \geq 4.18879020478639 \wedge x \leq 5.23598775598299 x ≥ 4.18879020478639 ∧ x ≤ 5.23598775598299 x ≥ 8.37758040957278 ∧ x ≤ 10.471975511966 x \geq 8.37758040957278 \wedge x \leq 10.471975511966 x ≥ 8.37758040957278 ∧ x ≤ 10.471975511966 x ≥ 11.5191730631626 ∧ x ≤ 16.7551608191456 x \geq 11.5191730631626 \wedge x \leq 16.7551608191456 x ≥ 11.5191730631626 ∧ x ≤ 16.7551608191456 x ≥ 17.8023583703422 ∧ x ≤ 19.8967534727354 x \geq 17.8023583703422 \wedge x \leq 19.8967534727354 x ≥ 17.8023583703422 ∧ x ≤ 19.8967534727354 x ≥ 23.0383461263252 ∧ x ≤ 24.0855436775217 x \geq 23.0383461263252 \wedge x \leq 24.0855436775217 x ≥ 23.0383461263252 ∧ x ≤ 24.0855436775217 x ≥ 26.1799387799149 ∧ x ≤ 27.2271363311115 x \geq 26.1799387799149 \wedge x \leq 27.2271363311115 x ≥ 26.1799387799149 ∧ x ≤ 27.2271363311115 x ≥ 30.3687289847013 ∧ x ≤ 32.4631240870945 x \geq 30.3687289847013 \wedge x \leq 32.4631240870945 x ≥ 30.3687289847013 ∧ x ≤ 32.4631240870945 x ≥ 33.5103216382911 ∧ x ≤ 38.7463093942741 x \geq 33.5103216382911 \wedge x \leq 38.7463093942741 x ≥ 33.5103216382911 ∧ x ≤ 38.7463093942741 x ≥ 39.7935069454707 ∧ x ≤ 41.8879020478639 x \geq 39.7935069454707 \wedge x \leq 41.8879020478639 x ≥ 39.7935069454707 ∧ x ≤ 41.8879020478639 x ≥ 45.0294947014537 ∧ x ≤ 46.0766922526503 x \geq 45.0294947014537 \wedge x \leq 46.0766922526503 x ≥ 45.0294947014537 ∧ x ≤ 46.0766922526503 x ≥ 48.1710873550435 ∧ x ≤ 49.2182849062401 x \geq 48.1710873550435 \wedge x \leq 49.2182849062401 x ≥ 48.1710873550435 ∧ x ≤ 49.2182849062401 x ≥ 52.3598775598299 ∧ x ≤ 54.4542726622231 x \geq 52.3598775598299 \wedge x \leq 54.4542726622231 x ≥ 52.3598775598299 ∧ x ≤ 54.4542726622231 x ≥ 55.5014702134197 ∧ x ≤ 60.7374579694027 x \geq 55.5014702134197 \wedge x \leq 60.7374579694027 x ≥ 55.5014702134197 ∧ x ≤ 60.7374579694027 x ≥ 61.7846555205993 ∧ x ≤ 63.8790506229925 x \geq 61.7846555205993 \wedge x \leq 63.8790506229925 x ≥ 61.7846555205993 ∧ x ≤ 63.8790506229925 x ≥ 67.0206432765823 ∧ x ≤ 68.0678408277789 x \geq 67.0206432765823 \wedge x \leq 68.0678408277789 x ≥ 67.0206432765823 ∧ x ≤ 68.0678408277789 x ≥ 70.162235930172 ∧ x ≤ 71.2094334813686 x \geq 70.162235930172 \wedge x \leq 71.2094334813686 x ≥ 70.162235930172 ∧ x ≤ 71.2094334813686 x ≥ 74.3510261349584 ∧ x ≤ 76.4454212373516 x \geq 74.3510261349584 \wedge x \leq 76.4454212373516 x ≥ 74.3510261349584 ∧ x ≤ 76.4454212373516 x ≥ 77.4926187885482 ∧ x ≤ 82.7286065445312 x \geq 77.4926187885482 \wedge x \leq 82.7286065445312 x ≥ 77.4926187885482 ∧ x ≤ 82.7286065445312 x ≥ 83.7758040957278 ∧ x ≤ 85.870199198121 x \geq 83.7758040957278 \wedge x \leq 85.870199198121 x ≥ 83.7758040957278 ∧ x ≤ 85.870199198121 x ≥ 86.9173967493176 ∧ x ≤ 89.0117918517108 x \geq 86.9173967493176 \wedge x \leq 89.0117918517108 x ≥ 86.9173967493176 ∧ x ≤ 89.0117918517108 x ≥ 90.0589894029074 ∧ x ≤ 92.1533845053006 x \geq 90.0589894029074 \wedge x \leq 92.1533845053006 x ≥ 90.0589894029074 ∧ x ≤ 92.1533845053006 x ≥ 93.2005820564972 ∧ x ≤ 96.342174710087 x \geq 93.2005820564972 \wedge x \leq 96.342174710087 x ≥ 93.2005820564972 ∧ x ≤ 96.342174710087 x ≥ 98.4365698124802 ∧ x ≤ 99.4837673636768 x \geq 98.4365698124802 \wedge x \leq 99.4837673636768 x ≥ 98.4365698124802 ∧ x ≤ 99.4837673636768 
-x^2+3x-2<0
x^2+4x-5<0
x+1>0
x^2-6x-27>0