Se da la desigualdad:
∣ cos ( x ) ∣ < 3 2 \left|{\cos{\left(x \right)}}\right| < \frac{\sqrt{3}}{2} ∣ cos ( x ) ∣ < 2 3 Para resolver esta desigualdad primero hay que resolver la ecuación correspondiente:
∣ cos ( x ) ∣ = 3 2 \left|{\cos{\left(x \right)}}\right| = \frac{\sqrt{3}}{2} ∣ cos ( x ) ∣ = 2 3 Resolvemos:
Tenemos la ecuación
∣ cos ( x ) ∣ = 3 2 \left|{\cos{\left(x \right)}}\right| = \frac{\sqrt{3}}{2} ∣ cos ( x ) ∣ = 2 3 cambiamos
∣ cos ( x ) ∣ − 1 − 3 2 = 0 \left|{\cos{\left(x \right)}}\right| - 1 - \frac{\sqrt{3}}{2} = 0 ∣ cos ( x ) ∣ − 1 − 2 3 = 0 ∣ cos ( x ) ∣ − 1 − 3 2 = 0 \left|{\cos{\left(x \right)}}\right| - 1 - \frac{\sqrt{3}}{2} = 0 ∣ cos ( x ) ∣ − 1 − 2 3 = 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 - 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
∣ cos ( x ) ∣ = w \left|{\cos{\left(x \right)}}\right| = w ∣ cos ( x ) ∣ = w sustituimos w:
x 1 = 69.6386371545737 x_{1} = 69.6386371545737 x 1 = 69.6386371545737 x 2 = 25.6563400043166 x_{2} = 25.6563400043166 x 2 = 25.6563400043166 x 3 = 22.5147473507269 x_{3} = 22.5147473507269 x 3 = 22.5147473507269 x 4 = − 25.6563400043166 x_{4} = -25.6563400043166 x 4 = − 25.6563400043166 x 5 = − 60.2138591938044 x_{5} = -60.2138591938044 x 5 = − 60.2138591938044 x 6 = 41.3643032722656 x_{6} = 41.3643032722656 x 6 = 41.3643032722656 x 7 = 84.2994028713261 x_{7} = 84.2994028713261 x 7 = 84.2994028713261 x 8 = 60.2138591938044 x_{8} = 60.2138591938044 x 8 = 60.2138591938044 x 9 = 31.9395253114962 x_{9} = 31.9395253114962 x 9 = 31.9395253114962 x 10 = 85.3466004225227 x_{10} = 85.3466004225227 x 10 = 85.3466004225227 x 11 = 53.9306738866248 x_{11} = 53.9306738866248 x 11 = 53.9306738866248 x 12 = 18.3259571459405 x_{12} = 18.3259571459405 x 12 = 18.3259571459405 x 13 = 44.5058959258554 x_{13} = 44.5058959258554 x 13 = 44.5058959258554 x 14 = 91.6297857297023 x_{14} = 91.6297857297023 x 14 = 91.6297857297023 x 15 = − 91.6297857297023 x_{15} = -91.6297857297023 x 15 = − 91.6297857297023 x 16 = − 103.148958792865 x_{16} = -103.148958792865 x 16 = − 103.148958792865 x 17 = 5.75958653158129 x_{17} = 5.75958653158129 x 17 = 5.75958653158129 x 18 = 9.94837673636768 x_{18} = 9.94837673636768 x 18 = 9.94837673636768 x 19 = 49.7418836818384 x_{19} = 49.7418836818384 x 19 = 49.7418836818384 x 20 = − 49.7418836818384 x_{20} = -49.7418836818384 x 20 = − 49.7418836818384 x 21 = 74.8746249105567 x_{21} = 74.8746249105567 x 21 = 74.8746249105567 x 22 = 2.61799387799149 x_{22} = 2.61799387799149 x 22 = 2.61799387799149 x 23 = 8.90117918517108 x_{23} = 8.90117918517108 x 23 = 8.90117918517108 x 24 = − 65.4498469497874 x_{24} = -65.4498469497874 x 24 = − 65.4498469497874 x 25 = 27.7507351067098 x_{25} = 27.7507351067098 x 25 = 27.7507351067098 x 26 = − 85.3466004225227 x_{26} = -85.3466004225227 x 26 = − 85.3466004225227 x 27 = − 90.5825881785057 x_{27} = -90.5825881785057 x 27 = − 90.5825881785057 x 28 = − 79.0634151153431 x_{28} = -79.0634151153431 x 28 = − 79.0634151153431 x 29 = − 31.9395253114962 x_{29} = -31.9395253114962 x 29 = − 31.9395253114962 x 30 = − 97.9129710368819 x_{30} = -97.9129710368819 x 30 = − 97.9129710368819 x 31 = − 18.3259571459405 x_{31} = -18.3259571459405 x 31 = − 18.3259571459405 x 32 = − 43.4586983746588 x_{32} = -43.4586983746588 x 32 = − 43.4586983746588 x 33 = − 82.2050077689329 x_{33} = -82.2050077689329 x 33 = − 82.2050077689329 x 34 = 93.7241808320955 x_{34} = 93.7241808320955 x 34 = 93.7241808320955 x 35 = − 19.3731546971371 x_{35} = -19.3731546971371 x 35 = − 19.3731546971371 x 36 = 88.4881930761125 x_{36} = 88.4881930761125 x 36 = 88.4881930761125 x 37 = − 62.3082542961976 x_{37} = -62.3082542961976 x 37 = − 62.3082542961976 x 38 = 66.497044500984 x_{38} = 66.497044500984 x 38 = 66.497044500984 x 39 = − 78.0162175641465 x_{39} = -78.0162175641465 x 39 = − 78.0162175641465 x 40 = − 9.94837673636768 x_{40} = -9.94837673636768 x 40 = − 9.94837673636768 x 41 = 16.2315620435473 x_{41} = 16.2315620435473 x 41 = 16.2315620435473 x 42 = 46.6002910282486 x_{42} = 46.6002910282486 x 42 = 46.6002910282486 x 43 = − 5.75958653158129 x_{43} = -5.75958653158129 x 43 = − 5.75958653158129 x 44 = − 100.007366139275 x_{44} = -100.007366139275 x 44 = − 100.007366139275 x 45 = 52.8834763354282 x_{45} = 52.8834763354282 x 45 = 52.8834763354282 x 46 = − 87.4409955249159 x_{46} = -87.4409955249159 x 46 = − 87.4409955249159 x 47 = − 93.7241808320955 x_{47} = -93.7241808320955 x 47 = − 93.7241808320955 x 48 = − 40.317105721069 x_{48} = -40.317105721069 x 48 = − 40.317105721069 x 49 = 4868.94501428858 x_{49} = 4868.94501428858 x 49 = 4868.94501428858 x 50 = 78.0162175641465 x_{50} = 78.0162175641465 x 50 = 78.0162175641465 x 51 = − 21.4675497995303 x_{51} = -21.4675497995303 x 51 = − 21.4675497995303 x 52 = − 57.0722665402146 x_{52} = -57.0722665402146 x 52 = − 57.0722665402146 x 53 = 100.007366139275 x_{53} = 100.007366139275 x 53 = 100.007366139275 x 54 = 82.2050077689329 x_{54} = 82.2050077689329 x 54 = 82.2050077689329 x 55 = − 75.9218224617533 x_{55} = -75.9218224617533 x 55 = − 75.9218224617533 x 56 = − 53.9306738866248 x_{56} = -53.9306738866248 x 56 = − 53.9306738866248 x 57 = − 27.7507351067098 x_{57} = -27.7507351067098 x 57 = − 27.7507351067098 x 58 = − 2.61799387799149 x_{58} = -2.61799387799149 x 58 = − 2.61799387799149 x 59 = 63.3554518473942 x_{59} = 63.3554518473942 x 59 = 63.3554518473942 x 60 = 0.523598775598299 x_{60} = 0.523598775598299 x 60 = 0.523598775598299 x 61 = 3.66519142918809 x_{61} = 3.66519142918809 x 61 = 3.66519142918809 x 62 = − 16.2315620435473 x_{62} = -16.2315620435473 x 62 = − 16.2315620435473 x 63 = − 56.025068989018 x_{63} = -56.025068989018 x 63 = − 56.025068989018 x 64 = − 84.2994028713261 x_{64} = -84.2994028713261 x 64 = − 84.2994028713261 x 65 = 68.5914396033772 x_{65} = 68.5914396033772 x 65 = 68.5914396033772 x 66 = 24.60914245312 x_{66} = 24.60914245312 x 66 = 24.60914245312 x 67 = 47.6474885794452 x_{67} = 47.6474885794452 x 67 = 47.6474885794452 x 68 = − 34.0339204138894 x_{68} = -34.0339204138894 x 68 = − 34.0339204138894 x 69 = 75.9218224617533 x_{69} = 75.9218224617533 x 69 = 75.9218224617533 x 70 = 97.9129710368819 x_{70} = 97.9129710368819 x 70 = 97.9129710368819 x 71 = 12.0427718387609 x_{71} = 12.0427718387609 x 71 = 12.0427718387609 x 72 = − 46.6002910282486 x_{72} = -46.6002910282486 x 72 = − 46.6002910282486 x 73 = 30.8923277602996 x_{73} = 30.8923277602996 x 73 = 30.8923277602996 x 74 = − 35.081117965086 x_{74} = -35.081117965086 x 74 = − 35.081117965086 x 75 = − 63.3554518473942 x_{75} = -63.3554518473942 x 75 = − 63.3554518473942 x 76 = 19.3731546971371 x_{76} = 19.3731546971371 x 76 = 19.3731546971371 x 77 = 62.3082542961976 x_{77} = 62.3082542961976 x 77 = 62.3082542961976 x 78 = − 68.5914396033772 x_{78} = -68.5914396033772 x 78 = − 68.5914396033772 x 79 = 71.733032256967 x_{79} = 71.733032256967 x 79 = 71.733032256967 x 80 = − 12.0427718387609 x_{80} = -12.0427718387609 x 80 = − 12.0427718387609 x 81 = − 24.60914245312 x_{81} = -24.60914245312 x 81 = − 24.60914245312 x 82 = 90.5825881785057 x_{82} = 90.5825881785057 x 82 = 90.5825881785057 x 83 = 6.80678408277789 x_{83} = 6.80678408277789 x 83 = 6.80678408277789 x 84 = − 69.6386371545737 x_{84} = -69.6386371545737 x 84 = − 69.6386371545737 x 85 = − 71.733032256967 x_{85} = -71.733032256967 x 85 = − 71.733032256967 x 86 = 38.2227106186758 x_{86} = 38.2227106186758 x 86 = 38.2227106186758 x 87 = − 13.0899693899575 x_{87} = -13.0899693899575 x 87 = − 13.0899693899575 x 88 = 40.317105721069 x_{88} = 40.317105721069 x 88 = 40.317105721069 x 89 = − 41.3643032722656 x_{89} = -41.3643032722656 x 89 = − 41.3643032722656 x 90 = 56.025068989018 x_{90} = 56.025068989018 x 90 = 56.025068989018 x 91 = − 47.6474885794452 x_{91} = -47.6474885794452 x 91 = − 47.6474885794452 x 92 = − 3.66519142918809 x_{92} = -3.66519142918809 x 92 = − 3.66519142918809 x 93 = 96.8657734856853 x_{93} = 96.8657734856853 x 93 = 96.8657734856853 x 94 = 34.0339204138894 x_{94} = 34.0339204138894 x 94 = 34.0339204138894 x 95 = − 38.2227106186758 x_{95} = -38.2227106186758 x 95 = − 38.2227106186758 x 1 = 69.6386371545737 x_{1} = 69.6386371545737 x 1 = 69.6386371545737 x 2 = 25.6563400043166 x_{2} = 25.6563400043166 x 2 = 25.6563400043166 x 3 = 22.5147473507269 x_{3} = 22.5147473507269 x 3 = 22.5147473507269 x 4 = − 25.6563400043166 x_{4} = -25.6563400043166 x 4 = − 25.6563400043166 x 5 = − 60.2138591938044 x_{5} = -60.2138591938044 x 5 = − 60.2138591938044 x 6 = 41.3643032722656 x_{6} = 41.3643032722656 x 6 = 41.3643032722656 x 7 = 84.2994028713261 x_{7} = 84.2994028713261 x 7 = 84.2994028713261 x 8 = 60.2138591938044 x_{8} = 60.2138591938044 x 8 = 60.2138591938044 x 9 = 31.9395253114962 x_{9} = 31.9395253114962 x 9 = 31.9395253114962 x 10 = 85.3466004225227 x_{10} = 85.3466004225227 x 10 = 85.3466004225227 x 11 = 53.9306738866248 x_{11} = 53.9306738866248 x 11 = 53.9306738866248 x 12 = 18.3259571459405 x_{12} = 18.3259571459405 x 12 = 18.3259571459405 x 13 = 44.5058959258554 x_{13} = 44.5058959258554 x 13 = 44.5058959258554 x 14 = 91.6297857297023 x_{14} = 91.6297857297023 x 14 = 91.6297857297023 x 15 = − 91.6297857297023 x_{15} = -91.6297857297023 x 15 = − 91.6297857297023 x 16 = − 103.148958792865 x_{16} = -103.148958792865 x 16 = − 103.148958792865 x 17 = 5.75958653158129 x_{17} = 5.75958653158129 x 17 = 5.75958653158129 x 18 = 9.94837673636768 x_{18} = 9.94837673636768 x 18 = 9.94837673636768 x 19 = 49.7418836818384 x_{19} = 49.7418836818384 x 19 = 49.7418836818384 x 20 = − 49.7418836818384 x_{20} = -49.7418836818384 x 20 = − 49.7418836818384 x 21 = 74.8746249105567 x_{21} = 74.8746249105567 x 21 = 74.8746249105567 x 22 = 2.61799387799149 x_{22} = 2.61799387799149 x 22 = 2.61799387799149 x 23 = 8.90117918517108 x_{23} = 8.90117918517108 x 23 = 8.90117918517108 x 24 = − 65.4498469497874 x_{24} = -65.4498469497874 x 24 = − 65.4498469497874 x 25 = 27.7507351067098 x_{25} = 27.7507351067098 x 25 = 27.7507351067098 x 26 = − 85.3466004225227 x_{26} = -85.3466004225227 x 26 = − 85.3466004225227 x 27 = − 90.5825881785057 x_{27} = -90.5825881785057 x 27 = − 90.5825881785057 x 28 = − 79.0634151153431 x_{28} = -79.0634151153431 x 28 = − 79.0634151153431 x 29 = − 31.9395253114962 x_{29} = -31.9395253114962 x 29 = − 31.9395253114962 x 30 = − 97.9129710368819 x_{30} = -97.9129710368819 x 30 = − 97.9129710368819 x 31 = − 18.3259571459405 x_{31} = -18.3259571459405 x 31 = − 18.3259571459405 x 32 = − 43.4586983746588 x_{32} = -43.4586983746588 x 32 = − 43.4586983746588 x 33 = − 82.2050077689329 x_{33} = -82.2050077689329 x 33 = − 82.2050077689329 x 34 = 93.7241808320955 x_{34} = 93.7241808320955 x 34 = 93.7241808320955 x 35 = − 19.3731546971371 x_{35} = -19.3731546971371 x 35 = − 19.3731546971371 x 36 = 88.4881930761125 x_{36} = 88.4881930761125 x 36 = 88.4881930761125 x 37 = − 62.3082542961976 x_{37} = -62.3082542961976 x 37 = − 62.3082542961976 x 38 = 66.497044500984 x_{38} = 66.497044500984 x 38 = 66.497044500984 x 39 = − 78.0162175641465 x_{39} = -78.0162175641465 x 39 = − 78.0162175641465 x 40 = − 9.94837673636768 x_{40} = -9.94837673636768 x 40 = − 9.94837673636768 x 41 = 16.2315620435473 x_{41} = 16.2315620435473 x 41 = 16.2315620435473 x 42 = 46.6002910282486 x_{42} = 46.6002910282486 x 42 = 46.6002910282486 x 43 = − 5.75958653158129 x_{43} = -5.75958653158129 x 43 = − 5.75958653158129 x 44 = − 100.007366139275 x_{44} = -100.007366139275 x 44 = − 100.007366139275 x 45 = 52.8834763354282 x_{45} = 52.8834763354282 x 45 = 52.8834763354282 x 46 = − 87.4409955249159 x_{46} = -87.4409955249159 x 46 = − 87.4409955249159 x 47 = − 93.7241808320955 x_{47} = -93.7241808320955 x 47 = − 93.7241808320955 x 48 = − 40.317105721069 x_{48} = -40.317105721069 x 48 = − 40.317105721069 x 49 = 4868.94501428858 x_{49} = 4868.94501428858 x 49 = 4868.94501428858 x 50 = 78.0162175641465 x_{50} = 78.0162175641465 x 50 = 78.0162175641465 x 51 = − 21.4675497995303 x_{51} = -21.4675497995303 x 51 = − 21.4675497995303 x 52 = − 57.0722665402146 x_{52} = -57.0722665402146 x 52 = − 57.0722665402146 x 53 = 100.007366139275 x_{53} = 100.007366139275 x 53 = 100.007366139275 x 54 = 82.2050077689329 x_{54} = 82.2050077689329 x 54 = 82.2050077689329 x 55 = − 75.9218224617533 x_{55} = -75.9218224617533 x 55 = − 75.9218224617533 x 56 = − 53.9306738866248 x_{56} = -53.9306738866248 x 56 = − 53.9306738866248 x 57 = − 27.7507351067098 x_{57} = -27.7507351067098 x 57 = − 27.7507351067098 x 58 = − 2.61799387799149 x_{58} = -2.61799387799149 x 58 = − 2.61799387799149 x 59 = 63.3554518473942 x_{59} = 63.3554518473942 x 59 = 63.3554518473942 x 60 = 0.523598775598299 x_{60} = 0.523598775598299 x 60 = 0.523598775598299 x 61 = 3.66519142918809 x_{61} = 3.66519142918809 x 61 = 3.66519142918809 x 62 = − 16.2315620435473 x_{62} = -16.2315620435473 x 62 = − 16.2315620435473 x 63 = − 56.025068989018 x_{63} = -56.025068989018 x 63 = − 56.025068989018 x 64 = − 84.2994028713261 x_{64} = -84.2994028713261 x 64 = − 84.2994028713261 x 65 = 68.5914396033772 x_{65} = 68.5914396033772 x 65 = 68.5914396033772 x 66 = 24.60914245312 x_{66} = 24.60914245312 x 66 = 24.60914245312 x 67 = 47.6474885794452 x_{67} = 47.6474885794452 x 67 = 47.6474885794452 x 68 = − 34.0339204138894 x_{68} = -34.0339204138894 x 68 = − 34.0339204138894 x 69 = 75.9218224617533 x_{69} = 75.9218224617533 x 69 = 75.9218224617533 x 70 = 97.9129710368819 x_{70} = 97.9129710368819 x 70 = 97.9129710368819 x 71 = 12.0427718387609 x_{71} = 12.0427718387609 x 71 = 12.0427718387609 x 72 = − 46.6002910282486 x_{72} = -46.6002910282486 x 72 = − 46.6002910282486 x 73 = 30.8923277602996 x_{73} = 30.8923277602996 x 73 = 30.8923277602996 x 74 = − 35.081117965086 x_{74} = -35.081117965086 x 74 = − 35.081117965086 x 75 = − 63.3554518473942 x_{75} = -63.3554518473942 x 75 = − 63.3554518473942 x 76 = 19.3731546971371 x_{76} = 19.3731546971371 x 76 = 19.3731546971371 x 77 = 62.3082542961976 x_{77} = 62.3082542961976 x 77 = 62.3082542961976 x 78 = − 68.5914396033772 x_{78} = -68.5914396033772 x 78 = − 68.5914396033772 x 79 = 71.733032256967 x_{79} = 71.733032256967 x 79 = 71.733032256967 x 80 = − 12.0427718387609 x_{80} = -12.0427718387609 x 80 = − 12.0427718387609 x 81 = − 24.60914245312 x_{81} = -24.60914245312 x 81 = − 24.60914245312 x 82 = 90.5825881785057 x_{82} = 90.5825881785057 x 82 = 90.5825881785057 x 83 = 6.80678408277789 x_{83} = 6.80678408277789 x 83 = 6.80678408277789 x 84 = − 69.6386371545737 x_{84} = -69.6386371545737 x 84 = − 69.6386371545737 x 85 = − 71.733032256967 x_{85} = -71.733032256967 x 85 = − 71.733032256967 x 86 = 38.2227106186758 x_{86} = 38.2227106186758 x 86 = 38.2227106186758 x 87 = − 13.0899693899575 x_{87} = -13.0899693899575 x 87 = − 13.0899693899575 x 88 = 40.317105721069 x_{88} = 40.317105721069 x 88 = 40.317105721069 x 89 = − 41.3643032722656 x_{89} = -41.3643032722656 x 89 = − 41.3643032722656 x 90 = 56.025068989018 x_{90} = 56.025068989018 x 90 = 56.025068989018 x 91 = − 47.6474885794452 x_{91} = -47.6474885794452 x 91 = − 47.6474885794452 x 92 = − 3.66519142918809 x_{92} = -3.66519142918809 x 92 = − 3.66519142918809 x 93 = 96.8657734856853 x_{93} = 96.8657734856853 x 93 = 96.8657734856853 x 94 = 34.0339204138894 x_{94} = 34.0339204138894 x 94 = 34.0339204138894 x 95 = − 38.2227106186758 x_{95} = -38.2227106186758 x 95 = − 38.2227106186758 Las raíces dadas
x 16 = − 103.148958792865 x_{16} = -103.148958792865 x 16 = − 103.148958792865 x 44 = − 100.007366139275 x_{44} = -100.007366139275 x 44 = − 100.007366139275 x 30 = − 97.9129710368819 x_{30} = -97.9129710368819 x 30 = − 97.9129710368819 x 47 = − 93.7241808320955 x_{47} = -93.7241808320955 x 47 = − 93.7241808320955 x 15 = − 91.6297857297023 x_{15} = -91.6297857297023 x 15 = − 91.6297857297023 x 27 = − 90.5825881785057 x_{27} = -90.5825881785057 x 27 = − 90.5825881785057 x 46 = − 87.4409955249159 x_{46} = -87.4409955249159 x 46 = − 87.4409955249159 x 26 = − 85.3466004225227 x_{26} = -85.3466004225227 x 26 = − 85.3466004225227 x 64 = − 84.2994028713261 x_{64} = -84.2994028713261 x 64 = − 84.2994028713261 x 33 = − 82.2050077689329 x_{33} = -82.2050077689329 x 33 = − 82.2050077689329 x 28 = − 79.0634151153431 x_{28} = -79.0634151153431 x 28 = − 79.0634151153431 x 39 = − 78.0162175641465 x_{39} = -78.0162175641465 x 39 = − 78.0162175641465 x 55 = − 75.9218224617533 x_{55} = -75.9218224617533 x 55 = − 75.9218224617533 x 85 = − 71.733032256967 x_{85} = -71.733032256967 x 85 = − 71.733032256967 x 84 = − 69.6386371545737 x_{84} = -69.6386371545737 x 84 = − 69.6386371545737 x 78 = − 68.5914396033772 x_{78} = -68.5914396033772 x 78 = − 68.5914396033772 x 24 = − 65.4498469497874 x_{24} = -65.4498469497874 x 24 = − 65.4498469497874 x 75 = − 63.3554518473942 x_{75} = -63.3554518473942 x 75 = − 63.3554518473942 x 37 = − 62.3082542961976 x_{37} = -62.3082542961976 x 37 = − 62.3082542961976 x 5 = − 60.2138591938044 x_{5} = -60.2138591938044 x 5 = − 60.2138591938044 x 52 = − 57.0722665402146 x_{52} = -57.0722665402146 x 52 = − 57.0722665402146 x 63 = − 56.025068989018 x_{63} = -56.025068989018 x 63 = − 56.025068989018 x 56 = − 53.9306738866248 x_{56} = -53.9306738866248 x 56 = − 53.9306738866248 x 20 = − 49.7418836818384 x_{20} = -49.7418836818384 x 20 = − 49.7418836818384 x 91 = − 47.6474885794452 x_{91} = -47.6474885794452 x 91 = − 47.6474885794452 x 72 = − 46.6002910282486 x_{72} = -46.6002910282486 x 72 = − 46.6002910282486 x 32 = − 43.4586983746588 x_{32} = -43.4586983746588 x 32 = − 43.4586983746588 x 89 = − 41.3643032722656 x_{89} = -41.3643032722656 x 89 = − 41.3643032722656 x 48 = − 40.317105721069 x_{48} = -40.317105721069 x 48 = − 40.317105721069 x 95 = − 38.2227106186758 x_{95} = -38.2227106186758 x 95 = − 38.2227106186758 x 74 = − 35.081117965086 x_{74} = -35.081117965086 x 74 = − 35.081117965086 x 68 = − 34.0339204138894 x_{68} = -34.0339204138894 x 68 = − 34.0339204138894 x 29 = − 31.9395253114962 x_{29} = -31.9395253114962 x 29 = − 31.9395253114962 x 57 = − 27.7507351067098 x_{57} = -27.7507351067098 x 57 = − 27.7507351067098 x 4 = − 25.6563400043166 x_{4} = -25.6563400043166 x 4 = − 25.6563400043166 x 81 = − 24.60914245312 x_{81} = -24.60914245312 x 81 = − 24.60914245312 x 51 = − 21.4675497995303 x_{51} = -21.4675497995303 x 51 = − 21.4675497995303 x 35 = − 19.3731546971371 x_{35} = -19.3731546971371 x 35 = − 19.3731546971371 x 31 = − 18.3259571459405 x_{31} = -18.3259571459405 x 31 = − 18.3259571459405 x 62 = − 16.2315620435473 x_{62} = -16.2315620435473 x 62 = − 16.2315620435473 x 87 = − 13.0899693899575 x_{87} = -13.0899693899575 x 87 = − 13.0899693899575 x 80 = − 12.0427718387609 x_{80} = -12.0427718387609 x 80 = − 12.0427718387609 x 40 = − 9.94837673636768 x_{40} = -9.94837673636768 x 40 = − 9.94837673636768 x 43 = − 5.75958653158129 x_{43} = -5.75958653158129 x 43 = − 5.75958653158129 x 92 = − 3.66519142918809 x_{92} = -3.66519142918809 x 92 = − 3.66519142918809 x 58 = − 2.61799387799149 x_{58} = -2.61799387799149 x 58 = − 2.61799387799149 x 60 = 0.523598775598299 x_{60} = 0.523598775598299 x 60 = 0.523598775598299 x 22 = 2.61799387799149 x_{22} = 2.61799387799149 x 22 = 2.61799387799149 x 61 = 3.66519142918809 x_{61} = 3.66519142918809 x 61 = 3.66519142918809 x 17 = 5.75958653158129 x_{17} = 5.75958653158129 x 17 = 5.75958653158129 x 83 = 6.80678408277789 x_{83} = 6.80678408277789 x 83 = 6.80678408277789 x 23 = 8.90117918517108 x_{23} = 8.90117918517108 x 23 = 8.90117918517108 x 18 = 9.94837673636768 x_{18} = 9.94837673636768 x 18 = 9.94837673636768 x 71 = 12.0427718387609 x_{71} = 12.0427718387609 x 71 = 12.0427718387609 x 41 = 16.2315620435473 x_{41} = 16.2315620435473 x 41 = 16.2315620435473 x 12 = 18.3259571459405 x_{12} = 18.3259571459405 x 12 = 18.3259571459405 x 76 = 19.3731546971371 x_{76} = 19.3731546971371 x 76 = 19.3731546971371 x 3 = 22.5147473507269 x_{3} = 22.5147473507269 x 3 = 22.5147473507269 x 66 = 24.60914245312 x_{66} = 24.60914245312 x 66 = 24.60914245312 x 2 = 25.6563400043166 x_{2} = 25.6563400043166 x 2 = 25.6563400043166 x 25 = 27.7507351067098 x_{25} = 27.7507351067098 x 25 = 27.7507351067098 x 73 = 30.8923277602996 x_{73} = 30.8923277602996 x 73 = 30.8923277602996 x 9 = 31.9395253114962 x_{9} = 31.9395253114962 x 9 = 31.9395253114962 x 94 = 34.0339204138894 x_{94} = 34.0339204138894 x 94 = 34.0339204138894 x 86 = 38.2227106186758 x_{86} = 38.2227106186758 x 86 = 38.2227106186758 x 88 = 40.317105721069 x_{88} = 40.317105721069 x 88 = 40.317105721069 x 6 = 41.3643032722656 x_{6} = 41.3643032722656 x 6 = 41.3643032722656 x 13 = 44.5058959258554 x_{13} = 44.5058959258554 x 13 = 44.5058959258554 x 42 = 46.6002910282486 x_{42} = 46.6002910282486 x 42 = 46.6002910282486 x 67 = 47.6474885794452 x_{67} = 47.6474885794452 x 67 = 47.6474885794452 x 19 = 49.7418836818384 x_{19} = 49.7418836818384 x 19 = 49.7418836818384 x 45 = 52.8834763354282 x_{45} = 52.8834763354282 x 45 = 52.8834763354282 x 11 = 53.9306738866248 x_{11} = 53.9306738866248 x 11 = 53.9306738866248 x 90 = 56.025068989018 x_{90} = 56.025068989018 x 90 = 56.025068989018 x 8 = 60.2138591938044 x_{8} = 60.2138591938044 x 8 = 60.2138591938044 x 77 = 62.3082542961976 x_{77} = 62.3082542961976 x 77 = 62.3082542961976 x 59 = 63.3554518473942 x_{59} = 63.3554518473942 x 59 = 63.3554518473942 x 38 = 66.497044500984 x_{38} = 66.497044500984 x 38 = 66.497044500984 x 65 = 68.5914396033772 x_{65} = 68.5914396033772 x 65 = 68.5914396033772 x 1 = 69.6386371545737 x_{1} = 69.6386371545737 x 1 = 69.6386371545737 x 79 = 71.733032256967 x_{79} = 71.733032256967 x 79 = 71.733032256967 x 21 = 74.8746249105567 x_{21} = 74.8746249105567 x 21 = 74.8746249105567 x 69 = 75.9218224617533 x_{69} = 75.9218224617533 x 69 = 75.9218224617533 x 50 = 78.0162175641465 x_{50} = 78.0162175641465 x 50 = 78.0162175641465 x 54 = 82.2050077689329 x_{54} = 82.2050077689329 x 54 = 82.2050077689329 x 7 = 84.2994028713261 x_{7} = 84.2994028713261 x 7 = 84.2994028713261 x 10 = 85.3466004225227 x_{10} = 85.3466004225227 x 10 = 85.3466004225227 x 36 = 88.4881930761125 x_{36} = 88.4881930761125 x 36 = 88.4881930761125 x 82 = 90.5825881785057 x_{82} = 90.5825881785057 x 82 = 90.5825881785057 x 14 = 91.6297857297023 x_{14} = 91.6297857297023 x 14 = 91.6297857297023 x 34 = 93.7241808320955 x_{34} = 93.7241808320955 x 34 = 93.7241808320955 x 93 = 96.8657734856853 x_{93} = 96.8657734856853 x 93 = 96.8657734856853 x 70 = 97.9129710368819 x_{70} = 97.9129710368819 x 70 = 97.9129710368819 x 53 = 100.007366139275 x_{53} = 100.007366139275 x 53 = 100.007366139275 x 49 = 4868.94501428858 x_{49} = 4868.94501428858 x 49 = 4868.94501428858 son puntos de cambio del signo de desigualdad en las soluciones.
Primero definámonos con el signo hasta el punto extremo izquierdo:
x 0 < x 16 x_{0} < x_{16} x 0 < x 16 Consideremos, por ejemplo, el punto
x 0 = x 16 − 1 10 x_{0} = x_{16} - \frac{1}{10} x 0 = x 16 − 10 1 =
− 103.148958792865 + − 1 10 -103.148958792865 + - \frac{1}{10} − 103.148958792865 + − 10 1 =
− 103.248958792865 -103.248958792865 − 103.248958792865 lo sustituimos en la expresión
∣ cos ( x ) ∣ < 3 2 \left|{\cos{\left(x \right)}}\right| < \frac{\sqrt{3}}{2} ∣ cos ( x ) ∣ < 2 3 ∣ cos ( − 103.248958792865 ) ∣ < 3 2 \left|{\cos{\left(-103.248958792865 \right)}}\right| < \frac{\sqrt{3}}{2} ∣ cos ( − 103.248958792865 ) ∣ < 2 3 ___
\/ 3
0.911615592325511 < -----
2
pero
___
\/ 3
0.911615592325511 > -----
2
Entonces
x < − 103.148958792865 x < -103.148958792865 x < − 103.148958792865 no se cumple
significa que una de las soluciones de nuestra ecuación será con:
x > − 103.148958792865 ∧ x < − 100.007366139275 x > -103.148958792865 \wedge x < -100.007366139275 x > − 103.148958792865 ∧ x < − 100.007366139275 _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____
/ \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
-------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-------
x16 x44 x30 x47 x15 x27 x46 x26 x64 x33 x28 x39 x55 x85 x84 x78 x24 x75 x37 x5 x52 x63 x56 x20 x91 x72 x32 x89 x48 x95 x74 x68 x29 x57 x4 x81 x51 x35 x31 x62 x87 x80 x40 x43 x92 x58 x60 x22 x61 x17 x83 x23 x18 x71 x41 x12 x76 x3 x66 x2 x25 x73 x9 x94 x86 x88 x6 x13 x42 x67 x19 x45 x11 x90 x8 x77 x59 x38 x65 x1 x79 x21 x69 x50 x54 x7 x10 x36 x82 x14 x34 x93 x70 x53 x49 Recibiremos otras soluciones de la desigualdad pasando al polo siguiente etc.
etc.
Respuesta:
x > − 103.148958792865 ∧ x < − 100.007366139275 x > -103.148958792865 \wedge x < -100.007366139275 x > − 103.148958792865 ∧ x < − 100.007366139275 x > − 97.9129710368819 ∧ x < − 93.7241808320955 x > -97.9129710368819 \wedge x < -93.7241808320955 x > − 97.9129710368819 ∧ x < − 93.7241808320955 x > − 91.6297857297023 ∧ x < − 90.5825881785057 x > -91.6297857297023 \wedge x < -90.5825881785057 x > − 91.6297857297023 ∧ x < − 90.5825881785057 x > − 87.4409955249159 ∧ x < − 85.3466004225227 x > -87.4409955249159 \wedge x < -85.3466004225227 x > − 87.4409955249159 ∧ x < − 85.3466004225227 x > − 84.2994028713261 ∧ x < − 82.2050077689329 x > -84.2994028713261 \wedge x < -82.2050077689329 x > − 84.2994028713261 ∧ x < − 82.2050077689329 x > − 79.0634151153431 ∧ x < − 78.0162175641465 x > -79.0634151153431 \wedge x < -78.0162175641465 x > − 79.0634151153431 ∧ x < − 78.0162175641465 x > − 75.9218224617533 ∧ x < − 71.733032256967 x > -75.9218224617533 \wedge x < -71.733032256967 x > − 75.9218224617533 ∧ x < − 71.733032256967 x > − 69.6386371545737 ∧ x < − 68.5914396033772 x > -69.6386371545737 \wedge x < -68.5914396033772 x > − 69.6386371545737 ∧ x < − 68.5914396033772 x > − 65.4498469497874 ∧ x < − 63.3554518473942 x > -65.4498469497874 \wedge x < -63.3554518473942 x > − 65.4498469497874 ∧ x < − 63.3554518473942 x > − 62.3082542961976 ∧ x < − 60.2138591938044 x > -62.3082542961976 \wedge x < -60.2138591938044 x > − 62.3082542961976 ∧ x < − 60.2138591938044 x > − 57.0722665402146 ∧ x < − 56.025068989018 x > -57.0722665402146 \wedge x < -56.025068989018 x > − 57.0722665402146 ∧ x < − 56.025068989018 x > − 53.9306738866248 ∧ x < − 49.7418836818384 x > -53.9306738866248 \wedge x < -49.7418836818384 x > − 53.9306738866248 ∧ x < − 49.7418836818384 x > − 47.6474885794452 ∧ x < − 46.6002910282486 x > -47.6474885794452 \wedge x < -46.6002910282486 x > − 47.6474885794452 ∧ x < − 46.6002910282486 x > − 43.4586983746588 ∧ x < − 41.3643032722656 x > -43.4586983746588 \wedge x < -41.3643032722656 x > − 43.4586983746588 ∧ x < − 41.3643032722656 x > − 40.317105721069 ∧ x < − 38.2227106186758 x > -40.317105721069 \wedge x < -38.2227106186758 x > − 40.317105721069 ∧ x < − 38.2227106186758 x > − 35.081117965086 ∧ x < − 34.0339204138894 x > -35.081117965086 \wedge x < -34.0339204138894 x > − 35.081117965086 ∧ x < − 34.0339204138894 x > − 31.9395253114962 ∧ x < − 27.7507351067098 x > -31.9395253114962 \wedge x < -27.7507351067098 x > − 31.9395253114962 ∧ x < − 27.7507351067098 x > − 25.6563400043166 ∧ x < − 24.60914245312 x > -25.6563400043166 \wedge x < -24.60914245312 x > − 25.6563400043166 ∧ x < − 24.60914245312 x > − 21.4675497995303 ∧ x < − 19.3731546971371 x > -21.4675497995303 \wedge x < -19.3731546971371 x > − 21.4675497995303 ∧ x < − 19.3731546971371 x > − 18.3259571459405 ∧ x < − 16.2315620435473 x > -18.3259571459405 \wedge x < -16.2315620435473 x > − 18.3259571459405 ∧ x < − 16.2315620435473 x > − 13.0899693899575 ∧ x < − 12.0427718387609 x > -13.0899693899575 \wedge x < -12.0427718387609 x > − 13.0899693899575 ∧ x < − 12.0427718387609 x > − 9.94837673636768 ∧ x < − 5.75958653158129 x > -9.94837673636768 \wedge x < -5.75958653158129 x > − 9.94837673636768 ∧ x < − 5.75958653158129 x > − 3.66519142918809 ∧ x < − 2.61799387799149 x > -3.66519142918809 \wedge x < -2.61799387799149 x > − 3.66519142918809 ∧ x < − 2.61799387799149 x > 0.523598775598299 ∧ x < 2.61799387799149 x > 0.523598775598299 \wedge x < 2.61799387799149 x > 0.523598775598299 ∧ x < 2.61799387799149 x > 3.66519142918809 ∧ x < 5.75958653158129 x > 3.66519142918809 \wedge x < 5.75958653158129 x > 3.66519142918809 ∧ x < 5.75958653158129 x > 6.80678408277789 ∧ x < 8.90117918517108 x > 6.80678408277789 \wedge x < 8.90117918517108 x > 6.80678408277789 ∧ x < 8.90117918517108 x > 9.94837673636768 ∧ x < 12.0427718387609 x > 9.94837673636768 \wedge x < 12.0427718387609 x > 9.94837673636768 ∧ x < 12.0427718387609 x > 16.2315620435473 ∧ x < 18.3259571459405 x > 16.2315620435473 \wedge x < 18.3259571459405 x > 16.2315620435473 ∧ x < 18.3259571459405 x > 19.3731546971371 ∧ x < 22.5147473507269 x > 19.3731546971371 \wedge x < 22.5147473507269 x > 19.3731546971371 ∧ x < 22.5147473507269 x > 24.60914245312 ∧ x < 25.6563400043166 x > 24.60914245312 \wedge x < 25.6563400043166 x > 24.60914245312 ∧ x < 25.6563400043166 x > 27.7507351067098 ∧ x < 30.8923277602996 x > 27.7507351067098 \wedge x < 30.8923277602996 x > 27.7507351067098 ∧ x < 30.8923277602996 x > 31.9395253114962 ∧ x < 34.0339204138894 x > 31.9395253114962 \wedge x < 34.0339204138894 x > 31.9395253114962 ∧ x < 34.0339204138894 x > 38.2227106186758 ∧ x < 40.317105721069 x > 38.2227106186758 \wedge x < 40.317105721069 x > 38.2227106186758 ∧ x < 40.317105721069 x > 41.3643032722656 ∧ x < 44.5058959258554 x > 41.3643032722656 \wedge x < 44.5058959258554 x > 41.3643032722656 ∧ x < 44.5058959258554 x > 46.6002910282486 ∧ x < 47.6474885794452 x > 46.6002910282486 \wedge x < 47.6474885794452 x > 46.6002910282486 ∧ x < 47.6474885794452 x > 49.7418836818384 ∧ x < 52.8834763354282 x > 49.7418836818384 \wedge x < 52.8834763354282 x > 49.7418836818384 ∧ x < 52.8834763354282 x > 53.9306738866248 ∧ x < 56.025068989018 x > 53.9306738866248 \wedge x < 56.025068989018 x > 53.9306738866248 ∧ x < 56.025068989018 x > 60.2138591938044 ∧ x < 62.3082542961976 x > 60.2138591938044 \wedge x < 62.3082542961976 x > 60.2138591938044 ∧ x < 62.3082542961976 x > 63.3554518473942 ∧ x < 66.497044500984 x > 63.3554518473942 \wedge x < 66.497044500984 x > 63.3554518473942 ∧ x < 66.497044500984 x > 68.5914396033772 ∧ x < 69.6386371545737 x > 68.5914396033772 \wedge x < 69.6386371545737 x > 68.5914396033772 ∧ x < 69.6386371545737 x > 71.733032256967 ∧ x < 74.8746249105567 x > 71.733032256967 \wedge x < 74.8746249105567 x > 71.733032256967 ∧ x < 74.8746249105567 x > 75.9218224617533 ∧ x < 78.0162175641465 x > 75.9218224617533 \wedge x < 78.0162175641465 x > 75.9218224617533 ∧ x < 78.0162175641465 x > 82.2050077689329 ∧ x < 84.2994028713261 x > 82.2050077689329 \wedge x < 84.2994028713261 x > 82.2050077689329 ∧ x < 84.2994028713261 x > 85.3466004225227 ∧ x < 88.4881930761125 x > 85.3466004225227 \wedge x < 88.4881930761125 x > 85.3466004225227 ∧ x < 88.4881930761125 x > 90.5825881785057 ∧ x < 91.6297857297023 x > 90.5825881785057 \wedge x < 91.6297857297023 x > 90.5825881785057 ∧ x < 91.6297857297023 x > 93.7241808320955 ∧ x < 96.8657734856853 x > 93.7241808320955 \wedge x < 96.8657734856853 x > 93.7241808320955 ∧ x < 96.8657734856853 x > 97.9129710368819 ∧ x < 100.007366139275 x > 97.9129710368819 \wedge x < 100.007366139275 x > 97.9129710368819 ∧ x < 100.007366139275 x > 4868.94501428858 x > 4868.94501428858 x > 4868.94501428858 
-x^2-2x+3>0
x^2>9
x^2-10x<0
(x^2-x-1)*(x^2-x-7)<-5