Se da la desigualdad:
∣sin(x)∣<21Para resolver esta desigualdad primero hay que resolver la ecuación correspondiente:
∣sin(x)∣=21Resolvemos:
Tenemos la ecuación
∣sin(x)∣=21cambiamos
∣sin(x)∣−21=0∣sin(x)∣−21=0Sustituimos
w=∣sin(x)∣Transportamos los términos libres (sin w)
del miembro izquierdo al derecho, obtenemos:
w=21Obtenemos la respuesta: w = 1/2
hacemos cambio inverso
∣sin(x)∣=wsustituimos w:
x1=60.2138591938044x2=18.3259571459405x3=3.66519142918809x4=−62.3082542961976x5=−60.2138591938044x6=−31.9395253114962x7=−9.94837673636768x8=−78.0162175641465x9=−18.3259571459405x10=44.5058959258554x11=−68.5914396033772x12=9.94837673636768x13=−382.750704962356x14=−87.4409955249159x15=−97.9129710368819x16=−2.61799387799149x17=−66.497044500984x18=−71.733032256967x19=49.7418836818384x20=66.497044500984x21=0.523598775598299x22=−56.025068989018x23=71.733032256967x24=22.5147473507269x25=−27.7507351067098x26=84.2994028713261x27=−19.3731546971371x28=−93.7241808320955x29=−21.4675497995303x30=38.2227106186758x31=100.007366139275x32=93.7241808320955x33=16.2315620435473x34=−75.9218224617533x35=−137.706477982353x36=40.317105721069x37=68.5914396033772x38=−85.3466004225227x39=69.6386371545737x40=12.0427718387609x41=78.0162175641465x42=90.5825881785057x43=−12.0427718387609x44=−91.6297857297023x45=27.7507351067098x46=−57.0722665402146x47=−34.0339204138894x48=−38.2227106186758x49=−46.6002910282486x50=−69.6386371545737x51=−100.007366139275x52=53.9306738866248x53=−82.2050077689329x54=−84.2994028713261x55=−24.60914245312x56=−16.2315620435473x57=−53.9306738866248x58=96.8657734856853x59=−3.66519142918809x60=75.9218224617533x61=34.0339204138894x62=−25.6563400043166x63=31.9395253114962x64=47.6474885794452x65=−5.75958653158129x66=24.60914245312x67=−65.4498469497874x68=−41.3643032722656x69=−63.3554518473942x70=56.025068989018x71=91.6297857297023x72=−47.6474885794452x73=88.4881930761125x74=46.6002910282486x75=2.61799387799149x76=82.2050077689329x77=25.6563400043166x78=−43.4586983746588x79=−40.317105721069x80=−49.7418836818384x81=5.75958653158129x82=−90.5825881785057x83=62.3082542961976x84=97.9129710368819x1=60.2138591938044x2=18.3259571459405x3=3.66519142918809x4=−62.3082542961976x5=−60.2138591938044x6=−31.9395253114962x7=−9.94837673636768x8=−78.0162175641465x9=−18.3259571459405x10=44.5058959258554x11=−68.5914396033772x12=9.94837673636768x13=−382.750704962356x14=−87.4409955249159x15=−97.9129710368819x16=−2.61799387799149x17=−66.497044500984x18=−71.733032256967x19=49.7418836818384x20=66.497044500984x21=0.523598775598299x22=−56.025068989018x23=71.733032256967x24=22.5147473507269x25=−27.7507351067098x26=84.2994028713261x27=−19.3731546971371x28=−93.7241808320955x29=−21.4675497995303x30=38.2227106186758x31=100.007366139275x32=93.7241808320955x33=16.2315620435473x34=−75.9218224617533x35=−137.706477982353x36=40.317105721069x37=68.5914396033772x38=−85.3466004225227x39=69.6386371545737x40=12.0427718387609x41=78.0162175641465x42=90.5825881785057x43=−12.0427718387609x44=−91.6297857297023x45=27.7507351067098x46=−57.0722665402146x47=−34.0339204138894x48=−38.2227106186758x49=−46.6002910282486x50=−69.6386371545737x51=−100.007366139275x52=53.9306738866248x53=−82.2050077689329x54=−84.2994028713261x55=−24.60914245312x56=−16.2315620435473x57=−53.9306738866248x58=96.8657734856853x59=−3.66519142918809x60=75.9218224617533x61=34.0339204138894x62=−25.6563400043166x63=31.9395253114962x64=47.6474885794452x65=−5.75958653158129x66=24.60914245312x67=−65.4498469497874x68=−41.3643032722656x69=−63.3554518473942x70=56.025068989018x71=91.6297857297023x72=−47.6474885794452x73=88.4881930761125x74=46.6002910282486x75=2.61799387799149x76=82.2050077689329x77=25.6563400043166x78=−43.4586983746588x79=−40.317105721069x80=−49.7418836818384x81=5.75958653158129x82=−90.5825881785057x83=62.3082542961976x84=97.9129710368819Las raíces dadas
x13=−382.750704962356x35=−137.706477982353x51=−100.007366139275x15=−97.9129710368819x28=−93.7241808320955x44=−91.6297857297023x82=−90.5825881785057x14=−87.4409955249159x38=−85.3466004225227x54=−84.2994028713261x53=−82.2050077689329x8=−78.0162175641465x34=−75.9218224617533x18=−71.733032256967x50=−69.6386371545737x11=−68.5914396033772x17=−66.497044500984x67=−65.4498469497874x69=−63.3554518473942x4=−62.3082542961976x5=−60.2138591938044x46=−57.0722665402146x22=−56.025068989018x57=−53.9306738866248x80=−49.7418836818384x72=−47.6474885794452x49=−46.6002910282486x78=−43.4586983746588x68=−41.3643032722656x79=−40.317105721069x48=−38.2227106186758x47=−34.0339204138894x6=−31.9395253114962x25=−27.7507351067098x62=−25.6563400043166x55=−24.60914245312x29=−21.4675497995303x27=−19.3731546971371x9=−18.3259571459405x56=−16.2315620435473x43=−12.0427718387609x7=−9.94837673636768x65=−5.75958653158129x59=−3.66519142918809x16=−2.61799387799149x21=0.523598775598299x75=2.61799387799149x3=3.66519142918809x81=5.75958653158129x12=9.94837673636768x40=12.0427718387609x33=16.2315620435473x2=18.3259571459405x24=22.5147473507269x66=24.60914245312x77=25.6563400043166x45=27.7507351067098x63=31.9395253114962x61=34.0339204138894x30=38.2227106186758x36=40.317105721069x10=44.5058959258554x74=46.6002910282486x64=47.6474885794452x19=49.7418836818384x52=53.9306738866248x70=56.025068989018x1=60.2138591938044x83=62.3082542961976x20=66.497044500984x37=68.5914396033772x39=69.6386371545737x23=71.733032256967x60=75.9218224617533x41=78.0162175641465x76=82.2050077689329x26=84.2994028713261x73=88.4881930761125x42=90.5825881785057x71=91.6297857297023x32=93.7241808320955x58=96.8657734856853x84=97.9129710368819x31=100.007366139275son puntos de cambio del signo de desigualdad en las soluciones.
Primero definámonos con el signo hasta el punto extremo izquierdo:
x0<x13Consideremos, por ejemplo, el punto
x0=x13−101=
−382.750704962356+−101=
−382.850704962356lo sustituimos en la expresión
∣sin(x)∣<21∣sin(−382.850704962356)∣<210.411043807676265 < 1/2
significa que una de las soluciones de nuestra ecuación será con:
x<−382.750704962356 _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____
\ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
-------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-------
x13 x35 x51 x15 x28 x44 x82 x14 x38 x54 x53 x8 x34 x18 x50 x11 x17 x67 x69 x4 x5 x46 x22 x57 x80 x72 x49 x78 x68 x79 x48 x47 x6 x25 x62 x55 x29 x27 x9 x56 x43 x7 x65 x59 x16 x21 x75 x3 x81 x12 x40 x33 x2 x24 x66 x77 x45 x63 x61 x30 x36 x10 x74 x64 x19 x52 x70 x1 x83 x20 x37 x39 x23 x60 x41 x76 x26 x73 x42 x71 x32 x58 x84 x31Recibiremos otras soluciones de la desigualdad pasando al polo siguiente etc.
etc.
Respuesta:
x<−382.750704962356x>−137.706477982353∧x<−100.007366139275x>−97.9129710368819∧x<−93.7241808320955x>−91.6297857297023∧x<−90.5825881785057x>−87.4409955249159∧x<−85.3466004225227x>−84.2994028713261∧x<−82.2050077689329x>−78.0162175641465∧x<−75.9218224617533x>−71.733032256967∧x<−69.6386371545737x>−68.5914396033772∧x<−66.497044500984x>−65.4498469497874∧x<−63.3554518473942x>−62.3082542961976∧x<−60.2138591938044x>−57.0722665402146∧x<−56.025068989018x>−53.9306738866248∧x<−49.7418836818384x>−47.6474885794452∧x<−46.6002910282486x>−43.4586983746588∧x<−41.3643032722656x>−40.317105721069∧x<−38.2227106186758x>−34.0339204138894∧x<−31.9395253114962x>−27.7507351067098∧x<−25.6563400043166x>−24.60914245312∧x<−21.4675497995303x>−19.3731546971371∧x<−18.3259571459405x>−16.2315620435473∧x<−12.0427718387609x>−9.94837673636768∧x<−5.75958653158129x>−3.66519142918809∧x<−2.61799387799149x>0.523598775598299∧x<2.61799387799149x>3.66519142918809∧x<5.75958653158129x>9.94837673636768∧x<12.0427718387609x>16.2315620435473∧x<18.3259571459405x>22.5147473507269∧x<24.60914245312x>25.6563400043166∧x<27.7507351067098x>31.9395253114962∧x<34.0339204138894x>38.2227106186758∧x<40.317105721069x>44.5058959258554∧x<46.6002910282486x>47.6474885794452∧x<49.7418836818384x>53.9306738866248∧x<56.025068989018x>60.2138591938044∧x<62.3082542961976x>66.497044500984∧x<68.5914396033772x>69.6386371545737∧x<71.733032256967x>75.9218224617533∧x<78.0162175641465x>82.2050077689329∧x<84.2994028713261x>88.4881930761125∧x<90.5825881785057x>91.6297857297023∧x<93.7241808320955x>96.8657734856853∧x<97.9129710368819x>100.007366139275
x^2+x+1>0
x^2-5x+4>0
(x-3)*(x+4)>0
(x-2)*(x-3)<0
|sinx|
1/2
1/2