Se da la desigualdad:
∣cos(x)∣<21Para resolver esta desigualdad primero hay que resolver la ecuación correspondiente:
∣cos(x)∣=21Resolvemos:
Tenemos la ecuación
∣cos(x)∣=21cambiamos
∣cos(x)∣−21=0∣cos(x)∣−21=0Sustituimos
w=∣cos(x)∣Transportamos los términos libres (sin w)
del miembro izquierdo al derecho, obtenemos:
w=21Obtenemos la respuesta: w = 1/2
hacemos cambio inverso
∣cos(x)∣=wsustituimos w:
x1=−93.2005820564972x2=−68.0678408277789x3=−79.5870138909414x4=63.8790506229925x5=82.7286065445312x6=−85.870199198121x7=−4.18879020478639x8=74.3510261349584x9=61.7846555205993x10=−35.6047167406843x11=2.0943951023932x12=33.5103216382911x13=−76.4454212373516x14=−32.4631240870945x15=30.3687289847013x16=−63.8790506229925x17=39.7935069454707x18=−70.162235930172x19=70.162235930172x20=269.129770657526x21=38.7463093942741x22=76.4454212373516x23=−39.7935069454707x24=−77.4926187885482x25=−96.342174710087x26=85.870199198121x27=98.4365698124802x28=−24.0855436775217x29=−90.0589894029074x30=−41.8879020478639x31=−673.348025419412x32=60.7374579694027x33=90.0589894029074x34=275.412955964705x35=−2.0943951023932x36=41.8879020478639x37=−74.3510261349584x38=26.1799387799149x39=55.5014702134197x40=−61.7846555205993x41=68.0678408277789x42=−13.6135681655558x43=−55.5014702134197x44=−30.3687289847013x45=77.4926187885482x46=−26.1799387799149x47=8.37758040957278x48=52.3598775598299x49=−46.0766922526503x50=−33.5103216382911x51=16.7551608191456x52=−60.7374579694027x53=−99.4837673636768x54=−27.2271363311115x55=−5.23598775598299x56=96.342174710087x57=−11.5191730631626x58=−83.7758040957278x59=−54.4542726622231x60=3803.42150594604x61=19.8967534727354x62=48.1710873550435x63=−49.2182849062401x64=−10.471975511966x65=4.18879020478639x66=32.4631240870945x67=11.5191730631626x68=83.7758040957278x69=−42.9350995990605x70=−71.2094334813686x71=−19.8967534727354x72=54.4542726622231x73=99.4837673636768x74=92.1533845053006x75=−48.1710873550435x76=46.0766922526503x77=24.0855436775217x78=10.471975511966x79=17.8023583703422x80=−17.8023583703422x81=−57.5958653158129x82=−92.1533845053006x83=−52.3598775598299x84=−98.4365698124802x1=−93.2005820564972x2=−68.0678408277789x3=−79.5870138909414x4=63.8790506229925x5=82.7286065445312x6=−85.870199198121x7=−4.18879020478639x8=74.3510261349584x9=61.7846555205993x10=−35.6047167406843x11=2.0943951023932x12=33.5103216382911x13=−76.4454212373516x14=−32.4631240870945x15=30.3687289847013x16=−63.8790506229925x17=39.7935069454707x18=−70.162235930172x19=70.162235930172x20=269.129770657526x21=38.7463093942741x22=76.4454212373516x23=−39.7935069454707x24=−77.4926187885482x25=−96.342174710087x26=85.870199198121x27=98.4365698124802x28=−24.0855436775217x29=−90.0589894029074x30=−41.8879020478639x31=−673.348025419412x32=60.7374579694027x33=90.0589894029074x34=275.412955964705x35=−2.0943951023932x36=41.8879020478639x37=−74.3510261349584x38=26.1799387799149x39=55.5014702134197x40=−61.7846555205993x41=68.0678408277789x42=−13.6135681655558x43=−55.5014702134197x44=−30.3687289847013x45=77.4926187885482x46=−26.1799387799149x47=8.37758040957278x48=52.3598775598299x49=−46.0766922526503x50=−33.5103216382911x51=16.7551608191456x52=−60.7374579694027x53=−99.4837673636768x54=−27.2271363311115x55=−5.23598775598299x56=96.342174710087x57=−11.5191730631626x58=−83.7758040957278x59=−54.4542726622231x60=3803.42150594604x61=19.8967534727354x62=48.1710873550435x63=−49.2182849062401x64=−10.471975511966x65=4.18879020478639x66=32.4631240870945x67=11.5191730631626x68=83.7758040957278x69=−42.9350995990605x70=−71.2094334813686x71=−19.8967534727354x72=54.4542726622231x73=99.4837673636768x74=92.1533845053006x75=−48.1710873550435x76=46.0766922526503x77=24.0855436775217x78=10.471975511966x79=17.8023583703422x80=−17.8023583703422x81=−57.5958653158129x82=−92.1533845053006x83=−52.3598775598299x84=−98.4365698124802Las raíces dadas
x31=−673.348025419412x53=−99.4837673636768x84=−98.4365698124802x25=−96.342174710087x1=−93.2005820564972x82=−92.1533845053006x29=−90.0589894029074x6=−85.870199198121x58=−83.7758040957278x3=−79.5870138909414x24=−77.4926187885482x13=−76.4454212373516x37=−74.3510261349584x70=−71.2094334813686x18=−70.162235930172x2=−68.0678408277789x16=−63.8790506229925x40=−61.7846555205993x52=−60.7374579694027x81=−57.5958653158129x43=−55.5014702134197x59=−54.4542726622231x83=−52.3598775598299x63=−49.2182849062401x75=−48.1710873550435x49=−46.0766922526503x69=−42.9350995990605x30=−41.8879020478639x23=−39.7935069454707x10=−35.6047167406843x50=−33.5103216382911x14=−32.4631240870945x44=−30.3687289847013x54=−27.2271363311115x46=−26.1799387799149x28=−24.0855436775217x71=−19.8967534727354x80=−17.8023583703422x42=−13.6135681655558x57=−11.5191730631626x64=−10.471975511966x55=−5.23598775598299x7=−4.18879020478639x35=−2.0943951023932x11=2.0943951023932x65=4.18879020478639x47=8.37758040957278x78=10.471975511966x67=11.5191730631626x51=16.7551608191456x79=17.8023583703422x61=19.8967534727354x77=24.0855436775217x38=26.1799387799149x15=30.3687289847013x66=32.4631240870945x12=33.5103216382911x21=38.7463093942741x17=39.7935069454707x36=41.8879020478639x76=46.0766922526503x62=48.1710873550435x48=52.3598775598299x72=54.4542726622231x39=55.5014702134197x32=60.7374579694027x9=61.7846555205993x4=63.8790506229925x41=68.0678408277789x19=70.162235930172x8=74.3510261349584x22=76.4454212373516x45=77.4926187885482x5=82.7286065445312x68=83.7758040957278x26=85.870199198121x33=90.0589894029074x74=92.1533845053006x56=96.342174710087x27=98.4365698124802x73=99.4837673636768x20=269.129770657526x34=275.412955964705x60=3803.42150594604son puntos de cambio del signo de desigualdad en las soluciones.
Primero definámonos con el signo hasta el punto extremo izquierdo:
x0<x31Consideremos, por ejemplo, el punto
x0=x31−101=
−673.348025419412+−101=
−673.448025419412lo sustituimos en la expresión
∣cos(x)∣<21∣cos(−673.448025419412)∣<210.411043807676242 < 1/2
significa que una de las soluciones de nuestra ecuación será con:
x<−673.348025419412 _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____
\ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
-------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-------
x31 x53 x84 x25 x1 x82 x29 x6 x58 x3 x24 x13 x37 x70 x18 x2 x16 x40 x52 x81 x43 x59 x83 x63 x75 x49 x69 x30 x23 x10 x50 x14 x44 x54 x46 x28 x71 x80 x42 x57 x64 x55 x7 x35 x11 x65 x47 x78 x67 x51 x79 x61 x77 x38 x15 x66 x12 x21 x17 x36 x76 x62 x48 x72 x39 x32 x9 x4 x41 x19 x8 x22 x45 x5 x68 x26 x33 x74 x56 x27 x73 x20 x34 x60Recibiremos otras soluciones de la desigualdad pasando al polo siguiente etc.
etc.
Respuesta:
x<−673.348025419412x>−99.4837673636768∧x<−98.4365698124802x>−96.342174710087∧x<−93.2005820564972x>−92.1533845053006∧x<−90.0589894029074x>−85.870199198121∧x<−83.7758040957278x>−79.5870138909414∧x<−77.4926187885482x>−76.4454212373516∧x<−74.3510261349584x>−71.2094334813686∧x<−70.162235930172x>−68.0678408277789∧x<−63.8790506229925x>−61.7846555205993∧x<−60.7374579694027x>−57.5958653158129∧x<−55.5014702134197x>−54.4542726622231∧x<−52.3598775598299x>−49.2182849062401∧x<−48.1710873550435x>−46.0766922526503∧x<−42.9350995990605x>−41.8879020478639∧x<−39.7935069454707x>−35.6047167406843∧x<−33.5103216382911x>−32.4631240870945∧x<−30.3687289847013x>−27.2271363311115∧x<−26.1799387799149x>−24.0855436775217∧x<−19.8967534727354x>−17.8023583703422∧x<−13.6135681655558x>−11.5191730631626∧x<−10.471975511966x>−5.23598775598299∧x<−4.18879020478639x>−2.0943951023932∧x<2.0943951023932x>4.18879020478639∧x<8.37758040957278x>10.471975511966∧x<11.5191730631626x>16.7551608191456∧x<17.8023583703422x>19.8967534727354∧x<24.0855436775217x>26.1799387799149∧x<30.3687289847013x>32.4631240870945∧x<33.5103216382911x>38.7463093942741∧x<39.7935069454707x>41.8879020478639∧x<46.0766922526503x>48.1710873550435∧x<52.3598775598299x>54.4542726622231∧x<55.5014702134197x>60.7374579694027∧x<61.7846555205993x>63.8790506229925∧x<68.0678408277789x>70.162235930172∧x<74.3510261349584x>76.4454212373516∧x<77.4926187885482x>82.7286065445312∧x<83.7758040957278x>85.870199198121∧x<90.0589894029074x>92.1533845053006∧x<96.342174710087x>98.4365698124802∧x<99.4837673636768x>269.129770657526∧x<275.412955964705x>3803.42150594604
x^2+15x>0
x^2+3x>0
x^2+15>0
1/2
1/2