Se da la desigualdad:
cos(x)∣sin(x)∣>41Para resolver esta desigualdad primero hay que resolver la ecuación correspondiente:
cos(x)∣sin(x)∣=41Resolvemos:
x1=−13.8753675533549x2=56.2868683768171x3=−50.0036830696375x4=93.9859802198946x5=80.3724120543389x6=36.3901149040818x7=57.857664703612x8=−3707.34113062376x9=7.59218224617533x10=−55.2396708256205x11=−82.9904059323304x12=−30.1069295969022x13=−95.5567765466895x14=−74.0892267471593x15=12.30457122656x16=−43.720497762458x17=45.2912940892529x18=6.54498469497874x19=−12.8281700021583x20=−51.5744793964324x21=−231.168859426649x22=−70.4240353179712x23=−17.540558982543x24=−61.5228561328001x25=100.269165527074x26=−0.261799387799149x27=−12.30457122656x28=−6.02138591938044x29=51.5744793964324x30=−19.1113553093379x31=95.5567765466895x32=−20.1585528605345x33=−80.3724120543389x34=20.1585528605345x35=50.0036830696375x36=70.4240353179712x37=67.8060414399797x38=107.075949609852x39=−36.3901149040818x40=−68.8532389911763x41=−50.5272818452358x42=−1368.42540002615x43=0.261799387799149x44=−42.6733002112614x45=−75.6600230739542x46=−86.6555973615185x47=86.6555973615185x48=−87.7027949127151x49=−37.9609112308767x50=64.1408500107916x51=−67.8060414399797x52=−1.30899693899575x53=−100.792764302673x54=76.7072206251508x55=26.4417381677141x56=−37.4373124552784x57=23.8237442897226x58=−93.9859802198946x59=−31.6777259236971x60=30.1069295969022x61=81.9432083811338x62=6.02138591938044x63=−7.59218224617533x64=−81.4196096055355x65=−25.3945406165175x66=−81.9432083811338x67=55.2396708256205x68=74.0892267471593x69=37.9609112308767x70=48.9564855184409x71=88.2263936883134x72=−89.27359123951x73=44.2440965380563x74=114.406332468228x75=−64.1408500107916x76=−23.8237442897226x77=13.8753675533549x78=−56.8104671524154x79=−45.2912940892529x80=−57.857664703612x81=31.1541271480988x82=42.6733002112614x1=−13.8753675533549x2=56.2868683768171x3=−50.0036830696375x4=93.9859802198946x5=80.3724120543389x6=36.3901149040818x7=57.857664703612x8=−3707.34113062376x9=7.59218224617533x10=−55.2396708256205x11=−82.9904059323304x12=−30.1069295969022x13=−95.5567765466895x14=−74.0892267471593x15=12.30457122656x16=−43.720497762458x17=45.2912940892529x18=6.54498469497874x19=−12.8281700021583x20=−51.5744793964324x21=−231.168859426649x22=−70.4240353179712x23=−17.540558982543x24=−61.5228561328001x25=100.269165527074x26=−0.261799387799149x27=−12.30457122656x28=−6.02138591938044x29=51.5744793964324x30=−19.1113553093379x31=95.5567765466895x32=−20.1585528605345x33=−80.3724120543389x34=20.1585528605345x35=50.0036830696375x36=70.4240353179712x37=67.8060414399797x38=107.075949609852x39=−36.3901149040818x40=−68.8532389911763x41=−50.5272818452358x42=−1368.42540002615x43=0.261799387799149x44=−42.6733002112614x45=−75.6600230739542x46=−86.6555973615185x47=86.6555973615185x48=−87.7027949127151x49=−37.9609112308767x50=64.1408500107916x51=−67.8060414399797x52=−1.30899693899575x53=−100.792764302673x54=76.7072206251508x55=26.4417381677141x56=−37.4373124552784x57=23.8237442897226x58=−93.9859802198946x59=−31.6777259236971x60=30.1069295969022x61=81.9432083811338x62=6.02138591938044x63=−7.59218224617533x64=−81.4196096055355x65=−25.3945406165175x66=−81.9432083811338x67=55.2396708256205x68=74.0892267471593x69=37.9609112308767x70=48.9564855184409x71=88.2263936883134x72=−89.27359123951x73=44.2440965380563x74=114.406332468228x75=−64.1408500107916x76=−23.8237442897226x77=13.8753675533549x78=−56.8104671524154x79=−45.2912940892529x80=−57.857664703612x81=31.1541271480988x82=42.6733002112614Las raíces dadas
x8=−3707.34113062376x42=−1368.42540002615x21=−231.168859426649x53=−100.792764302673x13=−95.5567765466895x58=−93.9859802198946x72=−89.27359123951x48=−87.7027949127151x46=−86.6555973615185x11=−82.9904059323304x66=−81.9432083811338x64=−81.4196096055355x33=−80.3724120543389x45=−75.6600230739542x14=−74.0892267471593x22=−70.4240353179712x40=−68.8532389911763x51=−67.8060414399797x75=−64.1408500107916x24=−61.5228561328001x80=−57.857664703612x78=−56.8104671524154x10=−55.2396708256205x20=−51.5744793964324x41=−50.5272818452358x3=−50.0036830696375x79=−45.2912940892529x16=−43.720497762458x44=−42.6733002112614x49=−37.9609112308767x56=−37.4373124552784x39=−36.3901149040818x59=−31.6777259236971x12=−30.1069295969022x65=−25.3945406165175x76=−23.8237442897226x32=−20.1585528605345x30=−19.1113553093379x23=−17.540558982543x1=−13.8753675533549x19=−12.8281700021583x27=−12.30457122656x63=−7.59218224617533x28=−6.02138591938044x52=−1.30899693899575x26=−0.261799387799149x43=0.261799387799149x62=6.02138591938044x18=6.54498469497874x9=7.59218224617533x15=12.30457122656x77=13.8753675533549x34=20.1585528605345x57=23.8237442897226x55=26.4417381677141x60=30.1069295969022x81=31.1541271480988x6=36.3901149040818x69=37.9609112308767x82=42.6733002112614x73=44.2440965380563x17=45.2912940892529x70=48.9564855184409x35=50.0036830696375x29=51.5744793964324x67=55.2396708256205x2=56.2868683768171x7=57.857664703612x50=64.1408500107916x37=67.8060414399797x36=70.4240353179712x68=74.0892267471593x54=76.7072206251508x5=80.3724120543389x61=81.9432083811338x47=86.6555973615185x71=88.2263936883134x4=93.9859802198946x31=95.5567765466895x25=100.269165527074x38=107.075949609852x74=114.406332468228son puntos de cambio del signo de desigualdad en las soluciones.
Primero definámonos con el signo hasta el punto extremo izquierdo:
x0<x8Consideremos, por ejemplo, el punto
x0=x8−101=
−3707.34113062376+−101=
−3707.44113062376lo sustituimos en la expresión
cos(x)∣sin(x)∣>41cos(−3707.44113062376)∣sin(−3707.44113062376)∣>410.331042988171074 > 1/4
significa que una de las soluciones de nuestra ecuación será con:
x<−3707.34113062376 _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____
\ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
-------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-------
x8 x42 x21 x53 x13 x58 x72 x48 x46 x11 x66 x64 x33 x45 x14 x22 x40 x51 x75 x24 x80 x78 x10 x20 x41 x3 x79 x16 x44 x49 x56 x39 x59 x12 x65 x76 x32 x30 x23 x1 x19 x27 x63 x28 x52 x26 x43 x62 x18 x9 x15 x77 x34 x57 x55 x60 x81 x6 x69 x82 x73 x17 x70 x35 x29 x67 x2 x7 x50 x37 x36 x68 x54 x5 x61 x47 x71 x4 x31 x25 x38 x74Recibiremos otras soluciones de la desigualdad pasando al polo siguiente etc.
etc.
Respuesta:
x<−3707.34113062376x>−1368.42540002615∧x<−231.168859426649x>−100.792764302673∧x<−95.5567765466895x>−93.9859802198946∧x<−89.27359123951x>−87.7027949127151∧x<−86.6555973615185x>−82.9904059323304∧x<−81.9432083811338x>−81.4196096055355∧x<−80.3724120543389x>−75.6600230739542∧x<−74.0892267471593x>−70.4240353179712∧x<−68.8532389911763x>−67.8060414399797∧x<−64.1408500107916x>−61.5228561328001∧x<−57.857664703612x>−56.8104671524154∧x<−55.2396708256205x>−51.5744793964324∧x<−50.5272818452358x>−50.0036830696375∧x<−45.2912940892529x>−43.720497762458∧x<−42.6733002112614x>−37.9609112308767∧x<−37.4373124552784x>−36.3901149040818∧x<−31.6777259236971x>−30.1069295969022∧x<−25.3945406165175x>−23.8237442897226∧x<−20.1585528605345x>−19.1113553093379∧x<−17.540558982543x>−13.8753675533549∧x<−12.8281700021583x>−12.30457122656∧x<−7.59218224617533x>−6.02138591938044∧x<−1.30899693899575x>−0.261799387799149∧x<0.261799387799149x>6.02138591938044∧x<6.54498469497874x>7.59218224617533∧x<12.30457122656x>13.8753675533549∧x<20.1585528605345x>23.8237442897226∧x<26.4417381677141x>30.1069295969022∧x<31.1541271480988x>36.3901149040818∧x<37.9609112308767x>42.6733002112614∧x<44.2440965380563x>45.2912940892529∧x<48.9564855184409x>50.0036830696375∧x<51.5744793964324x>55.2396708256205∧x<56.2868683768171x>57.857664703612∧x<64.1408500107916x>67.8060414399797∧x<70.4240353179712x>74.0892267471593∧x<76.7072206251508x>80.3724120543389∧x<81.9432083811338x>86.6555973615185∧x<88.2263936883134x>93.9859802198946∧x<95.5567765466895x>100.269165527074∧x<107.075949609852x>114.406332468228
-3-5x<=x+3
(x-5,7)*(x-7,2)>0
(x-4)^2/(x-1)>0
x-4x^2/x-1>0
1/4
1/4