Se da la desigualdad:
sin(t)<−32Para resolver esta desigualdad primero hay que resolver la ecuación correspondiente:
sin(t)=−32Resolvemos:
Tenemos la ecuación
sin(t)=−32cambiamos
sin(t)+32=0sin(t)+32=0Sustituimos
w=sin(t)Transportamos los términos libres (sin w)
del miembro izquierdo al derecho, obtenemos:
w=−32Obtenemos la respuesta: w = -2/3
hacemos cambio inverso
sin(t)=wsustituimos w:
x1=412.27836527649x2=10.1545056169963x3=93.5180519514668x4=1216.52608459548x5=−32.1456541921249x6=−65.2437180691587x7=30.686198879671x8=−107.54387787828x9=−302.322622400847x10=−77.8100886835179x11=−8.69505030454241x12=−38.4288394993045x13=5.55345765095262x14=16.4376909241759x15=−88.6943219567412x16=18.1198282653118x17=−50.9952101136637x18=−57.2783954208432x19=24.4030135724914x20=55.8189401083893x21=−14.978235611722x22=85.5527293031514x23=72.9863586887922x24=−69.8447660352024x25=11.8366429581322x26=74.6684960299281x27=−33.8277915332608x28=−101.2606925711x29=−7.01291296340655x30=−0.729727656226966x31=−63.5615807280228x32=−84.0932739906975x33=68.3853107227485x34=60.419988074433x35=99.8012372586464x36=−13.2960982705861x37=−58.9605327619791x38=−21.2614209189016x39=29.0040615385351x40=87.2348666442873x41=−2.41186499736283x42=62.1021254155689x43=−27.5446062260812x44=−44.7120248064841x45=−94.9775072639208x46=54.1368027672534x47=−25.8624688849453x48=98.1190999175106x49=−836.393373511112x50=3.87132030981676x51=−46.3941621476199x52=22.7208762313555x53=36.9693841868506x54=−90.376459297877x55=175.199460944801x56=−82.4111366495616x57=91.835914610331x58=−1755.42056570047x59=−19.5792835777657x60=79.2695439959718x61=49.5357548012097x62=66.7031733816126x63=80.9516813371077x64=−40.1109768404403x65=47.8536174600739x66=−96.6596446050566x67=43.2525694940301x68=−1979.9330994178x69=−71.5269033763383x70=41.5704321528943x71=−52.6773474547995x72=35.2872468457147x73=−76.127951342382x1=412.27836527649x2=10.1545056169963x3=93.5180519514668x4=1216.52608459548x5=−32.1456541921249x6=−65.2437180691587x7=30.686198879671x8=−107.54387787828x9=−302.322622400847x10=−77.8100886835179x11=−8.69505030454241x12=−38.4288394993045x13=5.55345765095262x14=16.4376909241759x15=−88.6943219567412x16=18.1198282653118x17=−50.9952101136637x18=−57.2783954208432x19=24.4030135724914x20=55.8189401083893x21=−14.978235611722x22=85.5527293031514x23=72.9863586887922x24=−69.8447660352024x25=11.8366429581322x26=74.6684960299281x27=−33.8277915332608x28=−101.2606925711x29=−7.01291296340655x30=−0.729727656226966x31=−63.5615807280228x32=−84.0932739906975x33=68.3853107227485x34=60.419988074433x35=99.8012372586464x36=−13.2960982705861x37=−58.9605327619791x38=−21.2614209189016x39=29.0040615385351x40=87.2348666442873x41=−2.41186499736283x42=62.1021254155689x43=−27.5446062260812x44=−44.7120248064841x45=−94.9775072639208x46=54.1368027672534x47=−25.8624688849453x48=98.1190999175106x49=−836.393373511112x50=3.87132030981676x51=−46.3941621476199x52=22.7208762313555x53=36.9693841868506x54=−90.376459297877x55=175.199460944801x56=−82.4111366495616x57=91.835914610331x58=−1755.42056570047x59=−19.5792835777657x60=79.2695439959718x61=49.5357548012097x62=66.7031733816126x63=80.9516813371077x64=−40.1109768404403x65=47.8536174600739x66=−96.6596446050566x67=43.2525694940301x68=−1979.9330994178x69=−71.5269033763383x70=41.5704321528943x71=−52.6773474547995x72=35.2872468457147x73=−76.127951342382Las raíces dadas
x68=−1979.9330994178x58=−1755.42056570047x49=−836.393373511112x9=−302.322622400847x8=−107.54387787828x28=−101.2606925711x66=−96.6596446050566x45=−94.9775072639208x54=−90.376459297877x15=−88.6943219567412x32=−84.0932739906975x56=−82.4111366495616x10=−77.8100886835179x73=−76.127951342382x69=−71.5269033763383x24=−69.8447660352024x6=−65.2437180691587x31=−63.5615807280228x37=−58.9605327619791x18=−57.2783954208432x71=−52.6773474547995x17=−50.9952101136637x51=−46.3941621476199x44=−44.7120248064841x64=−40.1109768404403x12=−38.4288394993045x27=−33.8277915332608x5=−32.1456541921249x43=−27.5446062260812x47=−25.8624688849453x38=−21.2614209189016x59=−19.5792835777657x21=−14.978235611722x36=−13.2960982705861x11=−8.69505030454241x29=−7.01291296340655x41=−2.41186499736283x30=−0.729727656226966x50=3.87132030981676x13=5.55345765095262x2=10.1545056169963x25=11.8366429581322x14=16.4376909241759x16=18.1198282653118x52=22.7208762313555x19=24.4030135724914x39=29.0040615385351x7=30.686198879671x72=35.2872468457147x53=36.9693841868506x70=41.5704321528943x67=43.2525694940301x65=47.8536174600739x61=49.5357548012097x46=54.1368027672534x20=55.8189401083893x34=60.419988074433x42=62.1021254155689x62=66.7031733816126x33=68.3853107227485x23=72.9863586887922x26=74.6684960299281x60=79.2695439959718x63=80.9516813371077x22=85.5527293031514x40=87.2348666442873x57=91.835914610331x3=93.5180519514668x48=98.1190999175106x35=99.8012372586464x55=175.199460944801x1=412.27836527649x4=1216.52608459548son puntos de cambio del signo de desigualdad en las soluciones.
Primero definámonos con el signo hasta el punto extremo izquierdo:
x0<x68Consideremos, por ejemplo, el punto
x0=x68−101=
−1979.9330994178+−101=
−1980.0330994178lo sustituimos en la expresión
sin(t)<−32sin(t)<−32sin(t) < -2/3
Entonces
x<−1979.9330994178no se cumple
significa que una de las soluciones de nuestra ecuación será con:
x>−1979.9330994178∧x<−1755.42056570047 _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____
/ \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
-------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-------
x68 x58 x49 x9 x8 x28 x66 x45 x54 x15 x32 x56 x10 x73 x69 x24 x6 x31 x37 x18 x71 x17 x51 x44 x64 x12 x27 x5 x43 x47 x38 x59 x21 x36 x11 x29 x41 x30 x50 x13 x2 x25 x14 x16 x52 x19 x39 x7 x72 x53 x70 x67 x65 x61 x46 x20 x34 x42 x62 x33 x23 x26 x60 x63 x22 x40 x57 x3 x48 x35 x55 x1 x4Recibiremos otras soluciones de la desigualdad pasando al polo siguiente etc.
etc.
Respuesta:
x>−1979.9330994178∧x<−1755.42056570047x>−836.393373511112∧x<−302.322622400847x>−107.54387787828∧x<−101.2606925711x>−96.6596446050566∧x<−94.9775072639208x>−90.376459297877∧x<−88.6943219567412x>−84.0932739906975∧x<−82.4111366495616x>−77.8100886835179∧x<−76.127951342382x>−71.5269033763383∧x<−69.8447660352024x>−65.2437180691587∧x<−63.5615807280228x>−58.9605327619791∧x<−57.2783954208432x>−52.6773474547995∧x<−50.9952101136637x>−46.3941621476199∧x<−44.7120248064841x>−40.1109768404403∧x<−38.4288394993045x>−33.8277915332608∧x<−32.1456541921249x>−27.5446062260812∧x<−25.8624688849453x>−21.2614209189016∧x<−19.5792835777657x>−14.978235611722∧x<−13.2960982705861x>−8.69505030454241∧x<−7.01291296340655x>−2.41186499736283∧x<−0.729727656226966x>3.87132030981676∧x<5.55345765095262x>10.1545056169963∧x<11.8366429581322x>16.4376909241759∧x<18.1198282653118x>22.7208762313555∧x<24.4030135724914x>29.0040615385351∧x<30.686198879671x>35.2872468457147∧x<36.9693841868506x>41.5704321528943∧x<43.2525694940301x>47.8536174600739∧x<49.5357548012097x>54.1368027672534∧x<55.8189401083893x>60.419988074433∧x<62.1021254155689x>66.7031733816126∧x<68.3853107227485x>72.9863586887922∧x<74.6684960299281x>79.2695439959718∧x<80.9516813371077x>85.5527293031514∧x<87.2348666442873x>91.835914610331∧x<93.5180519514668x>98.1190999175106∧x<99.8012372586464x>175.199460944801∧x<412.27836527649x>1216.52608459548
(x-2)/(3-x)>=0
(2x+1)^2*(x^2-4x+3)>0
(x-4)^2>0
x^2-2x-3>0
sin(t)