Se da la desigualdad:
∣sin(x)∣<1Para resolver esta desigualdad primero hay que resolver la ecuación correspondiente:
∣sin(x)∣=1Resolvemos:
Tenemos la ecuación
∣sin(x)∣=1cambiamos
∣sin(x)∣−1=0∣sin(x)∣−1=0Sustituimos
w=∣sin(x)∣Transportamos los términos libres (sin w)
del miembro izquierdo al derecho, obtenemos:
w=1Obtenemos la respuesta: w = 1
hacemos cambio inverso
∣sin(x)∣=wsustituimos w:
x1=51.8362789031518x2=−17.2787590920677x3=−36.1283154173375x4=−39.2699084145515x5=−64.4026497466373x6=−48.6946857788076x7=76.9690204681432x8=83.2522056907544x9=1.57079657289894x10=83.252204888767x11=48.6946866365921x12=23.5619444059921x13=54.9778710948428x14=1.57079582971902x15=45.5530929823099x16=−76.9690203748894x17=−98.9601689530982x18=−32.9867224188086x19=−51.8362786893284x20=−10.9955738413568x21=70.6858352127237x22=32.9867225164981x23=76.9690196732095x24=20.4203521477756x25=−98.9601681513438x26=−4.71238942125338x27=−58.1194639976905x28=36.128315618579x29=4.7123894841958x30=−92.6769829355125x31=67.54424230971x32=32.9867233134552x33=98.9601690454399x34=10.9955739381756x35=−1.57079643188553x36=−89.535390750197x37=61.2610563112167x38=64.4026493072124x39=−23.5619450115115x40=89.5353901350773x41=−32.9867232184024x42=54.9778718908148x43=−4.71238862219396x44=67.5442415586719x45=−48.6946865760795x46=−45.5530935911043x47=4.71238874329685x48=73.8274274830848x49=−54.9778709962906x50=80.1106131368654x51=−83.2522055723275x52=−73.8274272798455x53=45.553093730794x54=58.1194643979608x55=26.703537322248x56=39.2699077336963x57=−95.8185758680502x58=−80.1106125781572x59=23.5619451518571x60=26.7035380604159x61=10.9955747360645x62=−86.3937982346452x63=−70.6858351534454x64=−39.2699076683741x65=48.6946859012172x66=92.6769830592094x67=−76.9690195738024x68=−92.6769837307794x69=70.6858344802043x70=−83.2522048211133x71=−61.2610569934486x72=17.2787591562062x73=29.8451303231501x74=−26.7035379986821x75=−14.1371668370864x76=−42.4115005850814x77=−29.8451300954883x78=7.85398174307326x79=95.8185760629547x80=−42.411501235316x81=−54.9778717966574x82=14.1371671100222x83=−17.2787598356363x84=−7.85398149665124x85=−20.4203520060805x86=92.6769837789098x87=89.535390888605x88=−70.6858343571487x89=42.4115007274741x90=−20.4203527082932x91=61.2610571125526x92=39.2699085343272x93=−10.9955746401247x94=−67.5442421706656x95=−64.4026491641039x96=−86.3937977431483x97=17.2787599560783x98=−26.7035372004893x99=98.9601682515978x100=86.3937978869933x101=−61.2610562447228x1=51.8362789031518x2=−17.2787590920677x3=−36.1283154173375x4=−39.2699084145515x5=−64.4026497466373x6=−48.6946857788076x7=76.9690204681432x8=83.2522056907544x9=1.57079657289894x10=83.252204888767x11=48.6946866365921x12=23.5619444059921x13=54.9778710948428x14=1.57079582971902x15=45.5530929823099x16=−76.9690203748894x17=−98.9601689530982x18=−32.9867224188086x19=−51.8362786893284x20=−10.9955738413568x21=70.6858352127237x22=32.9867225164981x23=76.9690196732095x24=20.4203521477756x25=−98.9601681513438x26=−4.71238942125338x27=−58.1194639976905x28=36.128315618579x29=4.7123894841958x30=−92.6769829355125x31=67.54424230971x32=32.9867233134552x33=98.9601690454399x34=10.9955739381756x35=−1.57079643188553x36=−89.535390750197x37=61.2610563112167x38=64.4026493072124x39=−23.5619450115115x40=89.5353901350773x41=−32.9867232184024x42=54.9778718908148x43=−4.71238862219396x44=67.5442415586719x45=−48.6946865760795x46=−45.5530935911043x47=4.71238874329685x48=73.8274274830848x49=−54.9778709962906x50=80.1106131368654x51=−83.2522055723275x52=−73.8274272798455x53=45.553093730794x54=58.1194643979608x55=26.703537322248x56=39.2699077336963x57=−95.8185758680502x58=−80.1106125781572x59=23.5619451518571x60=26.7035380604159x61=10.9955747360645x62=−86.3937982346452x63=−70.6858351534454x64=−39.2699076683741x65=48.6946859012172x66=92.6769830592094x67=−76.9690195738024x68=−92.6769837307794x69=70.6858344802043x70=−83.2522048211133x71=−61.2610569934486x72=17.2787591562062x73=29.8451303231501x74=−26.7035379986821x75=−14.1371668370864x76=−42.4115005850814x77=−29.8451300954883x78=7.85398174307326x79=95.8185760629547x80=−42.411501235316x81=−54.9778717966574x82=14.1371671100222x83=−17.2787598356363x84=−7.85398149665124x85=−20.4203520060805x86=92.6769837789098x87=89.535390888605x88=−70.6858343571487x89=42.4115007274741x90=−20.4203527082932x91=61.2610571125526x92=39.2699085343272x93=−10.9955746401247x94=−67.5442421706656x95=−64.4026491641039x96=−86.3937977431483x97=17.2787599560783x98=−26.7035372004893x99=98.9601682515978x100=86.3937978869933x101=−61.2610562447228Las raíces dadas
x17=−98.9601689530982x25=−98.9601681513438x57=−95.8185758680502x68=−92.6769837307794x30=−92.6769829355125x36=−89.535390750197x62=−86.3937982346452x96=−86.3937977431483x51=−83.2522055723275x70=−83.2522048211133x58=−80.1106125781572x16=−76.9690203748894x67=−76.9690195738024x52=−73.8274272798455x63=−70.6858351534454x88=−70.6858343571487x94=−67.5442421706656x5=−64.4026497466373x95=−64.4026491641039x71=−61.2610569934486x101=−61.2610562447228x27=−58.1194639976905x81=−54.9778717966574x49=−54.9778709962906x19=−51.8362786893284x45=−48.6946865760795x6=−48.6946857788076x46=−45.5530935911043x80=−42.411501235316x76=−42.4115005850814x4=−39.2699084145515x64=−39.2699076683741x3=−36.1283154173375x41=−32.9867232184024x18=−32.9867224188086x77=−29.8451300954883x74=−26.7035379986821x98=−26.7035372004893x39=−23.5619450115115x90=−20.4203527082932x85=−20.4203520060805x83=−17.2787598356363x2=−17.2787590920677x75=−14.1371668370864x93=−10.9955746401247x20=−10.9955738413568x84=−7.85398149665124x26=−4.71238942125338x43=−4.71238862219396x35=−1.57079643188553x14=1.57079582971902x9=1.57079657289894x47=4.71238874329685x29=4.7123894841958x78=7.85398174307326x34=10.9955739381756x61=10.9955747360645x82=14.1371671100222x72=17.2787591562062x97=17.2787599560783x24=20.4203521477756x12=23.5619444059921x59=23.5619451518571x55=26.703537322248x60=26.7035380604159x73=29.8451303231501x22=32.9867225164981x32=32.9867233134552x28=36.128315618579x56=39.2699077336963x92=39.2699085343272x89=42.4115007274741x15=45.5530929823099x53=45.553093730794x65=48.6946859012172x11=48.6946866365921x1=51.8362789031518x13=54.9778710948428x42=54.9778718908148x54=58.1194643979608x37=61.2610563112167x91=61.2610571125526x38=64.4026493072124x44=67.5442415586719x31=67.54424230971x69=70.6858344802043x21=70.6858352127237x48=73.8274274830848x23=76.9690196732095x7=76.9690204681432x50=80.1106131368654x10=83.252204888767x8=83.2522056907544x100=86.3937978869933x40=89.5353901350773x87=89.535390888605x66=92.6769830592094x86=92.6769837789098x79=95.8185760629547x99=98.9601682515978x33=98.9601690454399son puntos de cambio del signo de desigualdad en las soluciones.
Primero definámonos con el signo hasta el punto extremo izquierdo:
x0<x17Consideremos, por ejemplo, el punto
x0=x17−101=
−98.9601689530982+−101=
−99.0601689530982lo sustituimos en la expresión
∣sin(x)∣<1∣sin(−99.0601689530982)∣<10.995004128836795 < 1
significa que una de las soluciones de nuestra ecuación será con:
x<−98.9601689530982 _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____
\ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \
-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------
x17 x25 x57 x68 x30 x36 x62 x96 x51 x70 x58 x16 x67 x52 x63 x88 x94 x5 x95 x71 x101 x27 x81 x49 x19 x45 x6 x46 x80 x76 x4 x64 x3 x41 x18 x77 x74 x98 x39 x90 x85 x83 x2 x75 x93 x20 x84 x26 x43 x35 x14 x9 x47 x29 x78 x34 x61 x82 x72 x97 x24 x12 x59 x55 x60 x73 x22 x32 x28 x56 x92 x89 x15 x53 x65 x11 x1 x13 x42 x54 x37 x91 x38 x44 x31 x69 x21 x48 x23 x7 x50 x10 x8 x100 x40 x87 x66 x86 x79 x99 x33Recibiremos otras soluciones de la desigualdad pasando al polo siguiente etc.
etc.
Respuesta:
x<−98.9601689530982x>−98.9601681513438∧x<−95.8185758680502x>−92.6769837307794∧x<−92.6769829355125x>−89.535390750197∧x<−86.3937982346452x>−86.3937977431483∧x<−83.2522055723275x>−83.2522048211133∧x<−80.1106125781572x>−76.9690203748894∧x<−76.9690195738024x>−73.8274272798455∧x<−70.6858351534454x>−70.6858343571487∧x<−67.5442421706656x>−64.4026497466373∧x<−64.4026491641039x>−61.2610569934486∧x<−61.2610562447228x>−58.1194639976905∧x<−54.9778717966574x>−54.9778709962906∧x<−51.8362786893284x>−48.6946865760795∧x<−48.6946857788076x>−45.5530935911043∧x<−42.411501235316x>−42.4115005850814∧x<−39.2699084145515x>−39.2699076683741∧x<−36.1283154173375x>−32.9867232184024∧x<−32.9867224188086x>−29.8451300954883∧x<−26.7035379986821x>−26.7035372004893∧x<−23.5619450115115x>−20.4203527082932∧x<−20.4203520060805x>−17.2787598356363∧x<−17.2787590920677x>−14.1371668370864∧x<−10.9955746401247x>−10.9955738413568∧x<−7.85398149665124x>−4.71238942125338∧x<−4.71238862219396x>−1.57079643188553∧x<1.57079582971902x>1.57079657289894∧x<4.71238874329685x>4.7123894841958∧x<7.85398174307326x>10.9955739381756∧x<10.9955747360645x>14.1371671100222∧x<17.2787591562062x>17.2787599560783∧x<20.4203521477756x>23.5619444059921∧x<23.5619451518571x>26.703537322248∧x<26.7035380604159x>29.8451303231501∧x<32.9867225164981x>32.9867233134552∧x<36.128315618579x>39.2699077336963∧x<39.2699085343272x>42.4115007274741∧x<45.5530929823099x>45.553093730794∧x<48.6946859012172x>48.6946866365921∧x<51.8362789031518x>54.9778710948428∧x<54.9778718908148x>58.1194643979608∧x<61.2610563112167x>61.2610571125526∧x<64.4026493072124x>67.5442415586719∧x<67.54424230971x>70.6858344802043∧x<70.6858352127237x>73.8274274830848∧x<76.9690196732095x>76.9690204681432∧x<80.1106131368654x>83.252204888767∧x<83.2522056907544x>86.3937978869933∧x<89.5353901350773x>89.535390888605∧x<92.6769830592094x>92.6769837789098∧x<95.8185760629547x>98.9601682515978∧x<98.9601690454399
-x^2+3x-2<0
x^2+4x-5<0
x+1>0
x^2-6x-27>0