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=4.71238874329685x2=−7.85398149665124x3=61.2610563112167x4=54.9778718908148x5=83.252204888767x6=−54.9778709962906x7=1.57079582971902x8=−70.6858351534454x9=−23.5619450115115x10=−48.6946865760795x11=92.6769837789098x12=−83.2522055723275x13=89.535390888605x14=39.2699077336963x15=−61.2610569934486x16=−32.9867232184024x17=48.6946859012172x18=−64.4026491641039x19=−1.57079643188553x20=−29.8451300954883x21=17.2787599560783x22=−4.71238862219396x23=92.6769830592094x24=32.9867225164981x25=4.7123894841958x26=23.5619451518571x27=26.7035380604159x28=−54.9778717966574x29=98.9601690454399x30=42.4115007274741x31=1.57079657289894x32=−42.4115005850814x33=70.6858344802043x34=−42.411501235316x35=−10.9955738413568x36=−26.7035372004893x37=−95.8185758680502x38=14.1371671100222x39=80.1106131368654x40=−39.2699084145515x41=−48.6946857788076x42=23.5619444059921x43=29.8451303231501x44=95.8185760629547x45=83.2522056907544x46=76.9690196732095x47=−86.3937982346452x48=−76.9690195738024x49=48.6946866365921x50=67.5442415586719x51=−4.71238942125338x52=64.4026493072124x53=−89.535390750197x54=−98.9601681513438x55=86.3937978869933x56=54.9778710948428x57=39.2699085343272x58=−80.1106125781572x59=−67.5442421706656x60=89.5353901350773x61=45.5530929823099x62=76.9690204681432x63=36.128315618579x64=−20.4203527082932x65=−26.7035379986821x66=−32.9867224188086x67=−45.5530935911043x68=−70.6858343571487x69=10.9955747360645x70=−36.1283154173375x71=7.85398174307326x72=32.9867233134552x73=−17.2787590920677x74=−17.2787598356363x75=61.2610571125526x76=26.703537322248x77=−58.1194639976905x78=−76.9690203748894x79=10.9955739381756x80=67.54424230971x81=−86.3937977431483x82=−51.8362786893284x83=−61.2610562447228x84=17.2787591562062x85=45.553093730794x86=98.9601682515978x87=−92.6769829355125x88=−92.6769837307794x89=51.8362789031518x90=−64.4026497466373x91=−39.2699076683741x92=−10.9955746401247x93=−14.1371668370864x94=70.6858352127237x95=58.1194643979608x96=−98.9601689530982x97=−83.2522048211133x98=73.8274274830848x99=−73.8274272798455x100=20.4203521477756x101=−20.4203520060805x1=4.71238874329685x2=−7.85398149665124x3=61.2610563112167x4=54.9778718908148x5=83.252204888767x6=−54.9778709962906x7=1.57079582971902x8=−70.6858351534454x9=−23.5619450115115x10=−48.6946865760795x11=92.6769837789098x12=−83.2522055723275x13=89.535390888605x14=39.2699077336963x15=−61.2610569934486x16=−32.9867232184024x17=48.6946859012172x18=−64.4026491641039x19=−1.57079643188553x20=−29.8451300954883x21=17.2787599560783x22=−4.71238862219396x23=92.6769830592094x24=32.9867225164981x25=4.7123894841958x26=23.5619451518571x27=26.7035380604159x28=−54.9778717966574x29=98.9601690454399x30=42.4115007274741x31=1.57079657289894x32=−42.4115005850814x33=70.6858344802043x34=−42.411501235316x35=−10.9955738413568x36=−26.7035372004893x37=−95.8185758680502x38=14.1371671100222x39=80.1106131368654x40=−39.2699084145515x41=−48.6946857788076x42=23.5619444059921x43=29.8451303231501x44=95.8185760629547x45=83.2522056907544x46=76.9690196732095x47=−86.3937982346452x48=−76.9690195738024x49=48.6946866365921x50=67.5442415586719x51=−4.71238942125338x52=64.4026493072124x53=−89.535390750197x54=−98.9601681513438x55=86.3937978869933x56=54.9778710948428x57=39.2699085343272x58=−80.1106125781572x59=−67.5442421706656x60=89.5353901350773x61=45.5530929823099x62=76.9690204681432x63=36.128315618579x64=−20.4203527082932x65=−26.7035379986821x66=−32.9867224188086x67=−45.5530935911043x68=−70.6858343571487x69=10.9955747360645x70=−36.1283154173375x71=7.85398174307326x72=32.9867233134552x73=−17.2787590920677x74=−17.2787598356363x75=61.2610571125526x76=26.703537322248x77=−58.1194639976905x78=−76.9690203748894x79=10.9955739381756x80=67.54424230971x81=−86.3937977431483x82=−51.8362786893284x83=−61.2610562447228x84=17.2787591562062x85=45.553093730794x86=98.9601682515978x87=−92.6769829355125x88=−92.6769837307794x89=51.8362789031518x90=−64.4026497466373x91=−39.2699076683741x92=−10.9955746401247x93=−14.1371668370864x94=70.6858352127237x95=58.1194643979608x96=−98.9601689530982x97=−83.2522048211133x98=73.8274274830848x99=−73.8274272798455x100=20.4203521477756x101=−20.4203520060805Las raíces dadas
x96=−98.9601689530982x54=−98.9601681513438x37=−95.8185758680502x88=−92.6769837307794x87=−92.6769829355125x53=−89.535390750197x47=−86.3937982346452x81=−86.3937977431483x12=−83.2522055723275x97=−83.2522048211133x58=−80.1106125781572x78=−76.9690203748894x48=−76.9690195738024x99=−73.8274272798455x8=−70.6858351534454x68=−70.6858343571487x59=−67.5442421706656x90=−64.4026497466373x18=−64.4026491641039x15=−61.2610569934486x83=−61.2610562447228x77=−58.1194639976905x28=−54.9778717966574x6=−54.9778709962906x82=−51.8362786893284x10=−48.6946865760795x41=−48.6946857788076x67=−45.5530935911043x34=−42.411501235316x32=−42.4115005850814x40=−39.2699084145515x91=−39.2699076683741x70=−36.1283154173375x16=−32.9867232184024x66=−32.9867224188086x20=−29.8451300954883x65=−26.7035379986821x36=−26.7035372004893x9=−23.5619450115115x64=−20.4203527082932x101=−20.4203520060805x74=−17.2787598356363x73=−17.2787590920677x93=−14.1371668370864x92=−10.9955746401247x35=−10.9955738413568x2=−7.85398149665124x51=−4.71238942125338x22=−4.71238862219396x19=−1.57079643188553x7=1.57079582971902x31=1.57079657289894x1=4.71238874329685x25=4.7123894841958x71=7.85398174307326x79=10.9955739381756x69=10.9955747360645x38=14.1371671100222x84=17.2787591562062x21=17.2787599560783x100=20.4203521477756x42=23.5619444059921x26=23.5619451518571x76=26.703537322248x27=26.7035380604159x43=29.8451303231501x24=32.9867225164981x72=32.9867233134552x63=36.128315618579x14=39.2699077336963x57=39.2699085343272x30=42.4115007274741x61=45.5530929823099x85=45.553093730794x17=48.6946859012172x49=48.6946866365921x89=51.8362789031518x56=54.9778710948428x4=54.9778718908148x95=58.1194643979608x3=61.2610563112167x75=61.2610571125526x52=64.4026493072124x50=67.5442415586719x80=67.54424230971x33=70.6858344802043x94=70.6858352127237x98=73.8274274830848x46=76.9690196732095x62=76.9690204681432x39=80.1106131368654x5=83.252204888767x45=83.2522056907544x55=86.3937978869933x60=89.5353901350773x13=89.535390888605x23=92.6769830592094x11=92.6769837789098x44=95.8185760629547x86=98.9601682515978x29=98.9601690454399son puntos de cambio del signo de desigualdad en las soluciones.
Primero definámonos con el signo hasta el punto extremo izquierdo:
x0<x96Consideremos, por ejemplo, el punto
x0=x96−101=
−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-------
x96 x54 x37 x88 x87 x53 x47 x81 x12 x97 x58 x78 x48 x99 x8 x68 x59 x90 x18 x15 x83 x77 x28 x6 x82 x10 x41 x67 x34 x32 x40 x91 x70 x16 x66 x20 x65 x36 x9 x64 x101 x74 x73 x93 x92 x35 x2 x51 x22 x19 x7 x31 x1 x25 x71 x79 x69 x38 x84 x21 x100 x42 x26 x76 x27 x43 x24 x72 x63 x14 x57 x30 x61 x85 x17 x49 x89 x56 x4 x95 x3 x75 x52 x50 x80 x33 x94 x98 x46 x62 x39 x5 x45 x55 x60 x13 x23 x11 x44 x86 x29Recibiremos 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+4)(x-8)>0
6x-11*(x+2)>-8
(x-1)^2>0
|sinx|