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Resolver desigualdades/ sin(t)>=-1/2

sin(t)>=-1/2 desigualdades

En la desigualdad la incógnita

Solución

Ha introducido [src] 
sin(t) >= -1/2
sin⁡(t)≥−12\sin{\left(t \right)} \geq - \frac{1}{2} 
sin(t) >= -1/2
Solución detallada
Se da la desigualdad:
sin⁡(t)≥−12\sin{\left(t \right)} \geq - \frac{1}{2}
Para resolver esta desigualdad primero hay que resolver la ecuación correspondiente:
sin⁡(t)=−12\sin{\left(t \right)} = - \frac{1}{2}
Resolvemos:
Tenemos la ecuación
sin⁡(t)=−12\sin{\left(t \right)} = - \frac{1}{2}
cambiamos
sin⁡(t)+12=0\sin{\left(t \right)} + \frac{1}{2} = 0
sin⁡(t)+12=0\sin{\left(t \right)} + \frac{1}{2} = 0
Sustituimos
w=sin⁡(t)w = \sin{\left(t \right)}
Transportamos los términos libres (sin w)
del miembro izquierdo al derecho, obtenemos:
w=−12w = - \frac{1}{2}
Obtenemos la respuesta: w = -1/2
hacemos cambio inverso
sin⁡(t)=w\sin{\left(t \right)} = w
sustituimos w:
x1=47.6474885794452x_{1} = 47.6474885794452
x2=74.8746249105567x_{2} = 74.8746249105567
x3=81.1578102177363x_{3} = 81.1578102177363
x4=−27.7507351067098x_{4} = -27.7507351067098
x5=60.2138591938044x_{5} = 60.2138591938044
x6=−19.3731546971371x_{6} = -19.3731546971371
x7=−50.789081233035x_{7} = -50.789081233035
x8=437.20497762458x_{8} = 437.20497762458
x9=−21.4675497995303x_{9} = -21.4675497995303
x10=−44.5058959258554x_{10} = -44.5058959258554
x11=−57.0722665402146x_{11} = -57.0722665402146
x12=43.4586983746588x_{12} = 43.4586983746588
x13=18.3259571459405x_{13} = 18.3259571459405
x14=−34.0339204138894x_{14} = -34.0339204138894
x15=−151.320046147908x_{15} = -151.320046147908
x16=−38.2227106186758x_{16} = -38.2227106186758
x17=−59.1666616426078x_{17} = -59.1666616426078
x18=72.7802298081635x_{18} = 72.7802298081635
x19=−13.0899693899575x_{19} = -13.0899693899575
x20=3.66519142918809x_{20} = 3.66519142918809
x21=−6.80678408277789x_{21} = -6.80678408277789
x22=−52.8834763354282x_{22} = -52.8834763354282
x23=−65.4498469497874x_{23} = -65.4498469497874
x24=−46.6002910282486x_{24} = -46.6002910282486
x25=87.4409955249159x_{25} = 87.4409955249159
x26=−88.4881930761125x_{26} = -88.4881930761125
x27=24.60914245312x_{27} = 24.60914245312
x28=100.007366139275x_{28} = 100.007366139275
x29=−31.9395253114962x_{29} = -31.9395253114962
x30=79.0634151153431x_{30} = 79.0634151153431
x31=−63.3554518473942x_{31} = -63.3554518473942
x32=56.025068989018x_{32} = 56.025068989018
x33=41.3643032722656x_{33} = 41.3643032722656
x34=91.6297857297023x_{34} = 91.6297857297023
x35=28.7979326579064x_{35} = 28.7979326579064
x36=93.7241808320955x_{36} = 93.7241808320955
x37=−69.6386371545737x_{37} = -69.6386371545737
x38=37.1755130674792x_{38} = 37.1755130674792
x39=66400.1787274983x_{39} = 66400.1787274983
x40=53.9306738866248x_{40} = 53.9306738866248
x41=16.2315620435473x_{41} = 16.2315620435473
x42=−75.9218224617533x_{42} = -75.9218224617533
x43=−94.7713783832921x_{43} = -94.7713783832921
x44=−78.0162175641465x_{44} = -78.0162175641465
x45=68.5914396033772x_{45} = 68.5914396033772
x46=−82.2050077689329x_{46} = -82.2050077689329
x47=−96.8657734856853x_{47} = -96.8657734856853
x48=−84.2994028713261x_{48} = -84.2994028713261
x49=9.94837673636768x_{49} = 9.94837673636768
x50=−0.523598775598299x_{50} = -0.523598775598299
x51=12.0427718387609x_{51} = 12.0427718387609
x52=−2.61799387799149x_{52} = -2.61799387799149
x53=−195.302343298165x_{53} = -195.302343298165
x54=49.7418836818384x_{54} = 49.7418836818384
x55=−71.733032256967x_{55} = -71.733032256967
x56=−40.317105721069x_{56} = -40.317105721069
x57=−8.90117918517108x_{57} = -8.90117918517108
x58=66.497044500984x_{58} = 66.497044500984
x59=35.081117965086x_{59} = 35.081117965086
x60=85.3466004225227x_{60} = 85.3466004225227
x61=192.160750644576x_{61} = 192.160750644576
x62=5.75958653158129x_{62} = 5.75958653158129
x63=−90.5825881785057x_{63} = -90.5825881785057
x64=−25.6563400043166x_{64} = -25.6563400043166
x65=22.5147473507269x_{65} = 22.5147473507269
x66=−101.054563690472x_{66} = -101.054563690472
x67=−15.1843644923507x_{67} = -15.1843644923507
x68=62.3082542961976x_{68} = 62.3082542961976
x69=30.8923277602996x_{69} = 30.8923277602996
x70=97.9129710368819x_{70} = 97.9129710368819
x1=47.6474885794452x_{1} = 47.6474885794452
x2=74.8746249105567x_{2} = 74.8746249105567
x3=81.1578102177363x_{3} = 81.1578102177363
x4=−27.7507351067098x_{4} = -27.7507351067098
x5=60.2138591938044x_{5} = 60.2138591938044
x6=−19.3731546971371x_{6} = -19.3731546971371
x7=−50.789081233035x_{7} = -50.789081233035
x8=437.20497762458x_{8} = 437.20497762458
x9=−21.4675497995303x_{9} = -21.4675497995303
x10=−44.5058959258554x_{10} = -44.5058959258554
x11=−57.0722665402146x_{11} = -57.0722665402146
x12=43.4586983746588x_{12} = 43.4586983746588
x13=18.3259571459405x_{13} = 18.3259571459405
x14=−34.0339204138894x_{14} = -34.0339204138894
x15=−151.320046147908x_{15} = -151.320046147908
x16=−38.2227106186758x_{16} = -38.2227106186758
x17=−59.1666616426078x_{17} = -59.1666616426078
x18=72.7802298081635x_{18} = 72.7802298081635
x19=−13.0899693899575x_{19} = -13.0899693899575
x20=3.66519142918809x_{20} = 3.66519142918809
x21=−6.80678408277789x_{21} = -6.80678408277789
x22=−52.8834763354282x_{22} = -52.8834763354282
x23=−65.4498469497874x_{23} = -65.4498469497874
x24=−46.6002910282486x_{24} = -46.6002910282486
x25=87.4409955249159x_{25} = 87.4409955249159
x26=−88.4881930761125x_{26} = -88.4881930761125
x27=24.60914245312x_{27} = 24.60914245312
x28=100.007366139275x_{28} = 100.007366139275
x29=−31.9395253114962x_{29} = -31.9395253114962
x30=79.0634151153431x_{30} = 79.0634151153431
x31=−63.3554518473942x_{31} = -63.3554518473942
x32=56.025068989018x_{32} = 56.025068989018
x33=41.3643032722656x_{33} = 41.3643032722656
x34=91.6297857297023x_{34} = 91.6297857297023
x35=28.7979326579064x_{35} = 28.7979326579064
x36=93.7241808320955x_{36} = 93.7241808320955
x37=−69.6386371545737x_{37} = -69.6386371545737
x38=37.1755130674792x_{38} = 37.1755130674792
x39=66400.1787274983x_{39} = 66400.1787274983
x40=53.9306738866248x_{40} = 53.9306738866248
x41=16.2315620435473x_{41} = 16.2315620435473
x42=−75.9218224617533x_{42} = -75.9218224617533
x43=−94.7713783832921x_{43} = -94.7713783832921
x44=−78.0162175641465x_{44} = -78.0162175641465
x45=68.5914396033772x_{45} = 68.5914396033772
x46=−82.2050077689329x_{46} = -82.2050077689329
x47=−96.8657734856853x_{47} = -96.8657734856853
x48=−84.2994028713261x_{48} = -84.2994028713261
x49=9.94837673636768x_{49} = 9.94837673636768
x50=−0.523598775598299x_{50} = -0.523598775598299
x51=12.0427718387609x_{51} = 12.0427718387609
x52=−2.61799387799149x_{52} = -2.61799387799149
x53=−195.302343298165x_{53} = -195.302343298165
x54=49.7418836818384x_{54} = 49.7418836818384
x55=−71.733032256967x_{55} = -71.733032256967
x56=−40.317105721069x_{56} = -40.317105721069
x57=−8.90117918517108x_{57} = -8.90117918517108
x58=66.497044500984x_{58} = 66.497044500984
x59=35.081117965086x_{59} = 35.081117965086
x60=85.3466004225227x_{60} = 85.3466004225227
x61=192.160750644576x_{61} = 192.160750644576
x62=5.75958653158129x_{62} = 5.75958653158129
x63=−90.5825881785057x_{63} = -90.5825881785057
x64=−25.6563400043166x_{64} = -25.6563400043166
x65=22.5147473507269x_{65} = 22.5147473507269
x66=−101.054563690472x_{66} = -101.054563690472
x67=−15.1843644923507x_{67} = -15.1843644923507
x68=62.3082542961976x_{68} = 62.3082542961976
x69=30.8923277602996x_{69} = 30.8923277602996
x70=97.9129710368819x_{70} = 97.9129710368819
Las raíces dadas
x53=−195.302343298165x_{53} = -195.302343298165
x15=−151.320046147908x_{15} = -151.320046147908
x66=−101.054563690472x_{66} = -101.054563690472
x47=−96.8657734856853x_{47} = -96.8657734856853
x43=−94.7713783832921x_{43} = -94.7713783832921
x63=−90.5825881785057x_{63} = -90.5825881785057
x26=−88.4881930761125x_{26} = -88.4881930761125
x48=−84.2994028713261x_{48} = -84.2994028713261
x46=−82.2050077689329x_{46} = -82.2050077689329
x44=−78.0162175641465x_{44} = -78.0162175641465
x42=−75.9218224617533x_{42} = -75.9218224617533
x55=−71.733032256967x_{55} = -71.733032256967
x37=−69.6386371545737x_{37} = -69.6386371545737
x23=−65.4498469497874x_{23} = -65.4498469497874
x31=−63.3554518473942x_{31} = -63.3554518473942
x17=−59.1666616426078x_{17} = -59.1666616426078
x11=−57.0722665402146x_{11} = -57.0722665402146
x22=−52.8834763354282x_{22} = -52.8834763354282
x7=−50.789081233035x_{7} = -50.789081233035
x24=−46.6002910282486x_{24} = -46.6002910282486
x10=−44.5058959258554x_{10} = -44.5058959258554
x56=−40.317105721069x_{56} = -40.317105721069
x16=−38.2227106186758x_{16} = -38.2227106186758
x14=−34.0339204138894x_{14} = -34.0339204138894
x29=−31.9395253114962x_{29} = -31.9395253114962
x4=−27.7507351067098x_{4} = -27.7507351067098
x64=−25.6563400043166x_{64} = -25.6563400043166
x9=−21.4675497995303x_{9} = -21.4675497995303
x6=−19.3731546971371x_{6} = -19.3731546971371
x67=−15.1843644923507x_{67} = -15.1843644923507
x19=−13.0899693899575x_{19} = -13.0899693899575
x57=−8.90117918517108x_{57} = -8.90117918517108
x21=−6.80678408277789x_{21} = -6.80678408277789
x52=−2.61799387799149x_{52} = -2.61799387799149
x50=−0.523598775598299x_{50} = -0.523598775598299
x20=3.66519142918809x_{20} = 3.66519142918809
x62=5.75958653158129x_{62} = 5.75958653158129
x49=9.94837673636768x_{49} = 9.94837673636768
x51=12.0427718387609x_{51} = 12.0427718387609
x41=16.2315620435473x_{41} = 16.2315620435473
x13=18.3259571459405x_{13} = 18.3259571459405
x65=22.5147473507269x_{65} = 22.5147473507269
x27=24.60914245312x_{27} = 24.60914245312
x35=28.7979326579064x_{35} = 28.7979326579064
x69=30.8923277602996x_{69} = 30.8923277602996
x59=35.081117965086x_{59} = 35.081117965086
x38=37.1755130674792x_{38} = 37.1755130674792
x33=41.3643032722656x_{33} = 41.3643032722656
x12=43.4586983746588x_{12} = 43.4586983746588
x1=47.6474885794452x_{1} = 47.6474885794452
x54=49.7418836818384x_{54} = 49.7418836818384
x40=53.9306738866248x_{40} = 53.9306738866248
x32=56.025068989018x_{32} = 56.025068989018
x5=60.2138591938044x_{5} = 60.2138591938044
x68=62.3082542961976x_{68} = 62.3082542961976
x58=66.497044500984x_{58} = 66.497044500984
x45=68.5914396033772x_{45} = 68.5914396033772
x18=72.7802298081635x_{18} = 72.7802298081635
x2=74.8746249105567x_{2} = 74.8746249105567
x30=79.0634151153431x_{30} = 79.0634151153431
x3=81.1578102177363x_{3} = 81.1578102177363
x60=85.3466004225227x_{60} = 85.3466004225227
x25=87.4409955249159x_{25} = 87.4409955249159
x34=91.6297857297023x_{34} = 91.6297857297023
x36=93.7241808320955x_{36} = 93.7241808320955
x70=97.9129710368819x_{70} = 97.9129710368819
x28=100.007366139275x_{28} = 100.007366139275
x61=192.160750644576x_{61} = 192.160750644576
x8=437.20497762458x_{8} = 437.20497762458
x39=66400.1787274983x_{39} = 66400.1787274983
son puntos de cambio del signo de desigualdad en las soluciones.
Primero definámonos con el signo hasta el punto extremo izquierdo:
x0≤x53x_{0} \leq x_{53}
Consideremos, por ejemplo, el punto
x0=x53−110x_{0} = x_{53} - \frac{1}{10}
=
−195.302343298165+−110-195.302343298165 + - \frac{1}{10}
=
−195.402343298165-195.402343298165
lo sustituimos en la expresión
sin⁡(t)≥−12\sin{\left(t \right)} \geq - \frac{1}{2}
sin⁡(t)≥−12\sin{\left(t \right)} \geq - \frac{1}{2}
sin(t) >= -1/2

Entonces
x≤−195.302343298165x \leq -195.302343298165
no se cumple
significa que una de las soluciones de nuestra ecuación será con:
x≥−195.302343298165∧x≤−151.320046147908x \geq -195.302343298165 \wedge x \leq -151.320046147908
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       x53      x15      x66      x47      x43      x63      x26      x48      x46      x44      x42      x55      x37      x23      x31      x17      x11      x22      x7      x24      x10      x56      x16      x14      x29      x4      x64      x9      x6      x67      x19      x57      x21      x52      x50      x20      x62      x49      x51      x41      x13      x65      x27      x35      x69      x59      x38      x33      x12      x1      x54      x40      x32      x5      x68      x58      x45      x18      x2      x30      x3      x60      x25      x34      x36      x70      x28      x61      x8      x39

Recibiremos otras soluciones de la desigualdad pasando al polo siguiente etc.
etc.
Respuesta:
x≥−195.302343298165∧x≤−151.320046147908x \geq -195.302343298165 \wedge x \leq -151.320046147908
x≥−101.054563690472∧x≤−96.8657734856853x \geq -101.054563690472 \wedge x \leq -96.8657734856853
x≥−94.7713783832921∧x≤−90.5825881785057x \geq -94.7713783832921 \wedge x \leq -90.5825881785057
x≥−88.4881930761125∧x≤−84.2994028713261x \geq -88.4881930761125 \wedge x \leq -84.2994028713261
x≥−82.2050077689329∧x≤−78.0162175641465x \geq -82.2050077689329 \wedge x \leq -78.0162175641465
x≥−75.9218224617533∧x≤−71.733032256967x \geq -75.9218224617533 \wedge x \leq -71.733032256967
x≥−69.6386371545737∧x≤−65.4498469497874x \geq -69.6386371545737 \wedge x \leq -65.4498469497874
x≥−63.3554518473942∧x≤−59.1666616426078x \geq -63.3554518473942 \wedge x \leq -59.1666616426078
x≥−57.0722665402146∧x≤−52.8834763354282x \geq -57.0722665402146 \wedge x \leq -52.8834763354282
x≥−50.789081233035∧x≤−46.6002910282486x \geq -50.789081233035 \wedge x \leq -46.6002910282486
x≥−44.5058959258554∧x≤−40.317105721069x \geq -44.5058959258554 \wedge x \leq -40.317105721069
x≥−38.2227106186758∧x≤−34.0339204138894x \geq -38.2227106186758 \wedge x \leq -34.0339204138894
x≥−31.9395253114962∧x≤−27.7507351067098x \geq -31.9395253114962 \wedge x \leq -27.7507351067098
x≥−25.6563400043166∧x≤−21.4675497995303x \geq -25.6563400043166 \wedge x \leq -21.4675497995303
x≥−19.3731546971371∧x≤−15.1843644923507x \geq -19.3731546971371 \wedge x \leq -15.1843644923507
x≥−13.0899693899575∧x≤−8.90117918517108x \geq -13.0899693899575 \wedge x \leq -8.90117918517108
x≥−6.80678408277789∧x≤−2.61799387799149x \geq -6.80678408277789 \wedge x \leq -2.61799387799149
x≥−0.523598775598299∧x≤3.66519142918809x \geq -0.523598775598299 \wedge x \leq 3.66519142918809
x≥5.75958653158129∧x≤9.94837673636768x \geq 5.75958653158129 \wedge x \leq 9.94837673636768
x≥12.0427718387609∧x≤16.2315620435473x \geq 12.0427718387609 \wedge x \leq 16.2315620435473
x≥18.3259571459405∧x≤22.5147473507269x \geq 18.3259571459405 \wedge x \leq 22.5147473507269
x≥24.60914245312∧x≤28.7979326579064x \geq 24.60914245312 \wedge x \leq 28.7979326579064
x≥30.8923277602996∧x≤35.081117965086x \geq 30.8923277602996 \wedge x \leq 35.081117965086
x≥37.1755130674792∧x≤41.3643032722656x \geq 37.1755130674792 \wedge x \leq 41.3643032722656
x≥43.4586983746588∧x≤47.6474885794452x \geq 43.4586983746588 \wedge x \leq 47.6474885794452
x≥49.7418836818384∧x≤53.9306738866248x \geq 49.7418836818384 \wedge x \leq 53.9306738866248
x≥56.025068989018∧x≤60.2138591938044x \geq 56.025068989018 \wedge x \leq 60.2138591938044
x≥62.3082542961976∧x≤66.497044500984x \geq 62.3082542961976 \wedge x \leq 66.497044500984
x≥68.5914396033772∧x≤72.7802298081635x \geq 68.5914396033772 \wedge x \leq 72.7802298081635
x≥74.8746249105567∧x≤79.0634151153431x \geq 74.8746249105567 \wedge x \leq 79.0634151153431
x≥81.1578102177363∧x≤85.3466004225227x \geq 81.1578102177363 \wedge x \leq 85.3466004225227
x≥87.4409955249159∧x≤91.6297857297023x \geq 87.4409955249159 \wedge x \leq 91.6297857297023
x≥93.7241808320955∧x≤97.9129710368819x \geq 93.7241808320955 \wedge x \leq 97.9129710368819
x≥100.007366139275∧x≤192.160750644576x \geq 100.007366139275 \wedge x \leq 192.160750644576
x≥437.20497762458∧x≤66400.1787274983x \geq 437.20497762458 \wedge x \leq 66400.1787274983
Respuesta rápida 2 [src] 
    7*pi     11*pi       
[0, ----] U [-----, 2*pi]
     6         6
x in [0,7π6]∪[11π6,2π]x\ in\ \left[0, \frac{7 \pi}{6}\right] \cup \left[\frac{11 \pi}{6}, 2 \pi\right] 
x in Union(Interval(0, 7*pi/6), Interval(11*pi/6, 2*pi))
Respuesta rápida [src] 
  /   /             7*pi\     /11*pi                \\
Or|And|0 <= t, t <= ----|, And|----- <= t, t <= 2*pi||
  \   \              6  /     \  6                  //
(0≤t∧t≤7π6)∨(11π6≤t∧t≤2π)\left(0 \leq t \wedge t \leq \frac{7 \pi}{6}\right) \vee \left(\frac{11 \pi}{6} \leq t \wedge t \leq 2 \pi\right) 
((0 <= t)∧(t <= 7*pi/6))∨((11*pi/6 <= t)∧(t <= 2*pi))
    © Sr Examen – Калькулятор