Se da la desigualdad:
sin(t)<0Para resolver esta desigualdad primero hay que resolver la ecuación correspondiente:
sin(t)=0Resolvemos:
Tenemos la ecuación
sin(t)=0cambiamos
sin(t)−1=0sin(t)−1=0Sustituimos
w=sin(t)Transportamos los términos libres (sin w)
del miembro izquierdo al derecho, obtenemos:
w=1Obtenemos la respuesta: w = 1
hacemos cambio inverso
sin(t)=wsustituimos w:
x1=−59.6902604182061x2=−62.8318530717959x3=−97.3893722612836x4=87.9645943005142x5=−56.5486677646163x6=31.4159265358979x7=69.1150383789755x8=−37.6991118430775x9=−81.6814089933346x10=−84.8230016469244x11=−21.9911485751286x12=47.1238898038469x13=−113.097335529233x14=−12.5663706143592x15=−15.707963267949x16=12.5663706143592x17=−87.9645943005142x18=53.4070751110265x19=−267.035375555132x20=−100.530964914873x21=−3.14159265358979x22=72.2566310325652x23=−2642.07942166902x24=34.5575191894877x25=−94.2477796076938x26=6.28318530717959x27=−69.1150383789755x28=97.3893722612836x29=0x30=65.9734457253857x31=−50.2654824574367x32=15.707963267949x33=3.14159265358979x34=−25.1327412287183x35=−18.8495559215388x36=40.8407044966673x37=−53.4070751110265x38=37.6991118430775x39=−43.9822971502571x40=18.8495559215388x41=−78.5398163397448x42=−6.28318530717959x43=−232.477856365645x44=−40.8407044966673x45=43.9822971502571x46=56.5486677646163x47=−65.9734457253857x48=−28.2743338823081x49=78.5398163397448x50=25.1327412287183x51=75.398223686155x52=59.6902604182061x53=−34.5575191894877x54=81.6814089933346x55=−47.1238898038469x56=100.530964914873x57=−9.42477796076938x58=−75.398223686155x59=−72.2566310325652x60=−31.4159265358979x61=28.2743338823081x62=−91.106186954104x63=21.9911485751286x64=62.8318530717959x65=9.42477796076938x66=50.2654824574367x67=94.2477796076938x68=91.106186954104x69=84.8230016469244x1=−59.6902604182061x2=−62.8318530717959x3=−97.3893722612836x4=87.9645943005142x5=−56.5486677646163x6=31.4159265358979x7=69.1150383789755x8=−37.6991118430775x9=−81.6814089933346x10=−84.8230016469244x11=−21.9911485751286x12=47.1238898038469x13=−113.097335529233x14=−12.5663706143592x15=−15.707963267949x16=12.5663706143592x17=−87.9645943005142x18=53.4070751110265x19=−267.035375555132x20=−100.530964914873x21=−3.14159265358979x22=72.2566310325652x23=−2642.07942166902x24=34.5575191894877x25=−94.2477796076938x26=6.28318530717959x27=−69.1150383789755x28=97.3893722612836x29=0x30=65.9734457253857x31=−50.2654824574367x32=15.707963267949x33=3.14159265358979x34=−25.1327412287183x35=−18.8495559215388x36=40.8407044966673x37=−53.4070751110265x38=37.6991118430775x39=−43.9822971502571x40=18.8495559215388x41=−78.5398163397448x42=−6.28318530717959x43=−232.477856365645x44=−40.8407044966673x45=43.9822971502571x46=56.5486677646163x47=−65.9734457253857x48=−28.2743338823081x49=78.5398163397448x50=25.1327412287183x51=75.398223686155x52=59.6902604182061x53=−34.5575191894877x54=81.6814089933346x55=−47.1238898038469x56=100.530964914873x57=−9.42477796076938x58=−75.398223686155x59=−72.2566310325652x60=−31.4159265358979x61=28.2743338823081x62=−91.106186954104x63=21.9911485751286x64=62.8318530717959x65=9.42477796076938x66=50.2654824574367x67=94.2477796076938x68=91.106186954104x69=84.8230016469244Las raíces dadas
x23=−2642.07942166902x19=−267.035375555132x43=−232.477856365645x13=−113.097335529233x20=−100.530964914873x3=−97.3893722612836x25=−94.2477796076938x62=−91.106186954104x17=−87.9645943005142x10=−84.8230016469244x9=−81.6814089933346x41=−78.5398163397448x58=−75.398223686155x59=−72.2566310325652x27=−69.1150383789755x47=−65.9734457253857x2=−62.8318530717959x1=−59.6902604182061x5=−56.5486677646163x37=−53.4070751110265x31=−50.2654824574367x55=−47.1238898038469x39=−43.9822971502571x44=−40.8407044966673x8=−37.6991118430775x53=−34.5575191894877x60=−31.4159265358979x48=−28.2743338823081x34=−25.1327412287183x11=−21.9911485751286x35=−18.8495559215388x15=−15.707963267949x14=−12.5663706143592x57=−9.42477796076938x42=−6.28318530717959x21=−3.14159265358979x29=0x33=3.14159265358979x26=6.28318530717959x65=9.42477796076938x16=12.5663706143592x32=15.707963267949x40=18.8495559215388x63=21.9911485751286x50=25.1327412287183x61=28.2743338823081x6=31.4159265358979x24=34.5575191894877x38=37.6991118430775x36=40.8407044966673x45=43.9822971502571x12=47.1238898038469x66=50.2654824574367x18=53.4070751110265x46=56.5486677646163x52=59.6902604182061x64=62.8318530717959x30=65.9734457253857x7=69.1150383789755x22=72.2566310325652x51=75.398223686155x49=78.5398163397448x54=81.6814089933346x69=84.8230016469244x4=87.9645943005142x68=91.106186954104x67=94.2477796076938x28=97.3893722612836x56=100.530964914873son puntos de cambio del signo de desigualdad en las soluciones.
Primero definámonos con el signo hasta el punto extremo izquierdo:
x0<x23Consideremos, por ejemplo, el punto
x0=x23−101=
−2642.07942166902+−101=
−2642.17942166902lo sustituimos en la expresión
sin(t)<0sin(t)<0sin(t) < 0
Entonces
x<−2642.07942166902no se cumple
significa que una de las soluciones de nuestra ecuación será con:
x>−2642.07942166902∧x<−267.035375555132 _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____
/ \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------
x23 x19 x43 x13 x20 x3 x25 x62 x17 x10 x9 x41 x58 x59 x27 x47 x2 x1 x5 x37 x31 x55 x39 x44 x8 x53 x60 x48 x34 x11 x35 x15 x14 x57 x42 x21 x29 x33 x26 x65 x16 x32 x40 x63 x50 x61 x6 x24 x38 x36 x45 x12 x66 x18 x46 x52 x64 x30 x7 x22 x51 x49 x54 x69 x4 x68 x67 x28 x56Recibiremos otras soluciones de la desigualdad pasando al polo siguiente etc.
etc.
Respuesta:
x>−2642.07942166902∧x<−267.035375555132x>−232.477856365645∧x<−113.097335529233x>−100.530964914873∧x<−97.3893722612836x>−94.2477796076938∧x<−91.106186954104x>−87.9645943005142∧x<−84.8230016469244x>−81.6814089933346∧x<−78.5398163397448x>−75.398223686155∧x<−72.2566310325652x>−69.1150383789755∧x<−65.9734457253857x>−62.8318530717959∧x<−59.6902604182061x>−56.5486677646163∧x<−53.4070751110265x>−50.2654824574367∧x<−47.1238898038469x>−43.9822971502571∧x<−40.8407044966673x>−37.6991118430775∧x<−34.5575191894877x>−31.4159265358979∧x<−28.2743338823081x>−25.1327412287183∧x<−21.9911485751286x>−18.8495559215388∧x<−15.707963267949x>−12.5663706143592∧x<−9.42477796076938x>−6.28318530717959∧x<−3.14159265358979x>0∧x<3.14159265358979x>6.28318530717959∧x<9.42477796076938x>12.5663706143592∧x<15.707963267949x>18.8495559215388∧x<21.9911485751286x>25.1327412287183∧x<28.2743338823081x>31.4159265358979∧x<34.5575191894877x>37.6991118430775∧x<40.8407044966673x>43.9822971502571∧x<47.1238898038469x>50.2654824574367∧x<53.4070751110265x>56.5486677646163∧x<59.6902604182061x>62.8318530717959∧x<65.9734457253857x>69.1150383789755∧x<72.2566310325652x>75.398223686155∧x<78.5398163397448x>81.6814089933346∧x<84.8230016469244x>87.9645943005142∧x<91.106186954104x>94.2477796076938∧x<97.3893722612836x>100.530964914873
(x-2)/(x-4)>0
-x^2-10x-24>0
(x+8)(x-5)>0
sin(t)