Given the inequality:
cos(t)≤1To solve this inequality, we must first solve the corresponding equation:
cos(t)=1Solve:
Given the equation
cos(t)=1transform
cos(t)−1=0cos(t)−1=0Do replacement
w=cos(t)Move free summands (without w)
from left part to right part, we given:
w=1We get the answer: w = 1
do backward replacement
cos(t)=wsubstitute w:
x1=62.8318535568358x2=37.6991115173992x3=−31.4159260507536x4=−37.6991113479743x5=−50.2654829667315x6=−18.8495563230046x7=94.2477800892631x8=100.530965157364x9=31.4159268459961x10=−94.2477801171671x11=12.5663710110881x12=−43.9822976246252x13=−94.2477797298079x14=−87.964593928489x15=−62.8318534787248x16=81.6814084860076x17=6.28318579821791x18=100.530964759815x19=−75.3982238741744x20=56.5486668532011x21=87.964594335905x22=−56.5486682426592x23=−12.5663710889626x24=−43.9822971745392x25=81.6814085860518x26=−56.5486674685864x27=−69.1150379045123x28=−62.831852673202x29=−25.1327407505866x30=56.5486682809363x31=50.2654824463392x32=−31.4159267157965x33=56.5486680806249x34=6.28318626747926x35=37.6991120311338x36=−18.8495552124105x37=69.115038794053x38=−87.9645947692094x39=0x40=87.9645946044253x41=−50.265482641087x42=25.1327416384075x43=−12.5663703112531x44=−6.2831858160515x45=−75.3982231720141x46=−69.1150386869085x47=−25.1327415297174x48=37.6991113348642x49=69.1150379887504x50=12.5663704426592x51=−6.2831851275477x52=87.9645938121814x53=−37.6991121287155x54=−81.6814075578313x55=3.40772025642919⋅10−7x56=−100.530964626003x57=−81.6814092565354x58=−18.8495555173448x59=−37.6991118772631x60=75.3982232188727x61=−43.9822967932182x62=−31.4159260208155x63=−94.2477794452815x64=6.28318528420851x65=94.2477792651059x66=−81.6814090382277x67=94.2477796093523x68=−6.28318555849548x69=50.2654821322586x70=25.1327408328211x71=−50.2654822863493x72=62.8318527849002x73=43.9822974733639x74=81.6814091897036x75=6.28318500093652x76=−81.6814084945807x77=12.5663711301703x78=43.9822971694647x79=75.3982240031607x80=−75.3982231045728x81=43.9822966661001x82=31.4159260648825x83=18.8495556275525x84=18.8495564031971x85=50.2654829439723x86=56.5486676011951x87=−4.79511606158654⋅10−7x88=−87.9645943586158x1=62.8318535568358x2=37.6991115173992x3=−31.4159260507536x4=−37.6991113479743x5=−50.2654829667315x6=−18.8495563230046x7=94.2477800892631x8=100.530965157364x9=31.4159268459961x10=−94.2477801171671x11=12.5663710110881x12=−43.9822976246252x13=−94.2477797298079x14=−87.964593928489x15=−62.8318534787248x16=81.6814084860076x17=6.28318579821791x18=100.530964759815x19=−75.3982238741744x20=56.5486668532011x21=87.964594335905x22=−56.5486682426592x23=−12.5663710889626x24=−43.9822971745392x25=81.6814085860518x26=−56.5486674685864x27=−69.1150379045123x28=−62.831852673202x29=−25.1327407505866x30=56.5486682809363x31=50.2654824463392x32=−31.4159267157965x33=56.5486680806249x34=6.28318626747926x35=37.6991120311338x36=−18.8495552124105x37=69.115038794053x38=−87.9645947692094x39=0x40=87.9645946044253x41=−50.265482641087x42=25.1327416384075x43=−12.5663703112531x44=−6.2831858160515x45=−75.3982231720141x46=−69.1150386869085x47=−25.1327415297174x48=37.6991113348642x49=69.1150379887504x50=12.5663704426592x51=−6.2831851275477x52=87.9645938121814x53=−37.6991121287155x54=−81.6814075578313x55=3.40772025642919⋅10−7x56=−100.530964626003x57=−81.6814092565354x58=−18.8495555173448x59=−37.6991118772631x60=75.3982232188727x61=−43.9822967932182x62=−31.4159260208155x63=−94.2477794452815x64=6.28318528420851x65=94.2477792651059x66=−81.6814090382277x67=94.2477796093523x68=−6.28318555849548x69=50.2654821322586x70=25.1327408328211x71=−50.2654822863493x72=62.8318527849002x73=43.9822974733639x74=81.6814091897036x75=6.28318500093652x76=−81.6814084945807x77=12.5663711301703x78=43.9822971694647x79=75.3982240031607x80=−75.3982231045728x81=43.9822966661001x82=31.4159260648825x83=18.8495556275525x84=18.8495564031971x85=50.2654829439723x86=56.5486676011951x87=−4.79511606158654⋅10−7x88=−87.9645943586158This roots
x56=−100.530964626003x10=−94.2477801171671x13=−94.2477797298079x63=−94.2477794452815x38=−87.9645947692094x88=−87.9645943586158x14=−87.964593928489x57=−81.6814092565354x66=−81.6814090382277x76=−81.6814084945807x54=−81.6814075578313x19=−75.3982238741744x45=−75.3982231720141x80=−75.3982231045728x46=−69.1150386869085x27=−69.1150379045123x15=−62.8318534787248x28=−62.831852673202x22=−56.5486682426592x26=−56.5486674685864x5=−50.2654829667315x41=−50.265482641087x71=−50.2654822863493x12=−43.9822976246252x24=−43.9822971745392x61=−43.9822967932182x53=−37.6991121287155x59=−37.6991118772631x4=−37.6991113479743x32=−31.4159267157965x3=−31.4159260507536x62=−31.4159260208155x47=−25.1327415297174x29=−25.1327407505866x6=−18.8495563230046x58=−18.8495555173448x36=−18.8495552124105x23=−12.5663710889626x43=−12.5663703112531x44=−6.2831858160515x68=−6.28318555849548x51=−6.2831851275477x87=−4.79511606158654⋅10−7x39=0x55=3.40772025642919⋅10−7x75=6.28318500093652x64=6.28318528420851x17=6.28318579821791x34=6.28318626747926x50=12.5663704426592x11=12.5663710110881x77=12.5663711301703x83=18.8495556275525x84=18.8495564031971x70=25.1327408328211x42=25.1327416384075x82=31.4159260648825x9=31.4159268459961x48=37.6991113348642x2=37.6991115173992x35=37.6991120311338x81=43.9822966661001x78=43.9822971694647x73=43.9822974733639x69=50.2654821322586x31=50.2654824463392x85=50.2654829439723x20=56.5486668532011x86=56.5486676011951x33=56.5486680806249x30=56.5486682809363x72=62.8318527849002x1=62.8318535568358x49=69.1150379887504x37=69.115038794053x60=75.3982232188727x79=75.3982240031607x16=81.6814084860076x25=81.6814085860518x74=81.6814091897036x52=87.9645938121814x21=87.964594335905x40=87.9645946044253x65=94.2477792651059x67=94.2477796093523x7=94.2477800892631x18=100.530964759815x8=100.530965157364is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
x0≤x56For example, let's take the point
x0=x56−101=
−100.530964626003+−101=
−100.630964626003substitute to the expression
cos(t)≤1cos(t)≤1cos(t) <= 1
Then
x≤−100.530964626003no execute
one of the solutions of our inequality is:
x≥−100.530964626003∧x≤−94.2477801171671 _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____
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x56 x10 x13 x63 x38 x88 x14 x57 x66 x76 x54 x19 x45 x80 x46 x27 x15 x28 x22 x26 x5 x41 x71 x12 x24 x61 x53 x59 x4 x32 x3 x62 x47 x29 x6 x58 x36 x23 x43 x44 x68 x51 x87 x39 x55 x75 x64 x17 x34 x50 x11 x77 x83 x84 x70 x42 x82 x9 x48 x2 x35 x81 x78 x73 x69 x31 x85 x20 x86 x33 x30 x72 x1 x49 x37 x60 x79 x16 x25 x74 x52 x21 x40 x65 x67 x7 x18 x8Other solutions will get with the changeover to the next point
etc.
The answer:
x≥−100.530964626003∧x≤−94.2477801171671x≥−94.2477797298079∧x≤−94.2477794452815x≥−87.9645947692094∧x≤−87.9645943586158x≥−87.964593928489∧x≤−81.6814092565354x≥−81.6814090382277∧x≤−81.6814084945807x≥−81.6814075578313∧x≤−75.3982238741744x≥−75.3982231720141∧x≤−75.3982231045728x≥−69.1150386869085∧x≤−69.1150379045123x≥−62.8318534787248∧x≤−62.831852673202x≥−56.5486682426592∧x≤−56.5486674685864x≥−50.2654829667315∧x≤−50.265482641087x≥−50.2654822863493∧x≤−43.9822976246252x≥−43.9822971745392∧x≤−43.9822967932182x≥−37.6991121287155∧x≤−37.6991118772631x≥−37.6991113479743∧x≤−31.4159267157965x≥−31.4159260507536∧x≤−31.4159260208155x≥−25.1327415297174∧x≤−25.1327407505866x≥−18.8495563230046∧x≤−18.8495555173448x≥−18.8495552124105∧x≤−12.5663710889626x≥−12.5663703112531∧x≤−6.2831858160515x≥−6.28318555849548∧x≤−6.2831851275477x≥−4.79511606158654⋅10−7∧x≤0x≥3.40772025642919⋅10−7∧x≤6.28318500093652x≥6.28318528420851∧x≤6.28318579821791x≥6.28318626747926∧x≤12.5663704426592x≥12.5663710110881∧x≤12.5663711301703x≥18.8495556275525∧x≤18.8495564031971x≥25.1327408328211∧x≤25.1327416384075x≥31.4159260648825∧x≤31.4159268459961x≥37.6991113348642∧x≤37.6991115173992x≥37.6991120311338∧x≤43.9822966661001x≥43.9822971694647∧x≤43.9822974733639x≥50.2654821322586∧x≤50.2654824463392x≥50.2654829439723∧x≤56.5486668532011x≥56.5486676011951∧x≤56.5486680806249x≥56.5486682809363∧x≤62.8318527849002x≥62.8318535568358∧x≤69.1150379887504x≥69.115038794053∧x≤75.3982232188727x≥75.3982240031607∧x≤81.6814084860076x≥81.6814085860518∧x≤81.6814091897036x≥87.9645938121814∧x≤87.964594335905x≥87.9645946044253∧x≤94.2477792651059x≥94.2477796093523∧x≤94.2477800892631x≥100.530964759815∧x≤100.530965157364
cost
cost