Given the inequality:
sin(t)≤21To solve this inequality, we must first solve the corresponding equation:
sin(t)=21Solve:
Given the equation
sin(t)=21transform
sin(t)−21=0sin(t)−21=0Do replacement
w=sin(t)Move free summands (without w)
from left part to right part, we given:
w=21We get the answer: w = 1/2
do backward replacement
sin(t)=wsubstitute w:
x1=−74.8746249105567x2=−37.1755130674792x3=75.9218224617533x4=78.0162175641465x5=−60.2138591938044x6=88.4881930761125x7=−41.3643032722656x8=52.8834763354282x9=50.789081233035x10=2.61799387799149x11=−62.3082542961976x12=31.9395253114962x13=57.0722665402146x14=−66.497044500984x15=17438.4572213013x16=−24.60914245312x17=−49.7418836818384x18=8.90117918517108x19=−12.0427718387609x20=−16.2315620435473x21=−18.3259571459405x22=−87.4409955249159x23=25.6563400043166x24=−2650.98060085419x25=134.564885328763x26=−28.7979326579064x27=34.0339204138894x28=46.6002910282486x29=94.7713783832921x30=0.523598775598299x31=−93.7241808320955x32=138.753675533549x33=40.317105721069x34=−68.5914396033772x35=−53.9306738866248x36=−47.6474885794452x37=−627.79493194236x38=90.5825881785057x39=−56.025068989018x40=84.2994028713261x41=101.054563690472x42=−100.007366139275x43=−85.3466004225227x44=13.0899693899575x45=−81.1578102177363x46=27.7507351067098x47=−91.6297857297023x48=−4454.25478401473x49=21.4675497995303x50=−3.66519142918809x51=−30.8923277602996x52=82.2050077689329x53=65.4498469497874x54=69.6386371545737x55=−97.9129710368819x56=−72.7802298081635x57=15.1843644923507x58=19.3731546971371x59=38.2227106186758x60=−35.081117965086x61=44.5058959258554x62=−9.94837673636768x63=−79.0634151153431x64=−43.4586983746588x65=96.8657734856853x66=−5.75958653158129x67=59.1666616426078x68=71.733032256967x69=−22.5147473507269x70=6.80678408277789x71=63.3554518473942x1=−74.8746249105567x2=−37.1755130674792x3=75.9218224617533x4=78.0162175641465x5=−60.2138591938044x6=88.4881930761125x7=−41.3643032722656x8=52.8834763354282x9=50.789081233035x10=2.61799387799149x11=−62.3082542961976x12=31.9395253114962x13=57.0722665402146x14=−66.497044500984x15=17438.4572213013x16=−24.60914245312x17=−49.7418836818384x18=8.90117918517108x19=−12.0427718387609x20=−16.2315620435473x21=−18.3259571459405x22=−87.4409955249159x23=25.6563400043166x24=−2650.98060085419x25=134.564885328763x26=−28.7979326579064x27=34.0339204138894x28=46.6002910282486x29=94.7713783832921x30=0.523598775598299x31=−93.7241808320955x32=138.753675533549x33=40.317105721069x34=−68.5914396033772x35=−53.9306738866248x36=−47.6474885794452x37=−627.79493194236x38=90.5825881785057x39=−56.025068989018x40=84.2994028713261x41=101.054563690472x42=−100.007366139275x43=−85.3466004225227x44=13.0899693899575x45=−81.1578102177363x46=27.7507351067098x47=−91.6297857297023x48=−4454.25478401473x49=21.4675497995303x50=−3.66519142918809x51=−30.8923277602996x52=82.2050077689329x53=65.4498469497874x54=69.6386371545737x55=−97.9129710368819x56=−72.7802298081635x57=15.1843644923507x58=19.3731546971371x59=38.2227106186758x60=−35.081117965086x61=44.5058959258554x62=−9.94837673636768x63=−79.0634151153431x64=−43.4586983746588x65=96.8657734856853x66=−5.75958653158129x67=59.1666616426078x68=71.733032256967x69=−22.5147473507269x70=6.80678408277789x71=63.3554518473942This roots
x48=−4454.25478401473x24=−2650.98060085419x37=−627.79493194236x42=−100.007366139275x55=−97.9129710368819x31=−93.7241808320955x47=−91.6297857297023x22=−87.4409955249159x43=−85.3466004225227x45=−81.1578102177363x63=−79.0634151153431x1=−74.8746249105567x56=−72.7802298081635x34=−68.5914396033772x14=−66.497044500984x11=−62.3082542961976x5=−60.2138591938044x39=−56.025068989018x35=−53.9306738866248x17=−49.7418836818384x36=−47.6474885794452x64=−43.4586983746588x7=−41.3643032722656x2=−37.1755130674792x60=−35.081117965086x51=−30.8923277602996x26=−28.7979326579064x16=−24.60914245312x69=−22.5147473507269x21=−18.3259571459405x20=−16.2315620435473x19=−12.0427718387609x62=−9.94837673636768x66=−5.75958653158129x50=−3.66519142918809x30=0.523598775598299x10=2.61799387799149x70=6.80678408277789x18=8.90117918517108x44=13.0899693899575x57=15.1843644923507x58=19.3731546971371x49=21.4675497995303x23=25.6563400043166x46=27.7507351067098x12=31.9395253114962x27=34.0339204138894x59=38.2227106186758x33=40.317105721069x61=44.5058959258554x28=46.6002910282486x9=50.789081233035x8=52.8834763354282x13=57.0722665402146x67=59.1666616426078x71=63.3554518473942x53=65.4498469497874x54=69.6386371545737x68=71.733032256967x3=75.9218224617533x4=78.0162175641465x52=82.2050077689329x40=84.2994028713261x6=88.4881930761125x38=90.5825881785057x29=94.7713783832921x65=96.8657734856853x41=101.054563690472x25=134.564885328763x32=138.753675533549x15=17438.4572213013is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
x0≤x48For example, let's take the point
x0=x48−101=
−4454.25478401473−101=
−4454.35478401473substitute to the expression
sin(t)≤21sin(t)≤21sin(t) <= 1/2
Then
x≤−4454.25478401473no execute
one of the solutions of our inequality is:
x≥−4454.25478401473∧x≤−2650.98060085419 _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____
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x_48 x_24 x_37 x_42 x_55 x_31 x_47 x_22 x_43 x_45 x_63 x_1 x_56 x_34 x_14 x_11 x_5 x_39 x_35 x_17 x_36 x_64 x_7 x_2 x_60 x_51 x_26 x_16 x_69 x_21 x_20 x_19 x_62 x_66 x_50 x_30 x_10 x_70 x_18 x_44 x_57 x_58 x_49 x_23 x_46 x_12 x_27 x_59 x_33 x_61 x_28 x_9 x_8 x_13 x_67 x_71 x_53 x_54 x_68 x_3 x_4 x_52 x_40 x_6 x_38 x_29 x_65 x_41 x_25 x_32 x_15Other solutions will get with the changeover to the next point
etc.
The answer:
x≥−4454.25478401473∧x≤−2650.98060085419x≥−627.79493194236∧x≤−100.007366139275x≥−97.9129710368819∧x≤−93.7241808320955x≥−91.6297857297023∧x≤−87.4409955249159x≥−85.3466004225227∧x≤−81.1578102177363x≥−79.0634151153431∧x≤−74.8746249105567x≥−72.7802298081635∧x≤−68.5914396033772x≥−66.497044500984∧x≤−62.3082542961976x≥−60.2138591938044∧x≤−56.025068989018x≥−53.9306738866248∧x≤−49.7418836818384x≥−47.6474885794452∧x≤−43.4586983746588x≥−41.3643032722656∧x≤−37.1755130674792x≥−35.081117965086∧x≤−30.8923277602996x≥−28.7979326579064∧x≤−24.60914245312x≥−22.5147473507269∧x≤−18.3259571459405x≥−16.2315620435473∧x≤−12.0427718387609x≥−9.94837673636768∧x≤−5.75958653158129x≥−3.66519142918809∧x≤0.523598775598299x≥2.61799387799149∧x≤6.80678408277789x≥8.90117918517108∧x≤13.0899693899575x≥15.1843644923507∧x≤19.3731546971371x≥21.4675497995303∧x≤25.6563400043166x≥27.7507351067098∧x≤31.9395253114962x≥34.0339204138894∧x≤38.2227106186758x≥40.317105721069∧x≤44.5058959258554x≥46.6002910282486∧x≤50.789081233035x≥52.8834763354282∧x≤57.0722665402146x≥59.1666616426078∧x≤63.3554518473942x≥65.4498469497874∧x≤69.6386371545737x≥71.733032256967∧x≤75.9218224617533x≥78.0162175641465∧x≤82.2050077689329x≥84.2994028713261∧x≤88.4881930761125x≥90.5825881785057∧x≤94.7713783832921x≥96.8657734856853∧x≤101.054563690472x≥134.564885328763∧x≤138.753675533549x≥17438.4572213013
