Given the inequality:
sin(t)<31To solve this inequality, we must first solve the corresponding equation:
sin(t)=31Solve:
Given the equation
sin(t)=31transform
sin(t)−31=0sin(t)−31=0Do replacement
w=sin(t)Move free summands (without w)
from left part to right part, we given:
w=31We get the answer: w = 1/3
do backward replacement
sin(t)=wsubstitute w:
x1=94.5876165171479x2=21.6513116656744x3=44.3221340597112x4=−56.2088308551622x5=−5.94334839772546x6=172.447759037984x7=78.1999794302907x8=100.870801824328x9=−43.642460240803x10=69.4548752884296x11=−24.7929043192642x12=63.17168998125x13=19.1893928309929x14=53.0672382015724x15=84.4831647374703x16=−18.5097190120846x17=−53.7469120204806x18=−9.7646148702235x19=90.7663500446499x20=−112585.595763607x21=75.7380605956092x22=65.6336088159315x23=27.934496972854x24=−34.8973560989418x25=−12.2265337049051x26=−60.0300973276602x27=46.7840528943928x28=59.350423508752x29=40.5008675872132x30=−41.1805414061214x31=−87.6247573910601x32=−47.463726713301x33=−72.5964679420194x34=−68.7752014695213x35=−100.191128005419x36=−116.578765092276x37=−49.9256455479826x38=−78.879653249199x39=97.0495353518295x40=6.62302221663371x41=9.08494105131526x42=38.0389487525316x43=−66.3132826348398x44=−37.3592749336234x45=34.2176822800336x46=12.9062075238133x47=−3.48142956304392x48=−75.0583867767009x49=−85.1628385563785x50=−22.3309854845827x51=2.80175574413567x52=−97.7292091707377x53=951.562737128253x54=−81.3415720838805x55=25.4725781381725x56=−28.6141707917623x57=−31.0760896264438x58=−93.9079426982397x59=50.6053193668908x60=15.3681263584948x61=−91.4460238635581x62=0.339836909454122x63=71.9167941231111x64=88.3044312099683x65=31.7557634453521x66=82.0212459027887x67=56.8885046740704x68=−16.0478001774031x69=−62.4920161623417x1=94.5876165171479x2=21.6513116656744x3=44.3221340597112x4=−56.2088308551622x5=−5.94334839772546x6=172.447759037984x7=78.1999794302907x8=100.870801824328x9=−43.642460240803x10=69.4548752884296x11=−24.7929043192642x12=63.17168998125x13=19.1893928309929x14=53.0672382015724x15=84.4831647374703x16=−18.5097190120846x17=−53.7469120204806x18=−9.7646148702235x19=90.7663500446499x20=−112585.595763607x21=75.7380605956092x22=65.6336088159315x23=27.934496972854x24=−34.8973560989418x25=−12.2265337049051x26=−60.0300973276602x27=46.7840528943928x28=59.350423508752x29=40.5008675872132x30=−41.1805414061214x31=−87.6247573910601x32=−47.463726713301x33=−72.5964679420194x34=−68.7752014695213x35=−100.191128005419x36=−116.578765092276x37=−49.9256455479826x38=−78.879653249199x39=97.0495353518295x40=6.62302221663371x41=9.08494105131526x42=38.0389487525316x43=−66.3132826348398x44=−37.3592749336234x45=34.2176822800336x46=12.9062075238133x47=−3.48142956304392x48=−75.0583867767009x49=−85.1628385563785x50=−22.3309854845827x51=2.80175574413567x52=−97.7292091707377x53=951.562737128253x54=−81.3415720838805x55=25.4725781381725x56=−28.6141707917623x57=−31.0760896264438x58=−93.9079426982397x59=50.6053193668908x60=15.3681263584948x61=−91.4460238635581x62=0.339836909454122x63=71.9167941231111x64=88.3044312099683x65=31.7557634453521x66=82.0212459027887x67=56.8885046740704x68=−16.0478001774031x69=−62.4920161623417This roots
x20=−112585.595763607x36=−116.578765092276x35=−100.191128005419x52=−97.7292091707377x58=−93.9079426982397x61=−91.4460238635581x31=−87.6247573910601x49=−85.1628385563785x54=−81.3415720838805x38=−78.879653249199x48=−75.0583867767009x33=−72.5964679420194x34=−68.7752014695213x43=−66.3132826348398x69=−62.4920161623417x26=−60.0300973276602x4=−56.2088308551622x17=−53.7469120204806x37=−49.9256455479826x32=−47.463726713301x9=−43.642460240803x30=−41.1805414061214x44=−37.3592749336234x24=−34.8973560989418x57=−31.0760896264438x56=−28.6141707917623x11=−24.7929043192642x50=−22.3309854845827x16=−18.5097190120846x68=−16.0478001774031x25=−12.2265337049051x18=−9.7646148702235x5=−5.94334839772546x47=−3.48142956304392x62=0.339836909454122x51=2.80175574413567x40=6.62302221663371x41=9.08494105131526x46=12.9062075238133x60=15.3681263584948x13=19.1893928309929x2=21.6513116656744x55=25.4725781381725x23=27.934496972854x65=31.7557634453521x45=34.2176822800336x42=38.0389487525316x29=40.5008675872132x3=44.3221340597112x27=46.7840528943928x59=50.6053193668908x14=53.0672382015724x67=56.8885046740704x28=59.350423508752x12=63.17168998125x22=65.6336088159315x10=69.4548752884296x63=71.9167941231111x21=75.7380605956092x7=78.1999794302907x66=82.0212459027887x15=84.4831647374703x64=88.3044312099683x19=90.7663500446499x1=94.5876165171479x39=97.0495353518295x8=100.870801824328x6=172.447759037984x53=951.562737128253is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
x0<x20For example, let's take the point
x0=x20−101=
−112585.595763607+−101=
−112585.695763607substitute to the expression
sin(t)<31sin(t)<31sin(t) < 1/3
Then
x<−112585.595763607no execute
one of the solutions of our inequality is:
x>−112585.595763607∧x<−116.578765092276 _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____
/ \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
-------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-------
x20 x36 x35 x52 x58 x61 x31 x49 x54 x38 x48 x33 x34 x43 x69 x26 x4 x17 x37 x32 x9 x30 x44 x24 x57 x56 x11 x50 x16 x68 x25 x18 x5 x47 x62 x51 x40 x41 x46 x60 x13 x2 x55 x23 x65 x45 x42 x29 x3 x27 x59 x14 x67 x28 x12 x22 x10 x63 x21 x7 x66 x15 x64 x19 x1 x39 x8 x6 x53Other solutions will get with the changeover to the next point
etc.
The answer:
x>−112585.595763607∧x<−116.578765092276x>−100.191128005419∧x<−97.7292091707377x>−93.9079426982397∧x<−91.4460238635581x>−87.6247573910601∧x<−85.1628385563785x>−81.3415720838805∧x<−78.879653249199x>−75.0583867767009∧x<−72.5964679420194x>−68.7752014695213∧x<−66.3132826348398x>−62.4920161623417∧x<−60.0300973276602x>−56.2088308551622∧x<−53.7469120204806x>−49.9256455479826∧x<−47.463726713301x>−43.642460240803∧x<−41.1805414061214x>−37.3592749336234∧x<−34.8973560989418x>−31.0760896264438∧x<−28.6141707917623x>−24.7929043192642∧x<−22.3309854845827x>−18.5097190120846∧x<−16.0478001774031x>−12.2265337049051∧x<−9.7646148702235x>−5.94334839772546∧x<−3.48142956304392x>0.339836909454122∧x<2.80175574413567x>6.62302221663371∧x<9.08494105131526x>12.9062075238133∧x<15.3681263584948x>19.1893928309929∧x<21.6513116656744x>25.4725781381725∧x<27.934496972854x>31.7557634453521∧x<34.2176822800336x>38.0389487525316∧x<40.5008675872132x>44.3221340597112∧x<46.7840528943928x>50.6053193668908∧x<53.0672382015724x>56.8885046740704∧x<59.350423508752x>63.17168998125∧x<65.6336088159315x>69.4548752884296∧x<71.9167941231111x>75.7380605956092∧x<78.1999794302907x>82.0212459027887∧x<84.4831647374703x>88.3044312099683∧x<90.7663500446499x>94.5876165171479∧x<97.0495353518295x>100.870801824328∧x<172.447759037984x>951.562737128253
49-25x<0
sint
sint