Given the inequality:
sin(t)<−32To solve this inequality, we must first solve the corresponding equation:
sin(t)=−32Solve:
Given the equation
sin(t)=−32transform
sin(t)+32=0sin(t)+32=0Do replacement
w=sin(t)Move free summands (without w)
from left part to right part, we given:
w=−32We get the answer: w = -2/3
do backward replacement
sin(t)=wsubstitute w:
x1=412.27836527649x2=10.1545056169963x3=93.5180519514668x4=1216.52608459548x5=−32.1456541921249x6=−65.2437180691587x7=30.686198879671x8=−107.54387787828x9=−302.322622400847x10=−77.8100886835179x11=−8.69505030454241x12=−38.4288394993045x13=5.55345765095262x14=16.4376909241759x15=−88.6943219567412x16=18.1198282653118x17=−50.9952101136637x18=−57.2783954208432x19=24.4030135724914x20=55.8189401083893x21=−14.978235611722x22=85.5527293031514x23=72.9863586887922x24=−69.8447660352024x25=11.8366429581322x26=74.6684960299281x27=−33.8277915332608x28=−101.2606925711x29=−7.01291296340655x30=−0.729727656226966x31=−63.5615807280228x32=−84.0932739906975x33=68.3853107227485x34=60.419988074433x35=99.8012372586464x36=−13.2960982705861x37=−58.9605327619791x38=−21.2614209189016x39=29.0040615385351x40=87.2348666442873x41=−2.41186499736283x42=62.1021254155689x43=−27.5446062260812x44=−44.7120248064841x45=−94.9775072639208x46=54.1368027672534x47=−25.8624688849453x48=98.1190999175106x49=−836.393373511112x50=3.87132030981676x51=−46.3941621476199x52=22.7208762313555x53=36.9693841868506x54=−90.376459297877x55=175.199460944801x56=−82.4111366495616x57=91.835914610331x58=−1755.42056570047x59=−19.5792835777657x60=79.2695439959718x61=49.5357548012097x62=66.7031733816126x63=80.9516813371077x64=−40.1109768404403x65=47.8536174600739x66=−96.6596446050566x67=43.2525694940301x68=−1979.9330994178x69=−71.5269033763383x70=41.5704321528943x71=−52.6773474547995x72=35.2872468457147x73=−76.127951342382x1=412.27836527649x2=10.1545056169963x3=93.5180519514668x4=1216.52608459548x5=−32.1456541921249x6=−65.2437180691587x7=30.686198879671x8=−107.54387787828x9=−302.322622400847x10=−77.8100886835179x11=−8.69505030454241x12=−38.4288394993045x13=5.55345765095262x14=16.4376909241759x15=−88.6943219567412x16=18.1198282653118x17=−50.9952101136637x18=−57.2783954208432x19=24.4030135724914x20=55.8189401083893x21=−14.978235611722x22=85.5527293031514x23=72.9863586887922x24=−69.8447660352024x25=11.8366429581322x26=74.6684960299281x27=−33.8277915332608x28=−101.2606925711x29=−7.01291296340655x30=−0.729727656226966x31=−63.5615807280228x32=−84.0932739906975x33=68.3853107227485x34=60.419988074433x35=99.8012372586464x36=−13.2960982705861x37=−58.9605327619791x38=−21.2614209189016x39=29.0040615385351x40=87.2348666442873x41=−2.41186499736283x42=62.1021254155689x43=−27.5446062260812x44=−44.7120248064841x45=−94.9775072639208x46=54.1368027672534x47=−25.8624688849453x48=98.1190999175106x49=−836.393373511112x50=3.87132030981676x51=−46.3941621476199x52=22.7208762313555x53=36.9693841868506x54=−90.376459297877x55=175.199460944801x56=−82.4111366495616x57=91.835914610331x58=−1755.42056570047x59=−19.5792835777657x60=79.2695439959718x61=49.5357548012097x62=66.7031733816126x63=80.9516813371077x64=−40.1109768404403x65=47.8536174600739x66=−96.6596446050566x67=43.2525694940301x68=−1979.9330994178x69=−71.5269033763383x70=41.5704321528943x71=−52.6773474547995x72=35.2872468457147x73=−76.127951342382This roots
x68=−1979.9330994178x58=−1755.42056570047x49=−836.393373511112x9=−302.322622400847x8=−107.54387787828x28=−101.2606925711x66=−96.6596446050566x45=−94.9775072639208x54=−90.376459297877x15=−88.6943219567412x32=−84.0932739906975x56=−82.4111366495616x10=−77.8100886835179x73=−76.127951342382x69=−71.5269033763383x24=−69.8447660352024x6=−65.2437180691587x31=−63.5615807280228x37=−58.9605327619791x18=−57.2783954208432x71=−52.6773474547995x17=−50.9952101136637x51=−46.3941621476199x44=−44.7120248064841x64=−40.1109768404403x12=−38.4288394993045x27=−33.8277915332608x5=−32.1456541921249x43=−27.5446062260812x47=−25.8624688849453x38=−21.2614209189016x59=−19.5792835777657x21=−14.978235611722x36=−13.2960982705861x11=−8.69505030454241x29=−7.01291296340655x41=−2.41186499736283x30=−0.729727656226966x50=3.87132030981676x13=5.55345765095262x2=10.1545056169963x25=11.8366429581322x14=16.4376909241759x16=18.1198282653118x52=22.7208762313555x19=24.4030135724914x39=29.0040615385351x7=30.686198879671x72=35.2872468457147x53=36.9693841868506x70=41.5704321528943x67=43.2525694940301x65=47.8536174600739x61=49.5357548012097x46=54.1368027672534x20=55.8189401083893x34=60.419988074433x42=62.1021254155689x62=66.7031733816126x33=68.3853107227485x23=72.9863586887922x26=74.6684960299281x60=79.2695439959718x63=80.9516813371077x22=85.5527293031514x40=87.2348666442873x57=91.835914610331x3=93.5180519514668x48=98.1190999175106x35=99.8012372586464x55=175.199460944801x1=412.27836527649x4=1216.52608459548is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
x0<x68For example, let's take the point
x0=x68−101=
−1979.9330994178+−101=
−1980.0330994178substitute to the expression
sin(t)<−32sin(t)<−32sin(t) < -2/3
Then
x<−1979.9330994178no execute
one of the solutions of our inequality is:
x>−1979.9330994178∧x<−1755.42056570047 _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____
/ \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
-------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-------ο-------ο-------ο-------ο-------
x68 x58 x49 x9 x8 x28 x66 x45 x54 x15 x32 x56 x10 x73 x69 x24 x6 x31 x37 x18 x71 x17 x51 x44 x64 x12 x27 x5 x43 x47 x38 x59 x21 x36 x11 x29 x41 x30 x50 x13 x2 x25 x14 x16 x52 x19 x39 x7 x72 x53 x70 x67 x65 x61 x46 x20 x34 x42 x62 x33 x23 x26 x60 x63 x22 x40 x57 x3 x48 x35 x55 x1 x4Other solutions will get with the changeover to the next point
etc.
The answer:
x>−1979.9330994178∧x<−1755.42056570047x>−836.393373511112∧x<−302.322622400847x>−107.54387787828∧x<−101.2606925711x>−96.6596446050566∧x<−94.9775072639208x>−90.376459297877∧x<−88.6943219567412x>−84.0932739906975∧x<−82.4111366495616x>−77.8100886835179∧x<−76.127951342382x>−71.5269033763383∧x<−69.8447660352024x>−65.2437180691587∧x<−63.5615807280228x>−58.9605327619791∧x<−57.2783954208432x>−52.6773474547995∧x<−50.9952101136637x>−46.3941621476199∧x<−44.7120248064841x>−40.1109768404403∧x<−38.4288394993045x>−33.8277915332608∧x<−32.1456541921249x>−27.5446062260812∧x<−25.8624688849453x>−21.2614209189016∧x<−19.5792835777657x>−14.978235611722∧x<−13.2960982705861x>−8.69505030454241∧x<−7.01291296340655x>−2.41186499736283∧x<−0.729727656226966x>3.87132030981676∧x<5.55345765095262x>10.1545056169963∧x<11.8366429581322x>16.4376909241759∧x<18.1198282653118x>22.7208762313555∧x<24.4030135724914x>29.0040615385351∧x<30.686198879671x>35.2872468457147∧x<36.9693841868506x>41.5704321528943∧x<43.2525694940301x>47.8536174600739∧x<49.5357548012097x>54.1368027672534∧x<55.8189401083893x>60.419988074433∧x<62.1021254155689x>66.7031733816126∧x<68.3853107227485x>72.9863586887922∧x<74.6684960299281x>79.2695439959718∧x<80.9516813371077x>85.5527293031514∧x<87.2348666442873x>91.835914610331∧x<93.5180519514668x>98.1190999175106∧x<99.8012372586464x>175.199460944801∧x<412.27836527649x>1216.52608459548
sin(t)
sin(t)
sin(t)