Given the inequality:
sin(t)<−53To solve this inequality, we must first solve the corresponding equation:
sin(t)=−53Solve:
Given the equation
sin(t)=−53transform
sin(t)+53=0sin(t)+53=0Do replacement
w=sin(t)Move free summands (without w)
from left part to right part, we given:
w=−53We get the answer: w = -3/5
do backward replacement
sin(t)=wsubstitute w:
x1=−0.643501108793284x2=−19.493057030332x3=66.6169468341789x4=37.0556107342842x5=−15.0644621591557x6=−27.6308327735149x7=−107.457651330846x8=−94.8912807164871x9=275.816652407109x10=81.0379078845413x11=−13.2098717231525x12=54.0505762198198x13=−63.4753541805892x14=85.4665027557177x15=−40.197203387874x16=−46.4803886950536x17=−71.613129923772x18=−84.1795005381311x19=−33.9140180806944x20=−25.7762423375116x21=−50.90898356623x22=91.7496880628973x23=18.2060548127455x24=−101.174466023667x25=16.3514643767422x26=35.201020298281x27=10.0682790695627x28=87.3210931917209x29=11.9228695055659x30=139936.605069959x31=−8.7812768519761x32=−65.3299446165924x33=−77.8963152309515x34=−32.0594276446912x35=−59.0467593094128x36=−52.7635740022332x37=93.6042784989005x38=3280205.36115037x39=60.3337615269994x40=−6.92668641597287x41=79.1833174485381x42=−57.1921688734096x43=−2.49809154479651x44=24.4892401199251x45=30.7724254271046x46=22.6346496839218x47=−38.3426129518708x48=28.9178349911014x49=68.4715372701822x50=−44.6257982590504x51=98.0328733700769x52=−90.4626858453107x53=−21.3476474663353x54=−96.7458711524903x55=−76.0417247949483x56=55.905166655823x57=99.8874638060801x58=72.9001321413585x59=74.7547225773617x60=5.6396841983863x61=−88.6080954093075x62=468.740806493672x63=47.7673909126402x64=62.1883519630026x65=41.4842056054606x66=49.6219813486434x67=−69.7585394877687x68=43.3387960414638x69=−82.3249101021279x70=3.78509376238308x1=−0.643501108793284x2=−19.493057030332x3=66.6169468341789x4=37.0556107342842x5=−15.0644621591557x6=−27.6308327735149x7=−107.457651330846x8=−94.8912807164871x9=275.816652407109x10=81.0379078845413x11=−13.2098717231525x12=54.0505762198198x13=−63.4753541805892x14=85.4665027557177x15=−40.197203387874x16=−46.4803886950536x17=−71.613129923772x18=−84.1795005381311x19=−33.9140180806944x20=−25.7762423375116x21=−50.90898356623x22=91.7496880628973x23=18.2060548127455x24=−101.174466023667x25=16.3514643767422x26=35.201020298281x27=10.0682790695627x28=87.3210931917209x29=11.9228695055659x30=139936.605069959x31=−8.7812768519761x32=−65.3299446165924x33=−77.8963152309515x34=−32.0594276446912x35=−59.0467593094128x36=−52.7635740022332x37=93.6042784989005x38=3280205.36115037x39=60.3337615269994x40=−6.92668641597287x41=79.1833174485381x42=−57.1921688734096x43=−2.49809154479651x44=24.4892401199251x45=30.7724254271046x46=22.6346496839218x47=−38.3426129518708x48=28.9178349911014x49=68.4715372701822x50=−44.6257982590504x51=98.0328733700769x52=−90.4626858453107x53=−21.3476474663353x54=−96.7458711524903x55=−76.0417247949483x56=55.905166655823x57=99.8874638060801x58=72.9001321413585x59=74.7547225773617x60=5.6396841983863x61=−88.6080954093075x62=468.740806493672x63=47.7673909126402x64=62.1883519630026x65=41.4842056054606x66=49.6219813486434x67=−69.7585394877687x68=43.3387960414638x69=−82.3249101021279x70=3.78509376238308This roots
x7=−107.457651330846x24=−101.174466023667x54=−96.7458711524903x8=−94.8912807164871x52=−90.4626858453107x61=−88.6080954093075x18=−84.1795005381311x69=−82.3249101021279x33=−77.8963152309515x55=−76.0417247949483x17=−71.613129923772x67=−69.7585394877687x32=−65.3299446165924x13=−63.4753541805892x35=−59.0467593094128x42=−57.1921688734096x36=−52.7635740022332x21=−50.90898356623x16=−46.4803886950536x50=−44.6257982590504x15=−40.197203387874x47=−38.3426129518708x19=−33.9140180806944x34=−32.0594276446912x6=−27.6308327735149x20=−25.7762423375116x53=−21.3476474663353x2=−19.493057030332x5=−15.0644621591557x11=−13.2098717231525x31=−8.7812768519761x40=−6.92668641597287x43=−2.49809154479651x1=−0.643501108793284x70=3.78509376238308x60=5.6396841983863x27=10.0682790695627x29=11.9228695055659x25=16.3514643767422x23=18.2060548127455x46=22.6346496839218x44=24.4892401199251x48=28.9178349911014x45=30.7724254271046x26=35.201020298281x4=37.0556107342842x65=41.4842056054606x68=43.3387960414638x63=47.7673909126402x66=49.6219813486434x12=54.0505762198198x56=55.905166655823x39=60.3337615269994x64=62.1883519630026x3=66.6169468341789x49=68.4715372701822x58=72.9001321413585x59=74.7547225773617x41=79.1833174485381x10=81.0379078845413x14=85.4665027557177x28=87.3210931917209x22=91.7496880628973x37=93.6042784989005x51=98.0328733700769x57=99.8874638060801x9=275.816652407109x62=468.740806493672x30=139936.605069959x38=3280205.36115037is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
x0<x7For example, let's take the point
x0=x7−101=
−107.457651330846+−101=
−107.557651330846substitute to the expression
sin(t)<−53sin(t)<−53sin(t) < -3/5
Then
x<−107.457651330846no execute
one of the solutions of our inequality is:
x>−107.457651330846∧x<−101.174466023667 _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____
/ \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \
-------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-------
x7 x24 x54 x8 x52 x61 x18 x69 x33 x55 x17 x67 x32 x13 x35 x42 x36 x21 x16 x50 x15 x47 x19 x34 x6 x20 x53 x2 x5 x11 x31 x40 x43 x1 x70 x60 x27 x29 x25 x23 x46 x44 x48 x45 x26 x4 x65 x68 x63 x66 x12 x56 x39 x64 x3 x49 x58 x59 x41 x10 x14 x28 x22 x37 x51 x57 x9 x62 x30 x38Other solutions will get with the changeover to the next point
etc.
The answer:
x>−107.457651330846∧x<−101.174466023667x>−96.7458711524903∧x<−94.8912807164871x>−90.4626858453107∧x<−88.6080954093075x>−84.1795005381311∧x<−82.3249101021279x>−77.8963152309515∧x<−76.0417247949483x>−71.613129923772∧x<−69.7585394877687x>−65.3299446165924∧x<−63.4753541805892x>−59.0467593094128∧x<−57.1921688734096x>−52.7635740022332∧x<−50.90898356623x>−46.4803886950536∧x<−44.6257982590504x>−40.197203387874∧x<−38.3426129518708x>−33.9140180806944∧x<−32.0594276446912x>−27.6308327735149∧x<−25.7762423375116x>−21.3476474663353∧x<−19.493057030332x>−15.0644621591557∧x<−13.2098717231525x>−8.7812768519761∧x<−6.92668641597287x>−2.49809154479651∧x<−0.643501108793284x>3.78509376238308∧x<5.6396841983863x>10.0682790695627∧x<11.9228695055659x>16.3514643767422∧x<18.2060548127455x>22.6346496839218∧x<24.4892401199251x>28.9178349911014∧x<30.7724254271046x>35.201020298281∧x<37.0556107342842x>41.4842056054606∧x<43.3387960414638x>47.7673909126402∧x<49.6219813486434x>54.0505762198198∧x<55.905166655823x>60.3337615269994∧x<62.1883519630026x>66.6169468341789∧x<68.4715372701822x>72.9001321413585∧x<74.7547225773617x>79.1833174485381∧x<81.0379078845413x>85.4665027557177∧x<87.3210931917209x>91.7496880628973∧x<93.6042784989005x>98.0328733700769∧x<99.8874638060801x>275.816652407109∧x<468.740806493672x>139936.605069959∧x<3280205.36115037
4sin^2x+8sinx+3≤0
sin(t)
sin(t)
sin(t)
-0,6