Given the inequality:
cos(t)≤0To solve this inequality, we must first solve the corresponding equation:
cos(t)=0Solve:
Given the equation
cos(t)=0transform
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=48.6946861306418x2=54.9778714378214x3=−98.9601685880785x4=67.5442420521806x5=76.9690200129499x6=36.1283155162826x7=58.1194640914112x8=14.1371669411541x9=−29.845130209103x10=61.261056745001x11=−36.1283155162826x12=−4.71238898038469x13=−39.2699081698724x14=1.5707963267949x15=−168.075206967054x16=−14.1371669411541x17=−64.4026493985908x18=−67.5442420521806x19=92.6769832808989x20=−51.8362787842316x21=−86.3937979737193x22=42.4115008234622x23=−17.2787595947439x24=−45.553093477052x25=−89.5353906273091x26=−1.5707963267949x27=39.2699081698724x28=23.5619449019235x29=7.85398163397448x30=−58.1194640914112x31=−61.261056745001x32=−73.8274273593601x33=73.8274273593601x34=29.845130209103x35=4.71238898038469x36=86.3937979737193x37=64.4026493985908x38=89.5353906273091x39=−20.4203522483337x40=−387.986692718339x41=−26.7035375555132x42=98.9601685880785x43=51.8362787842316x44=83.2522053201295x45=−48.6946861306418x46=−54.9778714378214x47=70.6858347057703x48=−95.8185759344887x49=26.7035375555132x50=80.1106126665397x51=−23.5619449019235x52=−7.85398163397448x53=−83.2522053201295x54=−76.9690200129499x55=−42.4115008234622x56=−32.9867228626928x57=17.2787595947439x58=32.9867228626928x59=20.4203522483337x60=−70.6858347057703x61=−10.9955742875643x62=−92.6769832808989x63=45.553093477052x64=10.9955742875643x65=−80.1106126665397x66=95.8185759344887x67=−2266.65909956504x1=48.6946861306418x2=54.9778714378214x3=−98.9601685880785x4=67.5442420521806x5=76.9690200129499x6=36.1283155162826x7=58.1194640914112x8=14.1371669411541x9=−29.845130209103x10=61.261056745001x11=−36.1283155162826x12=−4.71238898038469x13=−39.2699081698724x14=1.5707963267949x15=−168.075206967054x16=−14.1371669411541x17=−64.4026493985908x18=−67.5442420521806x19=92.6769832808989x20=−51.8362787842316x21=−86.3937979737193x22=42.4115008234622x23=−17.2787595947439x24=−45.553093477052x25=−89.5353906273091x26=−1.5707963267949x27=39.2699081698724x28=23.5619449019235x29=7.85398163397448x30=−58.1194640914112x31=−61.261056745001x32=−73.8274273593601x33=73.8274273593601x34=29.845130209103x35=4.71238898038469x36=86.3937979737193x37=64.4026493985908x38=89.5353906273091x39=−20.4203522483337x40=−387.986692718339x41=−26.7035375555132x42=98.9601685880785x43=51.8362787842316x44=83.2522053201295x45=−48.6946861306418x46=−54.9778714378214x47=70.6858347057703x48=−95.8185759344887x49=26.7035375555132x50=80.1106126665397x51=−23.5619449019235x52=−7.85398163397448x53=−83.2522053201295x54=−76.9690200129499x55=−42.4115008234622x56=−32.9867228626928x57=17.2787595947439x58=32.9867228626928x59=20.4203522483337x60=−70.6858347057703x61=−10.9955742875643x62=−92.6769832808989x63=45.553093477052x64=10.9955742875643x65=−80.1106126665397x66=95.8185759344887x67=−2266.65909956504This roots
x67=−2266.65909956504x40=−387.986692718339x15=−168.075206967054x3=−98.9601685880785x48=−95.8185759344887x62=−92.6769832808989x25=−89.5353906273091x21=−86.3937979737193x53=−83.2522053201295x65=−80.1106126665397x54=−76.9690200129499x32=−73.8274273593601x60=−70.6858347057703x18=−67.5442420521806x17=−64.4026493985908x31=−61.261056745001x30=−58.1194640914112x46=−54.9778714378214x20=−51.8362787842316x45=−48.6946861306418x24=−45.553093477052x55=−42.4115008234622x13=−39.2699081698724x11=−36.1283155162826x56=−32.9867228626928x9=−29.845130209103x41=−26.7035375555132x51=−23.5619449019235x39=−20.4203522483337x23=−17.2787595947439x16=−14.1371669411541x61=−10.9955742875643x52=−7.85398163397448x12=−4.71238898038469x26=−1.5707963267949x14=1.5707963267949x35=4.71238898038469x29=7.85398163397448x64=10.9955742875643x8=14.1371669411541x57=17.2787595947439x59=20.4203522483337x28=23.5619449019235x49=26.7035375555132x34=29.845130209103x58=32.9867228626928x6=36.1283155162826x27=39.2699081698724x22=42.4115008234622x63=45.553093477052x1=48.6946861306418x43=51.8362787842316x2=54.9778714378214x7=58.1194640914112x10=61.261056745001x37=64.4026493985908x4=67.5442420521806x47=70.6858347057703x33=73.8274273593601x5=76.9690200129499x50=80.1106126665397x44=83.2522053201295x36=86.3937979737193x38=89.5353906273091x19=92.6769832808989x66=95.8185759344887x42=98.9601685880785is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
x0≤x67For example, let's take the point
x0=x67−101=
−2266.65909956504+−101=
−2266.75909956504substitute to the expression
cos(t)≤0cos(t)≤0cos(t) <= 0
Then
x≤−2266.65909956504no execute
one of the solutions of our inequality is:
x≥−2266.65909956504∧x≤−387.986692718339 _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____
/ \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
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x67 x40 x15 x3 x48 x62 x25 x21 x53 x65 x54 x32 x60 x18 x17 x31 x30 x46 x20 x45 x24 x55 x13 x11 x56 x9 x41 x51 x39 x23 x16 x61 x52 x12 x26 x14 x35 x29 x64 x8 x57 x59 x28 x49 x34 x58 x6 x27 x22 x63 x1 x43 x2 x7 x10 x37 x4 x47 x33 x5 x50 x44 x36 x38 x19 x66 x42Other solutions will get with the changeover to the next point
etc.
The answer:
x≥−2266.65909956504∧x≤−387.986692718339x≥−168.075206967054∧x≤−98.9601685880785x≥−95.8185759344887∧x≤−92.6769832808989x≥−89.5353906273091∧x≤−86.3937979737193x≥−83.2522053201295∧x≤−80.1106126665397x≥−76.9690200129499∧x≤−73.8274273593601x≥−70.6858347057703∧x≤−67.5442420521806x≥−64.4026493985908∧x≤−61.261056745001x≥−58.1194640914112∧x≤−54.9778714378214x≥−51.8362787842316∧x≤−48.6946861306418x≥−45.553093477052∧x≤−42.4115008234622x≥−39.2699081698724∧x≤−36.1283155162826x≥−32.9867228626928∧x≤−29.845130209103x≥−26.7035375555132∧x≤−23.5619449019235x≥−20.4203522483337∧x≤−17.2787595947439x≥−14.1371669411541∧x≤−10.9955742875643x≥−7.85398163397448∧x≤−4.71238898038469x≥−1.5707963267949∧x≤1.5707963267949x≥4.71238898038469∧x≤7.85398163397448x≥10.9955742875643∧x≤14.1371669411541x≥17.2787595947439∧x≤20.4203522483337x≥23.5619449019235∧x≤26.7035375555132x≥29.845130209103∧x≤32.9867228626928x≥36.1283155162826∧x≤39.2699081698724x≥42.4115008234622∧x≤45.553093477052x≥48.6946861306418∧x≤51.8362787842316x≥54.9778714378214∧x≤58.1194640914112x≥61.261056745001∧x≤64.4026493985908x≥67.5442420521806∧x≤70.6858347057703x≥73.8274273593601∧x≤76.9690200129499x≥80.1106126665397∧x≤83.2522053201295x≥86.3937979737193∧x≤89.5353906273091x≥92.6769832808989∧x≤95.8185759344887x≥98.9601685880785
cos(t)
cos(t)