Mister Exam
|sin(x)|>=1/2 inequation
The solution
You have entered
[src]
|sin(x)| >= 1/2
$$\left|{\sin{\left(x \right)}}\right| \geq \frac{1}{2}$$
Abs(sin(x)) >= 1/2
Detail solution
Given the inequality:
$$\left|{\sin{\left(x \right)}}\right| \geq \frac{1}{2}$$
To solve this inequality, we must first solve the corresponding equation:
$$\left|{\sin{\left(x \right)}}\right| = \frac{1}{2}$$
Solve:
Given the equation
$$\left|{\sin{\left(x \right)}}\right| = \frac{1}{2}$$
transform
$$\left|{\sin{\left(x \right)}}\right| - \frac{1}{2} = 0$$
$$\left|{\sin{\left(x \right)}}\right| - \frac{1}{2} = 0$$
Do replacement
$$w = \left|{\sin{\left(x \right)}}\right|$$
Move free summands (without w)
from left part to right part, we given:
$$w = \frac{1}{2}$$
We get the answer: w = 1/2
do backward replacement
$$\left|{\sin{\left(x \right)}}\right| = w$$
substitute w:
$$x_{1} = 96.8657734856853$$
$$x_{2} = -100.007366139275$$
$$x_{3} = -71.733032256967$$
$$x_{4} = -65.4498469497874$$
$$x_{5} = -91.6297857297023$$
$$x_{6} = 22.5147473507269$$
$$x_{7} = 93.7241808320955$$
$$x_{8} = 66.497044500984$$
$$x_{9} = -75.9218224617533$$
$$x_{10} = -31.9395253114962$$
$$x_{11} = -87.4409955249159$$
$$x_{12} = -90.5825881785057$$
$$x_{13} = 60.2138591938044$$
$$x_{14} = -49.7418836818384$$
$$x_{15} = -25.6563400043166$$
$$x_{16} = -34.0339204138894$$
$$x_{17} = -84.2994028713261$$
$$x_{18} = -68.5914396033772$$
$$x_{19} = 97.9129710368819$$
$$x_{20} = -60.2138591938044$$
$$x_{21} = -53.9306738866248$$
$$x_{22} = -62.3082542961976$$
$$x_{23} = -63.3554518473942$$
$$x_{24} = -2.61799387799149$$
$$x_{25} = 34.0339204138894$$
$$x_{26} = -19.3731546971371$$
$$x_{27} = 100.007366139275$$
$$x_{28} = -12.0427718387609$$
$$x_{29} = 0.523598775598299$$
$$x_{30} = 12.0427718387609$$
$$x_{31} = 75.9218224617533$$
$$x_{32} = 62.3082542961976$$
$$x_{33} = 47.6474885794452$$
$$x_{34} = 90.5825881785057$$
$$x_{35} = 71.733032256967$$
$$x_{36} = 18.3259571459405$$
$$x_{37} = 38.2227106186758$$
$$x_{38} = -3.66519142918809$$
$$x_{39} = 31.9395253114962$$
$$x_{40} = -85.3466004225227$$
$$x_{41} = -21.4675497995303$$
$$x_{42} = -57.0722665402146$$
$$x_{43} = -82.2050077689329$$
$$x_{44} = 68.5914396033772$$
$$x_{45} = -18.3259571459405$$
$$x_{46} = 44.5058959258554$$
$$x_{47} = 82.2050077689329$$
$$x_{48} = -46.6002910282486$$
$$x_{49} = 84.2994028713261$$
$$x_{50} = -16.2315620435473$$
$$x_{51} = 25.6563400043166$$
$$x_{52} = -69.6386371545737$$
$$x_{53} = -27.7507351067098$$
$$x_{54} = -5.75958653158129$$
$$x_{55} = 3.66519142918809$$
$$x_{56} = -40.317105721069$$
$$x_{57} = 49.7418836818384$$
$$x_{58} = -41.3643032722656$$
$$x_{59} = -382.750704962356$$
$$x_{60} = 24.60914245312$$
$$x_{61} = 40.317105721069$$
$$x_{62} = -78.0162175641465$$
$$x_{63} = 46.6002910282486$$
$$x_{64} = 69.6386371545737$$
$$x_{65} = 2.61799387799149$$
$$x_{66} = -43.4586983746588$$
$$x_{67} = 27.7507351067098$$
$$x_{68} = -93.7241808320955$$
$$x_{69} = 88.4881930761125$$
$$x_{70} = -137.706477982353$$
$$x_{71} = 56.025068989018$$
$$x_{72} = 91.6297857297023$$
$$x_{73} = 78.0162175641465$$
$$x_{74} = -97.9129710368819$$
$$x_{75} = -9.94837673636768$$
$$x_{76} = -56.025068989018$$
$$x_{77} = -47.6474885794452$$
$$x_{78} = 16.2315620435473$$
$$x_{79} = -66.497044500984$$
$$x_{80} = 5.75958653158129$$
$$x_{81} = -24.60914245312$$
$$x_{82} = -38.2227106186758$$
$$x_{83} = 9.94837673636768$$
$$x_{84} = 53.9306738866248$$
$$x_{1} = 96.8657734856853$$
$$x_{2} = -100.007366139275$$
$$x_{3} = -71.733032256967$$
$$x_{4} = -65.4498469497874$$
$$x_{5} = -91.6297857297023$$
$$x_{6} = 22.5147473507269$$
$$x_{7} = 93.7241808320955$$
$$x_{8} = 66.497044500984$$
$$x_{9} = -75.9218224617533$$
$$x_{10} = -31.9395253114962$$
$$x_{11} = -87.4409955249159$$
$$x_{12} = -90.5825881785057$$
$$x_{13} = 60.2138591938044$$
$$x_{14} = -49.7418836818384$$
$$x_{15} = -25.6563400043166$$
$$x_{16} = -34.0339204138894$$
$$x_{17} = -84.2994028713261$$
$$x_{18} = -68.5914396033772$$
$$x_{19} = 97.9129710368819$$
$$x_{20} = -60.2138591938044$$
$$x_{21} = -53.9306738866248$$
$$x_{22} = -62.3082542961976$$
$$x_{23} = -63.3554518473942$$
$$x_{24} = -2.61799387799149$$
$$x_{25} = 34.0339204138894$$
$$x_{26} = -19.3731546971371$$
$$x_{27} = 100.007366139275$$
$$x_{28} = -12.0427718387609$$
$$x_{29} = 0.523598775598299$$
$$x_{30} = 12.0427718387609$$
$$x_{31} = 75.9218224617533$$
$$x_{32} = 62.3082542961976$$
$$x_{33} = 47.6474885794452$$
$$x_{34} = 90.5825881785057$$
$$x_{35} = 71.733032256967$$
$$x_{36} = 18.3259571459405$$
$$x_{37} = 38.2227106186758$$
$$x_{38} = -3.66519142918809$$
$$x_{39} = 31.9395253114962$$
$$x_{40} = -85.3466004225227$$
$$x_{41} = -21.4675497995303$$
$$x_{42} = -57.0722665402146$$
$$x_{43} = -82.2050077689329$$
$$x_{44} = 68.5914396033772$$
$$x_{45} = -18.3259571459405$$
$$x_{46} = 44.5058959258554$$
$$x_{47} = 82.2050077689329$$
$$x_{48} = -46.6002910282486$$
$$x_{49} = 84.2994028713261$$
$$x_{50} = -16.2315620435473$$
$$x_{51} = 25.6563400043166$$
$$x_{52} = -69.6386371545737$$
$$x_{53} = -27.7507351067098$$
$$x_{54} = -5.75958653158129$$
$$x_{55} = 3.66519142918809$$
$$x_{56} = -40.317105721069$$
$$x_{57} = 49.7418836818384$$
$$x_{58} = -41.3643032722656$$
$$x_{59} = -382.750704962356$$
$$x_{60} = 24.60914245312$$
$$x_{61} = 40.317105721069$$
$$x_{62} = -78.0162175641465$$
$$x_{63} = 46.6002910282486$$
$$x_{64} = 69.6386371545737$$
$$x_{65} = 2.61799387799149$$
$$x_{66} = -43.4586983746588$$
$$x_{67} = 27.7507351067098$$
$$x_{68} = -93.7241808320955$$
$$x_{69} = 88.4881930761125$$
$$x_{70} = -137.706477982353$$
$$x_{71} = 56.025068989018$$
$$x_{72} = 91.6297857297023$$
$$x_{73} = 78.0162175641465$$
$$x_{74} = -97.9129710368819$$
$$x_{75} = -9.94837673636768$$
$$x_{76} = -56.025068989018$$
$$x_{77} = -47.6474885794452$$
$$x_{78} = 16.2315620435473$$
$$x_{79} = -66.497044500984$$
$$x_{80} = 5.75958653158129$$
$$x_{81} = -24.60914245312$$
$$x_{82} = -38.2227106186758$$
$$x_{83} = 9.94837673636768$$
$$x_{84} = 53.9306738866248$$
This roots
$$x_{59} = -382.750704962356$$
$$x_{70} = -137.706477982353$$
$$x_{2} = -100.007366139275$$
$$x_{74} = -97.9129710368819$$
$$x_{68} = -93.7241808320955$$
$$x_{5} = -91.6297857297023$$
$$x_{12} = -90.5825881785057$$
$$x_{11} = -87.4409955249159$$
$$x_{40} = -85.3466004225227$$
$$x_{17} = -84.2994028713261$$
$$x_{43} = -82.2050077689329$$
$$x_{62} = -78.0162175641465$$
$$x_{9} = -75.9218224617533$$
$$x_{3} = -71.733032256967$$
$$x_{52} = -69.6386371545737$$
$$x_{18} = -68.5914396033772$$
$$x_{79} = -66.497044500984$$
$$x_{4} = -65.4498469497874$$
$$x_{23} = -63.3554518473942$$
$$x_{22} = -62.3082542961976$$
$$x_{20} = -60.2138591938044$$
$$x_{42} = -57.0722665402146$$
$$x_{76} = -56.025068989018$$
$$x_{21} = -53.9306738866248$$
$$x_{14} = -49.7418836818384$$
$$x_{77} = -47.6474885794452$$
$$x_{48} = -46.6002910282486$$
$$x_{66} = -43.4586983746588$$
$$x_{58} = -41.3643032722656$$
$$x_{56} = -40.317105721069$$
$$x_{82} = -38.2227106186758$$
$$x_{16} = -34.0339204138894$$
$$x_{10} = -31.9395253114962$$
$$x_{53} = -27.7507351067098$$
$$x_{15} = -25.6563400043166$$
$$x_{81} = -24.60914245312$$
$$x_{41} = -21.4675497995303$$
$$x_{26} = -19.3731546971371$$
$$x_{45} = -18.3259571459405$$
$$x_{50} = -16.2315620435473$$
$$x_{28} = -12.0427718387609$$
$$x_{75} = -9.94837673636768$$
$$x_{54} = -5.75958653158129$$
$$x_{38} = -3.66519142918809$$
$$x_{24} = -2.61799387799149$$
$$x_{29} = 0.523598775598299$$
$$x_{65} = 2.61799387799149$$
$$x_{55} = 3.66519142918809$$
$$x_{80} = 5.75958653158129$$
$$x_{83} = 9.94837673636768$$
$$x_{30} = 12.0427718387609$$
$$x_{78} = 16.2315620435473$$
$$x_{36} = 18.3259571459405$$
$$x_{6} = 22.5147473507269$$
$$x_{60} = 24.60914245312$$
$$x_{51} = 25.6563400043166$$
$$x_{67} = 27.7507351067098$$
$$x_{39} = 31.9395253114962$$
$$x_{25} = 34.0339204138894$$
$$x_{37} = 38.2227106186758$$
$$x_{61} = 40.317105721069$$
$$x_{46} = 44.5058959258554$$
$$x_{63} = 46.6002910282486$$
$$x_{33} = 47.6474885794452$$
$$x_{57} = 49.7418836818384$$
$$x_{84} = 53.9306738866248$$
$$x_{71} = 56.025068989018$$
$$x_{13} = 60.2138591938044$$
$$x_{32} = 62.3082542961976$$
$$x_{8} = 66.497044500984$$
$$x_{44} = 68.5914396033772$$
$$x_{64} = 69.6386371545737$$
$$x_{35} = 71.733032256967$$
$$x_{31} = 75.9218224617533$$
$$x_{73} = 78.0162175641465$$
$$x_{47} = 82.2050077689329$$
$$x_{49} = 84.2994028713261$$
$$x_{69} = 88.4881930761125$$
$$x_{34} = 90.5825881785057$$
$$x_{72} = 91.6297857297023$$
$$x_{7} = 93.7241808320955$$
$$x_{1} = 96.8657734856853$$
$$x_{19} = 97.9129710368819$$
$$x_{27} = 100.007366139275$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} \leq x_{59}$$
For example, let's take the point
$$x_{0} = x_{59} - \frac{1}{10}$$
=
$$-382.750704962356 + - \frac{1}{10}$$
=
$$-382.850704962356$$
substitute to the expression
$$\left|{\sin{\left(x \right)}}\right| \geq \frac{1}{2}$$
$$\left|{\sin{\left(-382.850704962356 \right)}}\right| \geq \frac{1}{2}$$
but
Then
$$x \leq -382.750704962356$$
no execute
one of the solutions of our inequality is:
$$x \geq -382.750704962356 \wedge x \leq -137.706477982353$$
Other solutions will get with the changeover to the next point
etc.
The answer:
$$x \geq -382.750704962356 \wedge x \leq -137.706477982353$$
$$x \geq -100.007366139275 \wedge x \leq -97.9129710368819$$
$$x \geq -93.7241808320955 \wedge x \leq -91.6297857297023$$
$$x \geq -90.5825881785057 \wedge x \leq -87.4409955249159$$
$$x \geq -85.3466004225227 \wedge x \leq -84.2994028713261$$
$$x \geq -82.2050077689329 \wedge x \leq -78.0162175641465$$
$$x \geq -75.9218224617533 \wedge x \leq -71.733032256967$$
$$x \geq -69.6386371545737 \wedge x \leq -68.5914396033772$$
$$x \geq -66.497044500984 \wedge x \leq -65.4498469497874$$
$$x \geq -63.3554518473942 \wedge x \leq -62.3082542961976$$
$$x \geq -60.2138591938044 \wedge x \leq -57.0722665402146$$
$$x \geq -56.025068989018 \wedge x \leq -53.9306738866248$$
$$x \geq -49.7418836818384 \wedge x \leq -47.6474885794452$$
$$x \geq -46.6002910282486 \wedge x \leq -43.4586983746588$$
$$x \geq -41.3643032722656 \wedge x \leq -40.317105721069$$
$$x \geq -38.2227106186758 \wedge x \leq -34.0339204138894$$
$$x \geq -31.9395253114962 \wedge x \leq -27.7507351067098$$
$$x \geq -25.6563400043166 \wedge x \leq -24.60914245312$$
$$x \geq -21.4675497995303 \wedge x \leq -19.3731546971371$$
$$x \geq -18.3259571459405 \wedge x \leq -16.2315620435473$$
$$x \geq -12.0427718387609 \wedge x \leq -9.94837673636768$$
$$x \geq -5.75958653158129 \wedge x \leq -3.66519142918809$$
$$x \geq -2.61799387799149 \wedge x \leq 0.523598775598299$$
$$x \geq 2.61799387799149 \wedge x \leq 3.66519142918809$$
$$x \geq 5.75958653158129 \wedge x \leq 9.94837673636768$$
$$x \geq 12.0427718387609 \wedge x \leq 16.2315620435473$$
$$x \geq 18.3259571459405 \wedge x \leq 22.5147473507269$$
$$x \geq 24.60914245312 \wedge x \leq 25.6563400043166$$
$$x \geq 27.7507351067098 \wedge x \leq 31.9395253114962$$
$$x \geq 34.0339204138894 \wedge x \leq 38.2227106186758$$
$$x \geq 40.317105721069 \wedge x \leq 44.5058959258554$$
$$x \geq 46.6002910282486 \wedge x \leq 47.6474885794452$$
$$x \geq 49.7418836818384 \wedge x \leq 53.9306738866248$$
$$x \geq 56.025068989018 \wedge x \leq 60.2138591938044$$
$$x \geq 62.3082542961976 \wedge x \leq 66.497044500984$$
$$x \geq 68.5914396033772 \wedge x \leq 69.6386371545737$$
$$x \geq 71.733032256967 \wedge x \leq 75.9218224617533$$
$$x \geq 78.0162175641465 \wedge x \leq 82.2050077689329$$
$$x \geq 84.2994028713261 \wedge x \leq 88.4881930761125$$
$$x \geq 90.5825881785057 \wedge x \leq 91.6297857297023$$
$$x \geq 93.7241808320955 \wedge x \leq 96.8657734856853$$
$$x \geq 97.9129710368819 \wedge x \leq 100.007366139275$$
$$\left|{\sin{\left(x \right)}}\right| \geq \frac{1}{2}$$
To solve this inequality, we must first solve the corresponding equation:
$$\left|{\sin{\left(x \right)}}\right| = \frac{1}{2}$$
Solve:
Given the equation
$$\left|{\sin{\left(x \right)}}\right| = \frac{1}{2}$$
transform
$$\left|{\sin{\left(x \right)}}\right| - \frac{1}{2} = 0$$
$$\left|{\sin{\left(x \right)}}\right| - \frac{1}{2} = 0$$
Do replacement
$$w = \left|{\sin{\left(x \right)}}\right|$$
Move free summands (without w)
from left part to right part, we given:
$$w = \frac{1}{2}$$
We get the answer: w = 1/2
do backward replacement
$$\left|{\sin{\left(x \right)}}\right| = w$$
substitute w:
$$x_{1} = 96.8657734856853$$
$$x_{2} = -100.007366139275$$
$$x_{3} = -71.733032256967$$
$$x_{4} = -65.4498469497874$$
$$x_{5} = -91.6297857297023$$
$$x_{6} = 22.5147473507269$$
$$x_{7} = 93.7241808320955$$
$$x_{8} = 66.497044500984$$
$$x_{9} = -75.9218224617533$$
$$x_{10} = -31.9395253114962$$
$$x_{11} = -87.4409955249159$$
$$x_{12} = -90.5825881785057$$
$$x_{13} = 60.2138591938044$$
$$x_{14} = -49.7418836818384$$
$$x_{15} = -25.6563400043166$$
$$x_{16} = -34.0339204138894$$
$$x_{17} = -84.2994028713261$$
$$x_{18} = -68.5914396033772$$
$$x_{19} = 97.9129710368819$$
$$x_{20} = -60.2138591938044$$
$$x_{21} = -53.9306738866248$$
$$x_{22} = -62.3082542961976$$
$$x_{23} = -63.3554518473942$$
$$x_{24} = -2.61799387799149$$
$$x_{25} = 34.0339204138894$$
$$x_{26} = -19.3731546971371$$
$$x_{27} = 100.007366139275$$
$$x_{28} = -12.0427718387609$$
$$x_{29} = 0.523598775598299$$
$$x_{30} = 12.0427718387609$$
$$x_{31} = 75.9218224617533$$
$$x_{32} = 62.3082542961976$$
$$x_{33} = 47.6474885794452$$
$$x_{34} = 90.5825881785057$$
$$x_{35} = 71.733032256967$$
$$x_{36} = 18.3259571459405$$
$$x_{37} = 38.2227106186758$$
$$x_{38} = -3.66519142918809$$
$$x_{39} = 31.9395253114962$$
$$x_{40} = -85.3466004225227$$
$$x_{41} = -21.4675497995303$$
$$x_{42} = -57.0722665402146$$
$$x_{43} = -82.2050077689329$$
$$x_{44} = 68.5914396033772$$
$$x_{45} = -18.3259571459405$$
$$x_{46} = 44.5058959258554$$
$$x_{47} = 82.2050077689329$$
$$x_{48} = -46.6002910282486$$
$$x_{49} = 84.2994028713261$$
$$x_{50} = -16.2315620435473$$
$$x_{51} = 25.6563400043166$$
$$x_{52} = -69.6386371545737$$
$$x_{53} = -27.7507351067098$$
$$x_{54} = -5.75958653158129$$
$$x_{55} = 3.66519142918809$$
$$x_{56} = -40.317105721069$$
$$x_{57} = 49.7418836818384$$
$$x_{58} = -41.3643032722656$$
$$x_{59} = -382.750704962356$$
$$x_{60} = 24.60914245312$$
$$x_{61} = 40.317105721069$$
$$x_{62} = -78.0162175641465$$
$$x_{63} = 46.6002910282486$$
$$x_{64} = 69.6386371545737$$
$$x_{65} = 2.61799387799149$$
$$x_{66} = -43.4586983746588$$
$$x_{67} = 27.7507351067098$$
$$x_{68} = -93.7241808320955$$
$$x_{69} = 88.4881930761125$$
$$x_{70} = -137.706477982353$$
$$x_{71} = 56.025068989018$$
$$x_{72} = 91.6297857297023$$
$$x_{73} = 78.0162175641465$$
$$x_{74} = -97.9129710368819$$
$$x_{75} = -9.94837673636768$$
$$x_{76} = -56.025068989018$$
$$x_{77} = -47.6474885794452$$
$$x_{78} = 16.2315620435473$$
$$x_{79} = -66.497044500984$$
$$x_{80} = 5.75958653158129$$
$$x_{81} = -24.60914245312$$
$$x_{82} = -38.2227106186758$$
$$x_{83} = 9.94837673636768$$
$$x_{84} = 53.9306738866248$$
$$x_{1} = 96.8657734856853$$
$$x_{2} = -100.007366139275$$
$$x_{3} = -71.733032256967$$
$$x_{4} = -65.4498469497874$$
$$x_{5} = -91.6297857297023$$
$$x_{6} = 22.5147473507269$$
$$x_{7} = 93.7241808320955$$
$$x_{8} = 66.497044500984$$
$$x_{9} = -75.9218224617533$$
$$x_{10} = -31.9395253114962$$
$$x_{11} = -87.4409955249159$$
$$x_{12} = -90.5825881785057$$
$$x_{13} = 60.2138591938044$$
$$x_{14} = -49.7418836818384$$
$$x_{15} = -25.6563400043166$$
$$x_{16} = -34.0339204138894$$
$$x_{17} = -84.2994028713261$$
$$x_{18} = -68.5914396033772$$
$$x_{19} = 97.9129710368819$$
$$x_{20} = -60.2138591938044$$
$$x_{21} = -53.9306738866248$$
$$x_{22} = -62.3082542961976$$
$$x_{23} = -63.3554518473942$$
$$x_{24} = -2.61799387799149$$
$$x_{25} = 34.0339204138894$$
$$x_{26} = -19.3731546971371$$
$$x_{27} = 100.007366139275$$
$$x_{28} = -12.0427718387609$$
$$x_{29} = 0.523598775598299$$
$$x_{30} = 12.0427718387609$$
$$x_{31} = 75.9218224617533$$
$$x_{32} = 62.3082542961976$$
$$x_{33} = 47.6474885794452$$
$$x_{34} = 90.5825881785057$$
$$x_{35} = 71.733032256967$$
$$x_{36} = 18.3259571459405$$
$$x_{37} = 38.2227106186758$$
$$x_{38} = -3.66519142918809$$
$$x_{39} = 31.9395253114962$$
$$x_{40} = -85.3466004225227$$
$$x_{41} = -21.4675497995303$$
$$x_{42} = -57.0722665402146$$
$$x_{43} = -82.2050077689329$$
$$x_{44} = 68.5914396033772$$
$$x_{45} = -18.3259571459405$$
$$x_{46} = 44.5058959258554$$
$$x_{47} = 82.2050077689329$$
$$x_{48} = -46.6002910282486$$
$$x_{49} = 84.2994028713261$$
$$x_{50} = -16.2315620435473$$
$$x_{51} = 25.6563400043166$$
$$x_{52} = -69.6386371545737$$
$$x_{53} = -27.7507351067098$$
$$x_{54} = -5.75958653158129$$
$$x_{55} = 3.66519142918809$$
$$x_{56} = -40.317105721069$$
$$x_{57} = 49.7418836818384$$
$$x_{58} = -41.3643032722656$$
$$x_{59} = -382.750704962356$$
$$x_{60} = 24.60914245312$$
$$x_{61} = 40.317105721069$$
$$x_{62} = -78.0162175641465$$
$$x_{63} = 46.6002910282486$$
$$x_{64} = 69.6386371545737$$
$$x_{65} = 2.61799387799149$$
$$x_{66} = -43.4586983746588$$
$$x_{67} = 27.7507351067098$$
$$x_{68} = -93.7241808320955$$
$$x_{69} = 88.4881930761125$$
$$x_{70} = -137.706477982353$$
$$x_{71} = 56.025068989018$$
$$x_{72} = 91.6297857297023$$
$$x_{73} = 78.0162175641465$$
$$x_{74} = -97.9129710368819$$
$$x_{75} = -9.94837673636768$$
$$x_{76} = -56.025068989018$$
$$x_{77} = -47.6474885794452$$
$$x_{78} = 16.2315620435473$$
$$x_{79} = -66.497044500984$$
$$x_{80} = 5.75958653158129$$
$$x_{81} = -24.60914245312$$
$$x_{82} = -38.2227106186758$$
$$x_{83} = 9.94837673636768$$
$$x_{84} = 53.9306738866248$$
This roots
$$x_{59} = -382.750704962356$$
$$x_{70} = -137.706477982353$$
$$x_{2} = -100.007366139275$$
$$x_{74} = -97.9129710368819$$
$$x_{68} = -93.7241808320955$$
$$x_{5} = -91.6297857297023$$
$$x_{12} = -90.5825881785057$$
$$x_{11} = -87.4409955249159$$
$$x_{40} = -85.3466004225227$$
$$x_{17} = -84.2994028713261$$
$$x_{43} = -82.2050077689329$$
$$x_{62} = -78.0162175641465$$
$$x_{9} = -75.9218224617533$$
$$x_{3} = -71.733032256967$$
$$x_{52} = -69.6386371545737$$
$$x_{18} = -68.5914396033772$$
$$x_{79} = -66.497044500984$$
$$x_{4} = -65.4498469497874$$
$$x_{23} = -63.3554518473942$$
$$x_{22} = -62.3082542961976$$
$$x_{20} = -60.2138591938044$$
$$x_{42} = -57.0722665402146$$
$$x_{76} = -56.025068989018$$
$$x_{21} = -53.9306738866248$$
$$x_{14} = -49.7418836818384$$
$$x_{77} = -47.6474885794452$$
$$x_{48} = -46.6002910282486$$
$$x_{66} = -43.4586983746588$$
$$x_{58} = -41.3643032722656$$
$$x_{56} = -40.317105721069$$
$$x_{82} = -38.2227106186758$$
$$x_{16} = -34.0339204138894$$
$$x_{10} = -31.9395253114962$$
$$x_{53} = -27.7507351067098$$
$$x_{15} = -25.6563400043166$$
$$x_{81} = -24.60914245312$$
$$x_{41} = -21.4675497995303$$
$$x_{26} = -19.3731546971371$$
$$x_{45} = -18.3259571459405$$
$$x_{50} = -16.2315620435473$$
$$x_{28} = -12.0427718387609$$
$$x_{75} = -9.94837673636768$$
$$x_{54} = -5.75958653158129$$
$$x_{38} = -3.66519142918809$$
$$x_{24} = -2.61799387799149$$
$$x_{29} = 0.523598775598299$$
$$x_{65} = 2.61799387799149$$
$$x_{55} = 3.66519142918809$$
$$x_{80} = 5.75958653158129$$
$$x_{83} = 9.94837673636768$$
$$x_{30} = 12.0427718387609$$
$$x_{78} = 16.2315620435473$$
$$x_{36} = 18.3259571459405$$
$$x_{6} = 22.5147473507269$$
$$x_{60} = 24.60914245312$$
$$x_{51} = 25.6563400043166$$
$$x_{67} = 27.7507351067098$$
$$x_{39} = 31.9395253114962$$
$$x_{25} = 34.0339204138894$$
$$x_{37} = 38.2227106186758$$
$$x_{61} = 40.317105721069$$
$$x_{46} = 44.5058959258554$$
$$x_{63} = 46.6002910282486$$
$$x_{33} = 47.6474885794452$$
$$x_{57} = 49.7418836818384$$
$$x_{84} = 53.9306738866248$$
$$x_{71} = 56.025068989018$$
$$x_{13} = 60.2138591938044$$
$$x_{32} = 62.3082542961976$$
$$x_{8} = 66.497044500984$$
$$x_{44} = 68.5914396033772$$
$$x_{64} = 69.6386371545737$$
$$x_{35} = 71.733032256967$$
$$x_{31} = 75.9218224617533$$
$$x_{73} = 78.0162175641465$$
$$x_{47} = 82.2050077689329$$
$$x_{49} = 84.2994028713261$$
$$x_{69} = 88.4881930761125$$
$$x_{34} = 90.5825881785057$$
$$x_{72} = 91.6297857297023$$
$$x_{7} = 93.7241808320955$$
$$x_{1} = 96.8657734856853$$
$$x_{19} = 97.9129710368819$$
$$x_{27} = 100.007366139275$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} \leq x_{59}$$
For example, let's take the point
$$x_{0} = x_{59} - \frac{1}{10}$$
=
$$-382.750704962356 + - \frac{1}{10}$$
=
$$-382.850704962356$$
substitute to the expression
$$\left|{\sin{\left(x \right)}}\right| \geq \frac{1}{2}$$
$$\left|{\sin{\left(-382.850704962356 \right)}}\right| \geq \frac{1}{2}$$
0.411043807676265 >= 1/2
but
0.411043807676265 < 1/2
Then
$$x \leq -382.750704962356$$
no execute
one of the solutions of our inequality is:
$$x \geq -382.750704962356 \wedge x \leq -137.706477982353$$
_____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____
/ \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \
-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------
x59 x70 x2 x74 x68 x5 x12 x11 x40 x17 x43 x62 x9 x3 x52 x18 x79 x4 x23 x22 x20 x42 x76 x21 x14 x77 x48 x66 x58 x56 x82 x16 x10 x53 x15 x81 x41 x26 x45 x50 x28 x75 x54 x38 x24 x29 x65 x55 x80 x83 x30 x78 x36 x6 x60 x51 x67 x39 x25 x37 x61 x46 x63 x33 x57 x84 x71 x13 x32 x8 x44 x64 x35 x31 x73 x47 x49 x69 x34 x72 x7 x1 x19 x27Other solutions will get with the changeover to the next point
etc.
The answer:
$$x \geq -382.750704962356 \wedge x \leq -137.706477982353$$
$$x \geq -100.007366139275 \wedge x \leq -97.9129710368819$$
$$x \geq -93.7241808320955 \wedge x \leq -91.6297857297023$$
$$x \geq -90.5825881785057 \wedge x \leq -87.4409955249159$$
$$x \geq -85.3466004225227 \wedge x \leq -84.2994028713261$$
$$x \geq -82.2050077689329 \wedge x \leq -78.0162175641465$$
$$x \geq -75.9218224617533 \wedge x \leq -71.733032256967$$
$$x \geq -69.6386371545737 \wedge x \leq -68.5914396033772$$
$$x \geq -66.497044500984 \wedge x \leq -65.4498469497874$$
$$x \geq -63.3554518473942 \wedge x \leq -62.3082542961976$$
$$x \geq -60.2138591938044 \wedge x \leq -57.0722665402146$$
$$x \geq -56.025068989018 \wedge x \leq -53.9306738866248$$
$$x \geq -49.7418836818384 \wedge x \leq -47.6474885794452$$
$$x \geq -46.6002910282486 \wedge x \leq -43.4586983746588$$
$$x \geq -41.3643032722656 \wedge x \leq -40.317105721069$$
$$x \geq -38.2227106186758 \wedge x \leq -34.0339204138894$$
$$x \geq -31.9395253114962 \wedge x \leq -27.7507351067098$$
$$x \geq -25.6563400043166 \wedge x \leq -24.60914245312$$
$$x \geq -21.4675497995303 \wedge x \leq -19.3731546971371$$
$$x \geq -18.3259571459405 \wedge x \leq -16.2315620435473$$
$$x \geq -12.0427718387609 \wedge x \leq -9.94837673636768$$
$$x \geq -5.75958653158129 \wedge x \leq -3.66519142918809$$
$$x \geq -2.61799387799149 \wedge x \leq 0.523598775598299$$
$$x \geq 2.61799387799149 \wedge x \leq 3.66519142918809$$
$$x \geq 5.75958653158129 \wedge x \leq 9.94837673636768$$
$$x \geq 12.0427718387609 \wedge x \leq 16.2315620435473$$
$$x \geq 18.3259571459405 \wedge x \leq 22.5147473507269$$
$$x \geq 24.60914245312 \wedge x \leq 25.6563400043166$$
$$x \geq 27.7507351067098 \wedge x \leq 31.9395253114962$$
$$x \geq 34.0339204138894 \wedge x \leq 38.2227106186758$$
$$x \geq 40.317105721069 \wedge x \leq 44.5058959258554$$
$$x \geq 46.6002910282486 \wedge x \leq 47.6474885794452$$
$$x \geq 49.7418836818384 \wedge x \leq 53.9306738866248$$
$$x \geq 56.025068989018 \wedge x \leq 60.2138591938044$$
$$x \geq 62.3082542961976 \wedge x \leq 66.497044500984$$
$$x \geq 68.5914396033772 \wedge x \leq 69.6386371545737$$
$$x \geq 71.733032256967 \wedge x \leq 75.9218224617533$$
$$x \geq 78.0162175641465 \wedge x \leq 82.2050077689329$$
$$x \geq 84.2994028713261 \wedge x \leq 88.4881930761125$$
$$x \geq 90.5825881785057 \wedge x \leq 91.6297857297023$$
$$x \geq 93.7241808320955 \wedge x \leq 96.8657734856853$$
$$x \geq 97.9129710368819 \wedge x \leq 100.007366139275$$
Solving inequality on a graph