Mister Exam
sint≤1/2 inequation
The solution
Detail solution
Given the inequality:
$$\sin{\left(t \right)} \leq \frac{1}{2}$$
To solve this inequality, we must first solve the corresponding equation:
$$\sin{\left(t \right)} = \frac{1}{2}$$
Solve:
Given the equation
$$\sin{\left(t \right)} = \frac{1}{2}$$
transform
$$\sin{\left(t \right)} - \frac{1}{2} = 0$$
$$\sin{\left(t \right)} - \frac{1}{2} = 0$$
Do replacement
$$w = \sin{\left(t \right)}$$
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
$$\sin{\left(t \right)} = w$$
substitute w:
$$x_{1} = -74.8746249105567$$
$$x_{2} = -37.1755130674792$$
$$x_{3} = 75.9218224617533$$
$$x_{4} = 78.0162175641465$$
$$x_{5} = -60.2138591938044$$
$$x_{6} = 88.4881930761125$$
$$x_{7} = -41.3643032722656$$
$$x_{8} = 52.8834763354282$$
$$x_{9} = 50.789081233035$$
$$x_{10} = 2.61799387799149$$
$$x_{11} = -62.3082542961976$$
$$x_{12} = 31.9395253114962$$
$$x_{13} = 57.0722665402146$$
$$x_{14} = -66.497044500984$$
$$x_{15} = 17438.4572213013$$
$$x_{16} = -24.60914245312$$
$$x_{17} = -49.7418836818384$$
$$x_{18} = 8.90117918517108$$
$$x_{19} = -12.0427718387609$$
$$x_{20} = -16.2315620435473$$
$$x_{21} = -18.3259571459405$$
$$x_{22} = -87.4409955249159$$
$$x_{23} = 25.6563400043166$$
$$x_{24} = -2650.98060085419$$
$$x_{25} = 134.564885328763$$
$$x_{26} = -28.7979326579064$$
$$x_{27} = 34.0339204138894$$
$$x_{28} = 46.6002910282486$$
$$x_{29} = 94.7713783832921$$
$$x_{30} = 0.523598775598299$$
$$x_{31} = -93.7241808320955$$
$$x_{32} = 138.753675533549$$
$$x_{33} = 40.317105721069$$
$$x_{34} = -68.5914396033772$$
$$x_{35} = -53.9306738866248$$
$$x_{36} = -47.6474885794452$$
$$x_{37} = -627.79493194236$$
$$x_{38} = 90.5825881785057$$
$$x_{39} = -56.025068989018$$
$$x_{40} = 84.2994028713261$$
$$x_{41} = 101.054563690472$$
$$x_{42} = -100.007366139275$$
$$x_{43} = -85.3466004225227$$
$$x_{44} = 13.0899693899575$$
$$x_{45} = -81.1578102177363$$
$$x_{46} = 27.7507351067098$$
$$x_{47} = -91.6297857297023$$
$$x_{48} = -4454.25478401473$$
$$x_{49} = 21.4675497995303$$
$$x_{50} = -3.66519142918809$$
$$x_{51} = -30.8923277602996$$
$$x_{52} = 82.2050077689329$$
$$x_{53} = 65.4498469497874$$
$$x_{54} = 69.6386371545737$$
$$x_{55} = -97.9129710368819$$
$$x_{56} = -72.7802298081635$$
$$x_{57} = 15.1843644923507$$
$$x_{58} = 19.3731546971371$$
$$x_{59} = 38.2227106186758$$
$$x_{60} = -35.081117965086$$
$$x_{61} = 44.5058959258554$$
$$x_{62} = -9.94837673636768$$
$$x_{63} = -79.0634151153431$$
$$x_{64} = -43.4586983746588$$
$$x_{65} = 96.8657734856853$$
$$x_{66} = -5.75958653158129$$
$$x_{67} = 59.1666616426078$$
$$x_{68} = 71.733032256967$$
$$x_{69} = -22.5147473507269$$
$$x_{70} = 6.80678408277789$$
$$x_{71} = 63.3554518473942$$
$$x_{1} = -74.8746249105567$$
$$x_{2} = -37.1755130674792$$
$$x_{3} = 75.9218224617533$$
$$x_{4} = 78.0162175641465$$
$$x_{5} = -60.2138591938044$$
$$x_{6} = 88.4881930761125$$
$$x_{7} = -41.3643032722656$$
$$x_{8} = 52.8834763354282$$
$$x_{9} = 50.789081233035$$
$$x_{10} = 2.61799387799149$$
$$x_{11} = -62.3082542961976$$
$$x_{12} = 31.9395253114962$$
$$x_{13} = 57.0722665402146$$
$$x_{14} = -66.497044500984$$
$$x_{15} = 17438.4572213013$$
$$x_{16} = -24.60914245312$$
$$x_{17} = -49.7418836818384$$
$$x_{18} = 8.90117918517108$$
$$x_{19} = -12.0427718387609$$
$$x_{20} = -16.2315620435473$$
$$x_{21} = -18.3259571459405$$
$$x_{22} = -87.4409955249159$$
$$x_{23} = 25.6563400043166$$
$$x_{24} = -2650.98060085419$$
$$x_{25} = 134.564885328763$$
$$x_{26} = -28.7979326579064$$
$$x_{27} = 34.0339204138894$$
$$x_{28} = 46.6002910282486$$
$$x_{29} = 94.7713783832921$$
$$x_{30} = 0.523598775598299$$
$$x_{31} = -93.7241808320955$$
$$x_{32} = 138.753675533549$$
$$x_{33} = 40.317105721069$$
$$x_{34} = -68.5914396033772$$
$$x_{35} = -53.9306738866248$$
$$x_{36} = -47.6474885794452$$
$$x_{37} = -627.79493194236$$
$$x_{38} = 90.5825881785057$$
$$x_{39} = -56.025068989018$$
$$x_{40} = 84.2994028713261$$
$$x_{41} = 101.054563690472$$
$$x_{42} = -100.007366139275$$
$$x_{43} = -85.3466004225227$$
$$x_{44} = 13.0899693899575$$
$$x_{45} = -81.1578102177363$$
$$x_{46} = 27.7507351067098$$
$$x_{47} = -91.6297857297023$$
$$x_{48} = -4454.25478401473$$
$$x_{49} = 21.4675497995303$$
$$x_{50} = -3.66519142918809$$
$$x_{51} = -30.8923277602996$$
$$x_{52} = 82.2050077689329$$
$$x_{53} = 65.4498469497874$$
$$x_{54} = 69.6386371545737$$
$$x_{55} = -97.9129710368819$$
$$x_{56} = -72.7802298081635$$
$$x_{57} = 15.1843644923507$$
$$x_{58} = 19.3731546971371$$
$$x_{59} = 38.2227106186758$$
$$x_{60} = -35.081117965086$$
$$x_{61} = 44.5058959258554$$
$$x_{62} = -9.94837673636768$$
$$x_{63} = -79.0634151153431$$
$$x_{64} = -43.4586983746588$$
$$x_{65} = 96.8657734856853$$
$$x_{66} = -5.75958653158129$$
$$x_{67} = 59.1666616426078$$
$$x_{68} = 71.733032256967$$
$$x_{69} = -22.5147473507269$$
$$x_{70} = 6.80678408277789$$
$$x_{71} = 63.3554518473942$$
This roots
$$x_{48} = -4454.25478401473$$
$$x_{24} = -2650.98060085419$$
$$x_{37} = -627.79493194236$$
$$x_{42} = -100.007366139275$$
$$x_{55} = -97.9129710368819$$
$$x_{31} = -93.7241808320955$$
$$x_{47} = -91.6297857297023$$
$$x_{22} = -87.4409955249159$$
$$x_{43} = -85.3466004225227$$
$$x_{45} = -81.1578102177363$$
$$x_{63} = -79.0634151153431$$
$$x_{1} = -74.8746249105567$$
$$x_{56} = -72.7802298081635$$
$$x_{34} = -68.5914396033772$$
$$x_{14} = -66.497044500984$$
$$x_{11} = -62.3082542961976$$
$$x_{5} = -60.2138591938044$$
$$x_{39} = -56.025068989018$$
$$x_{35} = -53.9306738866248$$
$$x_{17} = -49.7418836818384$$
$$x_{36} = -47.6474885794452$$
$$x_{64} = -43.4586983746588$$
$$x_{7} = -41.3643032722656$$
$$x_{2} = -37.1755130674792$$
$$x_{60} = -35.081117965086$$
$$x_{51} = -30.8923277602996$$
$$x_{26} = -28.7979326579064$$
$$x_{16} = -24.60914245312$$
$$x_{69} = -22.5147473507269$$
$$x_{21} = -18.3259571459405$$
$$x_{20} = -16.2315620435473$$
$$x_{19} = -12.0427718387609$$
$$x_{62} = -9.94837673636768$$
$$x_{66} = -5.75958653158129$$
$$x_{50} = -3.66519142918809$$
$$x_{30} = 0.523598775598299$$
$$x_{10} = 2.61799387799149$$
$$x_{70} = 6.80678408277789$$
$$x_{18} = 8.90117918517108$$
$$x_{44} = 13.0899693899575$$
$$x_{57} = 15.1843644923507$$
$$x_{58} = 19.3731546971371$$
$$x_{49} = 21.4675497995303$$
$$x_{23} = 25.6563400043166$$
$$x_{46} = 27.7507351067098$$
$$x_{12} = 31.9395253114962$$
$$x_{27} = 34.0339204138894$$
$$x_{59} = 38.2227106186758$$
$$x_{33} = 40.317105721069$$
$$x_{61} = 44.5058959258554$$
$$x_{28} = 46.6002910282486$$
$$x_{9} = 50.789081233035$$
$$x_{8} = 52.8834763354282$$
$$x_{13} = 57.0722665402146$$
$$x_{67} = 59.1666616426078$$
$$x_{71} = 63.3554518473942$$
$$x_{53} = 65.4498469497874$$
$$x_{54} = 69.6386371545737$$
$$x_{68} = 71.733032256967$$
$$x_{3} = 75.9218224617533$$
$$x_{4} = 78.0162175641465$$
$$x_{52} = 82.2050077689329$$
$$x_{40} = 84.2994028713261$$
$$x_{6} = 88.4881930761125$$
$$x_{38} = 90.5825881785057$$
$$x_{29} = 94.7713783832921$$
$$x_{65} = 96.8657734856853$$
$$x_{41} = 101.054563690472$$
$$x_{25} = 134.564885328763$$
$$x_{32} = 138.753675533549$$
$$x_{15} = 17438.4572213013$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} \leq x_{48}$$
For example, let's take the point
$$x_{0} = x_{48} - \frac{1}{10}$$
=
$$-4454.25478401473 - \frac{1}{10}$$
=
$$-4454.35478401473$$
substitute to the expression
$$\sin{\left(t \right)} \leq \frac{1}{2}$$
$$\sin{\left(t \right)} \leq \frac{1}{2}$$
Then
$$x \leq -4454.25478401473$$
no execute
one of the solutions of our inequality is:
$$x \geq -4454.25478401473 \wedge x \leq -2650.98060085419$$
Other solutions will get with the changeover to the next point
etc.
The answer:
$$x \geq -4454.25478401473 \wedge x \leq -2650.98060085419$$
$$x \geq -627.79493194236 \wedge x \leq -100.007366139275$$
$$x \geq -97.9129710368819 \wedge x \leq -93.7241808320955$$
$$x \geq -91.6297857297023 \wedge x \leq -87.4409955249159$$
$$x \geq -85.3466004225227 \wedge x \leq -81.1578102177363$$
$$x \geq -79.0634151153431 \wedge x \leq -74.8746249105567$$
$$x \geq -72.7802298081635 \wedge x \leq -68.5914396033772$$
$$x \geq -66.497044500984 \wedge x \leq -62.3082542961976$$
$$x \geq -60.2138591938044 \wedge x \leq -56.025068989018$$
$$x \geq -53.9306738866248 \wedge x \leq -49.7418836818384$$
$$x \geq -47.6474885794452 \wedge x \leq -43.4586983746588$$
$$x \geq -41.3643032722656 \wedge x \leq -37.1755130674792$$
$$x \geq -35.081117965086 \wedge x \leq -30.8923277602996$$
$$x \geq -28.7979326579064 \wedge x \leq -24.60914245312$$
$$x \geq -22.5147473507269 \wedge x \leq -18.3259571459405$$
$$x \geq -16.2315620435473 \wedge x \leq -12.0427718387609$$
$$x \geq -9.94837673636768 \wedge x \leq -5.75958653158129$$
$$x \geq -3.66519142918809 \wedge x \leq 0.523598775598299$$
$$x \geq 2.61799387799149 \wedge x \leq 6.80678408277789$$
$$x \geq 8.90117918517108 \wedge x \leq 13.0899693899575$$
$$x \geq 15.1843644923507 \wedge x \leq 19.3731546971371$$
$$x \geq 21.4675497995303 \wedge x \leq 25.6563400043166$$
$$x \geq 27.7507351067098 \wedge x \leq 31.9395253114962$$
$$x \geq 34.0339204138894 \wedge x \leq 38.2227106186758$$
$$x \geq 40.317105721069 \wedge x \leq 44.5058959258554$$
$$x \geq 46.6002910282486 \wedge x \leq 50.789081233035$$
$$x \geq 52.8834763354282 \wedge x \leq 57.0722665402146$$
$$x \geq 59.1666616426078 \wedge x \leq 63.3554518473942$$
$$x \geq 65.4498469497874 \wedge x \leq 69.6386371545737$$
$$x \geq 71.733032256967 \wedge x \leq 75.9218224617533$$
$$x \geq 78.0162175641465 \wedge x \leq 82.2050077689329$$
$$x \geq 84.2994028713261 \wedge x \leq 88.4881930761125$$
$$x \geq 90.5825881785057 \wedge x \leq 94.7713783832921$$
$$x \geq 96.8657734856853 \wedge x \leq 101.054563690472$$
$$x \geq 134.564885328763 \wedge x \leq 138.753675533549$$
$$x \geq 17438.4572213013$$
$$\sin{\left(t \right)} \leq \frac{1}{2}$$
To solve this inequality, we must first solve the corresponding equation:
$$\sin{\left(t \right)} = \frac{1}{2}$$
Solve:
Given the equation
$$\sin{\left(t \right)} = \frac{1}{2}$$
transform
$$\sin{\left(t \right)} - \frac{1}{2} = 0$$
$$\sin{\left(t \right)} - \frac{1}{2} = 0$$
Do replacement
$$w = \sin{\left(t \right)}$$
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
$$\sin{\left(t \right)} = w$$
substitute w:
$$x_{1} = -74.8746249105567$$
$$x_{2} = -37.1755130674792$$
$$x_{3} = 75.9218224617533$$
$$x_{4} = 78.0162175641465$$
$$x_{5} = -60.2138591938044$$
$$x_{6} = 88.4881930761125$$
$$x_{7} = -41.3643032722656$$
$$x_{8} = 52.8834763354282$$
$$x_{9} = 50.789081233035$$
$$x_{10} = 2.61799387799149$$
$$x_{11} = -62.3082542961976$$
$$x_{12} = 31.9395253114962$$
$$x_{13} = 57.0722665402146$$
$$x_{14} = -66.497044500984$$
$$x_{15} = 17438.4572213013$$
$$x_{16} = -24.60914245312$$
$$x_{17} = -49.7418836818384$$
$$x_{18} = 8.90117918517108$$
$$x_{19} = -12.0427718387609$$
$$x_{20} = -16.2315620435473$$
$$x_{21} = -18.3259571459405$$
$$x_{22} = -87.4409955249159$$
$$x_{23} = 25.6563400043166$$
$$x_{24} = -2650.98060085419$$
$$x_{25} = 134.564885328763$$
$$x_{26} = -28.7979326579064$$
$$x_{27} = 34.0339204138894$$
$$x_{28} = 46.6002910282486$$
$$x_{29} = 94.7713783832921$$
$$x_{30} = 0.523598775598299$$
$$x_{31} = -93.7241808320955$$
$$x_{32} = 138.753675533549$$
$$x_{33} = 40.317105721069$$
$$x_{34} = -68.5914396033772$$
$$x_{35} = -53.9306738866248$$
$$x_{36} = -47.6474885794452$$
$$x_{37} = -627.79493194236$$
$$x_{38} = 90.5825881785057$$
$$x_{39} = -56.025068989018$$
$$x_{40} = 84.2994028713261$$
$$x_{41} = 101.054563690472$$
$$x_{42} = -100.007366139275$$
$$x_{43} = -85.3466004225227$$
$$x_{44} = 13.0899693899575$$
$$x_{45} = -81.1578102177363$$
$$x_{46} = 27.7507351067098$$
$$x_{47} = -91.6297857297023$$
$$x_{48} = -4454.25478401473$$
$$x_{49} = 21.4675497995303$$
$$x_{50} = -3.66519142918809$$
$$x_{51} = -30.8923277602996$$
$$x_{52} = 82.2050077689329$$
$$x_{53} = 65.4498469497874$$
$$x_{54} = 69.6386371545737$$
$$x_{55} = -97.9129710368819$$
$$x_{56} = -72.7802298081635$$
$$x_{57} = 15.1843644923507$$
$$x_{58} = 19.3731546971371$$
$$x_{59} = 38.2227106186758$$
$$x_{60} = -35.081117965086$$
$$x_{61} = 44.5058959258554$$
$$x_{62} = -9.94837673636768$$
$$x_{63} = -79.0634151153431$$
$$x_{64} = -43.4586983746588$$
$$x_{65} = 96.8657734856853$$
$$x_{66} = -5.75958653158129$$
$$x_{67} = 59.1666616426078$$
$$x_{68} = 71.733032256967$$
$$x_{69} = -22.5147473507269$$
$$x_{70} = 6.80678408277789$$
$$x_{71} = 63.3554518473942$$
$$x_{1} = -74.8746249105567$$
$$x_{2} = -37.1755130674792$$
$$x_{3} = 75.9218224617533$$
$$x_{4} = 78.0162175641465$$
$$x_{5} = -60.2138591938044$$
$$x_{6} = 88.4881930761125$$
$$x_{7} = -41.3643032722656$$
$$x_{8} = 52.8834763354282$$
$$x_{9} = 50.789081233035$$
$$x_{10} = 2.61799387799149$$
$$x_{11} = -62.3082542961976$$
$$x_{12} = 31.9395253114962$$
$$x_{13} = 57.0722665402146$$
$$x_{14} = -66.497044500984$$
$$x_{15} = 17438.4572213013$$
$$x_{16} = -24.60914245312$$
$$x_{17} = -49.7418836818384$$
$$x_{18} = 8.90117918517108$$
$$x_{19} = -12.0427718387609$$
$$x_{20} = -16.2315620435473$$
$$x_{21} = -18.3259571459405$$
$$x_{22} = -87.4409955249159$$
$$x_{23} = 25.6563400043166$$
$$x_{24} = -2650.98060085419$$
$$x_{25} = 134.564885328763$$
$$x_{26} = -28.7979326579064$$
$$x_{27} = 34.0339204138894$$
$$x_{28} = 46.6002910282486$$
$$x_{29} = 94.7713783832921$$
$$x_{30} = 0.523598775598299$$
$$x_{31} = -93.7241808320955$$
$$x_{32} = 138.753675533549$$
$$x_{33} = 40.317105721069$$
$$x_{34} = -68.5914396033772$$
$$x_{35} = -53.9306738866248$$
$$x_{36} = -47.6474885794452$$
$$x_{37} = -627.79493194236$$
$$x_{38} = 90.5825881785057$$
$$x_{39} = -56.025068989018$$
$$x_{40} = 84.2994028713261$$
$$x_{41} = 101.054563690472$$
$$x_{42} = -100.007366139275$$
$$x_{43} = -85.3466004225227$$
$$x_{44} = 13.0899693899575$$
$$x_{45} = -81.1578102177363$$
$$x_{46} = 27.7507351067098$$
$$x_{47} = -91.6297857297023$$
$$x_{48} = -4454.25478401473$$
$$x_{49} = 21.4675497995303$$
$$x_{50} = -3.66519142918809$$
$$x_{51} = -30.8923277602996$$
$$x_{52} = 82.2050077689329$$
$$x_{53} = 65.4498469497874$$
$$x_{54} = 69.6386371545737$$
$$x_{55} = -97.9129710368819$$
$$x_{56} = -72.7802298081635$$
$$x_{57} = 15.1843644923507$$
$$x_{58} = 19.3731546971371$$
$$x_{59} = 38.2227106186758$$
$$x_{60} = -35.081117965086$$
$$x_{61} = 44.5058959258554$$
$$x_{62} = -9.94837673636768$$
$$x_{63} = -79.0634151153431$$
$$x_{64} = -43.4586983746588$$
$$x_{65} = 96.8657734856853$$
$$x_{66} = -5.75958653158129$$
$$x_{67} = 59.1666616426078$$
$$x_{68} = 71.733032256967$$
$$x_{69} = -22.5147473507269$$
$$x_{70} = 6.80678408277789$$
$$x_{71} = 63.3554518473942$$
This roots
$$x_{48} = -4454.25478401473$$
$$x_{24} = -2650.98060085419$$
$$x_{37} = -627.79493194236$$
$$x_{42} = -100.007366139275$$
$$x_{55} = -97.9129710368819$$
$$x_{31} = -93.7241808320955$$
$$x_{47} = -91.6297857297023$$
$$x_{22} = -87.4409955249159$$
$$x_{43} = -85.3466004225227$$
$$x_{45} = -81.1578102177363$$
$$x_{63} = -79.0634151153431$$
$$x_{1} = -74.8746249105567$$
$$x_{56} = -72.7802298081635$$
$$x_{34} = -68.5914396033772$$
$$x_{14} = -66.497044500984$$
$$x_{11} = -62.3082542961976$$
$$x_{5} = -60.2138591938044$$
$$x_{39} = -56.025068989018$$
$$x_{35} = -53.9306738866248$$
$$x_{17} = -49.7418836818384$$
$$x_{36} = -47.6474885794452$$
$$x_{64} = -43.4586983746588$$
$$x_{7} = -41.3643032722656$$
$$x_{2} = -37.1755130674792$$
$$x_{60} = -35.081117965086$$
$$x_{51} = -30.8923277602996$$
$$x_{26} = -28.7979326579064$$
$$x_{16} = -24.60914245312$$
$$x_{69} = -22.5147473507269$$
$$x_{21} = -18.3259571459405$$
$$x_{20} = -16.2315620435473$$
$$x_{19} = -12.0427718387609$$
$$x_{62} = -9.94837673636768$$
$$x_{66} = -5.75958653158129$$
$$x_{50} = -3.66519142918809$$
$$x_{30} = 0.523598775598299$$
$$x_{10} = 2.61799387799149$$
$$x_{70} = 6.80678408277789$$
$$x_{18} = 8.90117918517108$$
$$x_{44} = 13.0899693899575$$
$$x_{57} = 15.1843644923507$$
$$x_{58} = 19.3731546971371$$
$$x_{49} = 21.4675497995303$$
$$x_{23} = 25.6563400043166$$
$$x_{46} = 27.7507351067098$$
$$x_{12} = 31.9395253114962$$
$$x_{27} = 34.0339204138894$$
$$x_{59} = 38.2227106186758$$
$$x_{33} = 40.317105721069$$
$$x_{61} = 44.5058959258554$$
$$x_{28} = 46.6002910282486$$
$$x_{9} = 50.789081233035$$
$$x_{8} = 52.8834763354282$$
$$x_{13} = 57.0722665402146$$
$$x_{67} = 59.1666616426078$$
$$x_{71} = 63.3554518473942$$
$$x_{53} = 65.4498469497874$$
$$x_{54} = 69.6386371545737$$
$$x_{68} = 71.733032256967$$
$$x_{3} = 75.9218224617533$$
$$x_{4} = 78.0162175641465$$
$$x_{52} = 82.2050077689329$$
$$x_{40} = 84.2994028713261$$
$$x_{6} = 88.4881930761125$$
$$x_{38} = 90.5825881785057$$
$$x_{29} = 94.7713783832921$$
$$x_{65} = 96.8657734856853$$
$$x_{41} = 101.054563690472$$
$$x_{25} = 134.564885328763$$
$$x_{32} = 138.753675533549$$
$$x_{15} = 17438.4572213013$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} \leq x_{48}$$
For example, let's take the point
$$x_{0} = x_{48} - \frac{1}{10}$$
=
$$-4454.25478401473 - \frac{1}{10}$$
=
$$-4454.35478401473$$
substitute to the expression
$$\sin{\left(t \right)} \leq \frac{1}{2}$$
$$\sin{\left(t \right)} \leq \frac{1}{2}$$
sin(t) <= 1/2
Then
$$x \leq -4454.25478401473$$
no execute
one of the solutions of our inequality is:
$$x \geq -4454.25478401473 \wedge x \leq -2650.98060085419$$
_____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____
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x_48 x_24 x_37 x_42 x_55 x_31 x_47 x_22 x_43 x_45 x_63 x_1 x_56 x_34 x_14 x_11 x_5 x_39 x_35 x_17 x_36 x_64 x_7 x_2 x_60 x_51 x_26 x_16 x_69 x_21 x_20 x_19 x_62 x_66 x_50 x_30 x_10 x_70 x_18 x_44 x_57 x_58 x_49 x_23 x_46 x_12 x_27 x_59 x_33 x_61 x_28 x_9 x_8 x_13 x_67 x_71 x_53 x_54 x_68 x_3 x_4 x_52 x_40 x_6 x_38 x_29 x_65 x_41 x_25 x_32 x_15Other solutions will get with the changeover to the next point
etc.
The answer:
$$x \geq -4454.25478401473 \wedge x \leq -2650.98060085419$$
$$x \geq -627.79493194236 \wedge x \leq -100.007366139275$$
$$x \geq -97.9129710368819 \wedge x \leq -93.7241808320955$$
$$x \geq -91.6297857297023 \wedge x \leq -87.4409955249159$$
$$x \geq -85.3466004225227 \wedge x \leq -81.1578102177363$$
$$x \geq -79.0634151153431 \wedge x \leq -74.8746249105567$$
$$x \geq -72.7802298081635 \wedge x \leq -68.5914396033772$$
$$x \geq -66.497044500984 \wedge x \leq -62.3082542961976$$
$$x \geq -60.2138591938044 \wedge x \leq -56.025068989018$$
$$x \geq -53.9306738866248 \wedge x \leq -49.7418836818384$$
$$x \geq -47.6474885794452 \wedge x \leq -43.4586983746588$$
$$x \geq -41.3643032722656 \wedge x \leq -37.1755130674792$$
$$x \geq -35.081117965086 \wedge x \leq -30.8923277602996$$
$$x \geq -28.7979326579064 \wedge x \leq -24.60914245312$$
$$x \geq -22.5147473507269 \wedge x \leq -18.3259571459405$$
$$x \geq -16.2315620435473 \wedge x \leq -12.0427718387609$$
$$x \geq -9.94837673636768 \wedge x \leq -5.75958653158129$$
$$x \geq -3.66519142918809 \wedge x \leq 0.523598775598299$$
$$x \geq 2.61799387799149 \wedge x \leq 6.80678408277789$$
$$x \geq 8.90117918517108 \wedge x \leq 13.0899693899575$$
$$x \geq 15.1843644923507 \wedge x \leq 19.3731546971371$$
$$x \geq 21.4675497995303 \wedge x \leq 25.6563400043166$$
$$x \geq 27.7507351067098 \wedge x \leq 31.9395253114962$$
$$x \geq 34.0339204138894 \wedge x \leq 38.2227106186758$$
$$x \geq 40.317105721069 \wedge x \leq 44.5058959258554$$
$$x \geq 46.6002910282486 \wedge x \leq 50.789081233035$$
$$x \geq 52.8834763354282 \wedge x \leq 57.0722665402146$$
$$x \geq 59.1666616426078 \wedge x \leq 63.3554518473942$$
$$x \geq 65.4498469497874 \wedge x \leq 69.6386371545737$$
$$x \geq 71.733032256967 \wedge x \leq 75.9218224617533$$
$$x \geq 78.0162175641465 \wedge x \leq 82.2050077689329$$
$$x \geq 84.2994028713261 \wedge x \leq 88.4881930761125$$
$$x \geq 90.5825881785057 \wedge x \leq 94.7713783832921$$
$$x \geq 96.8657734856853 \wedge x \leq 101.054563690472$$
$$x \geq 134.564885328763 \wedge x \leq 138.753675533549$$
$$x \geq 17438.4572213013$$
Rapid solution
[src]
/ / pi\ /5*pi \\ Or|And|0 <= t, t <= --|, And|---- <= t, t < 2*pi|| \ \ 6 / \ 6 //
$$\left(0 \leq t \wedge t \leq \frac{\pi}{6}\right) \vee \left(\frac{5 \pi}{6} \leq t \wedge t < 2 \pi\right)$$
((0 <= t)∧(t <= pi/6))∨((5*pi/6 <= t)∧(t < 2*pi))
Rapid solution 2
[src]
pi 5*pi
[0, --] U [----, 2*pi)
6 6
$$x\ in\ \left[0, \frac{\pi}{6}\right] \cup \left[\frac{5 \pi}{6}, 2 \pi\right)$$
x in Union(Interval(0, pi/6), Interval.Ropen(5*pi/6, 2*pi))