sin(t)>=1/2 inequation

Given the inequality:
$$\sin{\left(t \right)} \geq \frac{1}{2}$$
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:
$$k_{1} = 38.2227106186758$$
$$k_{2} = -28.7979326579064$$
$$k_{3} = -47.6474885794452$$
$$k_{4} = 21.4675497995303$$
$$k_{5} = 34.0339204138894$$
$$k_{6} = -74.8746249105567$$
$$k_{7} = -627.79493194236$$
$$k_{8} = 71.733032256967$$
$$k_{9} = 78.0162175641465$$
$$k_{10} = 44.5058959258554$$
$$k_{11} = 40.317105721069$$
$$k_{12} = -16.2315620435473$$
$$k_{13} = -60.2138591938044$$
$$k_{14} = 84.2994028713261$$
$$k_{15} = -56.025068989018$$
$$k_{16} = 0.523598775598299$$
$$k_{17} = 8.90117918517108$$
$$k_{18} = 50.789081233035$$
$$k_{19} = 75.9218224617533$$
$$k_{20} = -37.1755130674792$$
$$k_{21} = 6.80678408277789$$
$$k_{22} = -41.3643032722656$$
$$k_{23} = 31.9395253114962$$
$$k_{24} = 63.3554518473942$$
$$k_{25} = -62.3082542961976$$
$$k_{26} = -24.60914245312$$
$$k_{27} = -5.75958653158129$$
$$k_{28} = -18.3259571459405$$
$$k_{29} = 90.5825881785057$$
$$k_{30} = -100.007366139275$$
$$k_{31} = -79.0634151153431$$
$$k_{32} = -85.3466004225227$$
$$k_{33} = 94.7713783832921$$
$$k_{34} = -97.9129710368819$$
$$k_{35} = -9.94837673636768$$
$$k_{36} = 25.6563400043166$$
$$k_{37} = -81.1578102177363$$
$$k_{38} = -30.8923277602996$$
$$k_{39} = -72.7802298081635$$
$$k_{40} = 46.6002910282486$$
$$k_{41} = -91.6297857297023$$
$$k_{42} = 13.0899693899575$$
$$k_{43} = 52.8834763354282$$
$$k_{44} = 96.8657734856853$$
$$k_{45} = -2650.98060085419$$
$$k_{46} = 19.3731546971371$$
$$k_{47} = 2.61799387799149$$
$$k_{48} = -35.081117965086$$
$$k_{49} = 57.0722665402146$$
$$k_{50} = 27.7507351067098$$
$$k_{51} = 138.753675533549$$
$$k_{52} = 101.054563690472$$
$$k_{53} = -4454.25478401473$$
$$k_{54} = -12.0427718387609$$
$$k_{55} = -68.5914396033772$$
$$k_{56} = -87.4409955249159$$
$$k_{57} = -22.5147473507269$$
$$k_{58} = 88.4881930761125$$
$$k_{59} = 15.1843644923507$$
$$k_{60} = -49.7418836818384$$
$$k_{61} = 65.4498469497874$$
$$k_{62} = 82.2050077689329$$
$$k_{63} = -53.9306738866248$$
$$k_{64} = 59.1666616426078$$
$$k_{65} = -3.66519142918809$$
$$k_{66} = 17438.4572213013$$
$$k_{67} = 69.6386371545737$$
$$k_{68} = -93.7241808320955$$
$$k_{69} = -66.497044500984$$
$$k_{70} = -43.4586983746588$$
$$k_{71} = 134.564885328763$$
$$k_{1} = 38.2227106186758$$
$$k_{2} = -28.7979326579064$$
$$k_{3} = -47.6474885794452$$
$$k_{4} = 21.4675497995303$$
$$k_{5} = 34.0339204138894$$
$$k_{6} = -74.8746249105567$$
$$k_{7} = -627.79493194236$$
$$k_{8} = 71.733032256967$$
$$k_{9} = 78.0162175641465$$
$$k_{10} = 44.5058959258554$$
$$k_{11} = 40.317105721069$$
$$k_{12} = -16.2315620435473$$
$$k_{13} = -60.2138591938044$$
$$k_{14} = 84.2994028713261$$
$$k_{15} = -56.025068989018$$
$$k_{16} = 0.523598775598299$$
$$k_{17} = 8.90117918517108$$
$$k_{18} = 50.789081233035$$
$$k_{19} = 75.9218224617533$$
$$k_{20} = -37.1755130674792$$
$$k_{21} = 6.80678408277789$$
$$k_{22} = -41.3643032722656$$
$$k_{23} = 31.9395253114962$$
$$k_{24} = 63.3554518473942$$
$$k_{25} = -62.3082542961976$$
$$k_{26} = -24.60914245312$$
$$k_{27} = -5.75958653158129$$
$$k_{28} = -18.3259571459405$$
$$k_{29} = 90.5825881785057$$
$$k_{30} = -100.007366139275$$
$$k_{31} = -79.0634151153431$$
$$k_{32} = -85.3466004225227$$
$$k_{33} = 94.7713783832921$$
$$k_{34} = -97.9129710368819$$
$$k_{35} = -9.94837673636768$$
$$k_{36} = 25.6563400043166$$
$$k_{37} = -81.1578102177363$$
$$k_{38} = -30.8923277602996$$
$$k_{39} = -72.7802298081635$$
$$k_{40} = 46.6002910282486$$
$$k_{41} = -91.6297857297023$$
$$k_{42} = 13.0899693899575$$
$$k_{43} = 52.8834763354282$$
$$k_{44} = 96.8657734856853$$
$$k_{45} = -2650.98060085419$$
$$k_{46} = 19.3731546971371$$
$$k_{47} = 2.61799387799149$$
$$k_{48} = -35.081117965086$$
$$k_{49} = 57.0722665402146$$
$$k_{50} = 27.7507351067098$$
$$k_{51} = 138.753675533549$$
$$k_{52} = 101.054563690472$$
$$k_{53} = -4454.25478401473$$
$$k_{54} = -12.0427718387609$$
$$k_{55} = -68.5914396033772$$
$$k_{56} = -87.4409955249159$$
$$k_{57} = -22.5147473507269$$
$$k_{58} = 88.4881930761125$$
$$k_{59} = 15.1843644923507$$
$$k_{60} = -49.7418836818384$$
$$k_{61} = 65.4498469497874$$
$$k_{62} = 82.2050077689329$$
$$k_{63} = -53.9306738866248$$
$$k_{64} = 59.1666616426078$$
$$k_{65} = -3.66519142918809$$
$$k_{66} = 17438.4572213013$$
$$k_{67} = 69.6386371545737$$
$$k_{68} = -93.7241808320955$$
$$k_{69} = -66.497044500984$$
$$k_{70} = -43.4586983746588$$
$$k_{71} = 134.564885328763$$
This roots
$$k_{53} = -4454.25478401473$$
$$k_{45} = -2650.98060085419$$
$$k_{7} = -627.79493194236$$
$$k_{30} = -100.007366139275$$
$$k_{34} = -97.9129710368819$$
$$k_{68} = -93.7241808320955$$
$$k_{41} = -91.6297857297023$$
$$k_{56} = -87.4409955249159$$
$$k_{32} = -85.3466004225227$$
$$k_{37} = -81.1578102177363$$
$$k_{31} = -79.0634151153431$$
$$k_{6} = -74.8746249105567$$
$$k_{39} = -72.7802298081635$$
$$k_{55} = -68.5914396033772$$
$$k_{69} = -66.497044500984$$
$$k_{25} = -62.3082542961976$$
$$k_{13} = -60.2138591938044$$
$$k_{15} = -56.025068989018$$
$$k_{63} = -53.9306738866248$$
$$k_{60} = -49.7418836818384$$
$$k_{3} = -47.6474885794452$$
$$k_{70} = -43.4586983746588$$
$$k_{22} = -41.3643032722656$$
$$k_{20} = -37.1755130674792$$
$$k_{48} = -35.081117965086$$
$$k_{38} = -30.8923277602996$$
$$k_{2} = -28.7979326579064$$
$$k_{26} = -24.60914245312$$
$$k_{57} = -22.5147473507269$$
$$k_{28} = -18.3259571459405$$
$$k_{12} = -16.2315620435473$$
$$k_{54} = -12.0427718387609$$
$$k_{35} = -9.94837673636768$$
$$k_{27} = -5.75958653158129$$
$$k_{65} = -3.66519142918809$$
$$k_{16} = 0.523598775598299$$
$$k_{47} = 2.61799387799149$$
$$k_{21} = 6.80678408277789$$
$$k_{17} = 8.90117918517108$$
$$k_{42} = 13.0899693899575$$
$$k_{59} = 15.1843644923507$$
$$k_{46} = 19.3731546971371$$
$$k_{4} = 21.4675497995303$$
$$k_{36} = 25.6563400043166$$
$$k_{50} = 27.7507351067098$$
$$k_{23} = 31.9395253114962$$
$$k_{5} = 34.0339204138894$$
$$k_{1} = 38.2227106186758$$
$$k_{11} = 40.317105721069$$
$$k_{10} = 44.5058959258554$$
$$k_{40} = 46.6002910282486$$
$$k_{18} = 50.789081233035$$
$$k_{43} = 52.8834763354282$$
$$k_{49} = 57.0722665402146$$
$$k_{64} = 59.1666616426078$$
$$k_{24} = 63.3554518473942$$
$$k_{61} = 65.4498469497874$$
$$k_{67} = 69.6386371545737$$
$$k_{8} = 71.733032256967$$
$$k_{19} = 75.9218224617533$$
$$k_{9} = 78.0162175641465$$
$$k_{62} = 82.2050077689329$$
$$k_{14} = 84.2994028713261$$
$$k_{58} = 88.4881930761125$$
$$k_{29} = 90.5825881785057$$
$$k_{33} = 94.7713783832921$$
$$k_{44} = 96.8657734856853$$
$$k_{52} = 101.054563690472$$
$$k_{71} = 134.564885328763$$
$$k_{51} = 138.753675533549$$
$$k_{66} = 17438.4572213013$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$k_{0} \leq k_{53}$$
For example, let's take the point
$$k_{0} = k_{53} - \frac{1}{10}$$
=
$$-4454.25478401473 + - \frac{1}{10}$$
=
$$-4454.35478401473$$
substitute to the expression
$$\sin{\left(t \right)} \geq \frac{1}{2}$$
$$\sin{\left(t \right)} \geq \frac{1}{2}$$

sin(t) >= 1/2

Then
$$k \leq -4454.25478401473$$
no execute
one of the solutions of our inequality is:
$$k \geq -4454.25478401473 \wedge k \leq -2650.98060085419$$

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       k53      k45      k7      k30      k34      k68      k41      k56      k32      k37      k31      k6      k39      k55      k69      k25      k13      k15      k63      k60      k3      k70      k22      k20      k48      k38      k2      k26      k57      k28      k12      k54      k35      k27      k65      k16      k47      k21      k17      k42      k59      k46      k4      k36      k50      k23      k5      k1      k11      k10      k40      k18      k43      k49      k64      k24      k61      k67      k8      k19      k9      k62      k14      k58      k29      k33      k44      k52      k71      k51      k66

Other solutions will get with the changeover to the next point
etc.
The answer:
$$k \geq -4454.25478401473 \wedge k \leq -2650.98060085419$$
$$k \geq -627.79493194236 \wedge k \leq -100.007366139275$$
$$k \geq -97.9129710368819 \wedge k \leq -93.7241808320955$$
$$k \geq -91.6297857297023 \wedge k \leq -87.4409955249159$$
$$k \geq -85.3466004225227 \wedge k \leq -81.1578102177363$$
$$k \geq -79.0634151153431 \wedge k \leq -74.8746249105567$$
$$k \geq -72.7802298081635 \wedge k \leq -68.5914396033772$$
$$k \geq -66.497044500984 \wedge k \leq -62.3082542961976$$
$$k \geq -60.2138591938044 \wedge k \leq -56.025068989018$$
$$k \geq -53.9306738866248 \wedge k \leq -49.7418836818384$$
$$k \geq -47.6474885794452 \wedge k \leq -43.4586983746588$$
$$k \geq -41.3643032722656 \wedge k \leq -37.1755130674792$$
$$k \geq -35.081117965086 \wedge k \leq -30.8923277602996$$
$$k \geq -28.7979326579064 \wedge k \leq -24.60914245312$$
$$k \geq -22.5147473507269 \wedge k \leq -18.3259571459405$$
$$k \geq -16.2315620435473 \wedge k \leq -12.0427718387609$$
$$k \geq -9.94837673636768 \wedge k \leq -5.75958653158129$$
$$k \geq -3.66519142918809 \wedge k \leq 0.523598775598299$$
$$k \geq 2.61799387799149 \wedge k \leq 6.80678408277789$$
$$k \geq 8.90117918517108 \wedge k \leq 13.0899693899575$$
$$k \geq 15.1843644923507 \wedge k \leq 19.3731546971371$$
$$k \geq 21.4675497995303 \wedge k \leq 25.6563400043166$$
$$k \geq 27.7507351067098 \wedge k \leq 31.9395253114962$$
$$k \geq 34.0339204138894 \wedge k \leq 38.2227106186758$$
$$k \geq 40.317105721069 \wedge k \leq 44.5058959258554$$
$$k \geq 46.6002910282486 \wedge k \leq 50.789081233035$$
$$k \geq 52.8834763354282 \wedge k \leq 57.0722665402146$$
$$k \geq 59.1666616426078 \wedge k \leq 63.3554518473942$$
$$k \geq 65.4498469497874 \wedge k \leq 69.6386371545737$$
$$k \geq 71.733032256967 \wedge k \leq 75.9218224617533$$
$$k \geq 78.0162175641465 \wedge k \leq 82.2050077689329$$
$$k \geq 84.2994028713261 \wedge k \leq 88.4881930761125$$
$$k \geq 90.5825881785057 \wedge k \leq 94.7713783832921$$
$$k \geq 96.8657734856853 \wedge k \leq 101.054563690472$$
$$k \geq 134.564885328763 \wedge k \leq 138.753675533549$$
$$k \geq 17438.4572213013$$