Mister Exam
Other calculators
- Square Inequality Step-by-Step
- Inequality with the module Step by Step
- Logarithmic inequality online
-
How to use it?
- Inequation:
-
49-25x<0
- (abs((x+2)/(x-6)))<=(abs((x+2)/(x-3)))
- 4(x-5)>=3/(x+2)
- -6*x^2+13*x-5>=0
- Derivative of:
-
sin(t)
- Limit of the function:
-
sin(t)
- Integral of d{x}:
-
sin(t)
-
Identical expressions
- sin(t)>= one / two
- sinus of (t) greater than or equal to 1 divide by 2
- sinus of (t) greater than or equal to one divide by two
- sint>=1/2
- sin(t)>=1 divide by 2
-
Similar expressions
- sint<√3/2
- sint>0,5
sin(t)>=1/2 inequation
The solution
Detail solution
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
$$\sin{\left(t \right)} \geq \frac{1}{2}$$
$$\sin{\left(t \right)} \geq \frac{1}{2}$$
Then
$$k \leq -4454.25478401473$$
no execute
one of the solutions of our inequality is:
$$k \geq -4454.25478401473 \wedge k \leq -2650.98060085419$$
Other solutions will get with the changeover to the next point
etc.
The answer:
$$k \geq -4454.25478401473 \wedge k \leq -2650.98060085419$$
$$k \geq -627.79493194236 \wedge k \leq -100.007366139275$$
$$k \geq -97.9129710368819 \wedge k \leq -93.7241808320955$$
$$k \geq -91.6297857297023 \wedge k \leq -87.4409955249159$$
$$k \geq -85.3466004225227 \wedge k \leq -81.1578102177363$$
$$k \geq -79.0634151153431 \wedge k \leq -74.8746249105567$$
$$k \geq -72.7802298081635 \wedge k \leq -68.5914396033772$$
$$k \geq -66.497044500984 \wedge k \leq -62.3082542961976$$
$$k \geq -60.2138591938044 \wedge k \leq -56.025068989018$$
$$k \geq -53.9306738866248 \wedge k \leq -49.7418836818384$$
$$k \geq -47.6474885794452 \wedge k \leq -43.4586983746588$$
$$k \geq -41.3643032722656 \wedge k \leq -37.1755130674792$$
$$k \geq -35.081117965086 \wedge k \leq -30.8923277602996$$
$$k \geq -28.7979326579064 \wedge k \leq -24.60914245312$$
$$k \geq -22.5147473507269 \wedge k \leq -18.3259571459405$$
$$k \geq -16.2315620435473 \wedge k \leq -12.0427718387609$$
$$k \geq -9.94837673636768 \wedge k \leq -5.75958653158129$$
$$k \geq -3.66519142918809 \wedge k \leq 0.523598775598299$$
$$k \geq 2.61799387799149 \wedge k \leq 6.80678408277789$$
$$k \geq 8.90117918517108 \wedge k \leq 13.0899693899575$$
$$k \geq 15.1843644923507 \wedge k \leq 19.3731546971371$$
$$k \geq 21.4675497995303 \wedge k \leq 25.6563400043166$$
$$k \geq 27.7507351067098 \wedge k \leq 31.9395253114962$$
$$k \geq 34.0339204138894 \wedge k \leq 38.2227106186758$$
$$k \geq 40.317105721069 \wedge k \leq 44.5058959258554$$
$$k \geq 46.6002910282486 \wedge k \leq 50.789081233035$$
$$k \geq 52.8834763354282 \wedge k \leq 57.0722665402146$$
$$k \geq 59.1666616426078 \wedge k \leq 63.3554518473942$$
$$k \geq 65.4498469497874 \wedge k \leq 69.6386371545737$$
$$k \geq 71.733032256967 \wedge k \leq 75.9218224617533$$
$$k \geq 78.0162175641465 \wedge k \leq 82.2050077689329$$
$$k \geq 84.2994028713261 \wedge k \leq 88.4881930761125$$
$$k \geq 90.5825881785057 \wedge k \leq 94.7713783832921$$
$$k \geq 96.8657734856853 \wedge k \leq 101.054563690472$$
$$k \geq 134.564885328763 \wedge k \leq 138.753675533549$$
$$k \geq 17438.4572213013$$
$$\sin{\left(t \right)} \geq \frac{1}{2}$$
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 k66Other solutions will get with the changeover to the next point
etc.
The answer:
$$k \geq -4454.25478401473 \wedge k \leq -2650.98060085419$$
$$k \geq -627.79493194236 \wedge k \leq -100.007366139275$$
$$k \geq -97.9129710368819 \wedge k \leq -93.7241808320955$$
$$k \geq -91.6297857297023 \wedge k \leq -87.4409955249159$$
$$k \geq -85.3466004225227 \wedge k \leq -81.1578102177363$$
$$k \geq -79.0634151153431 \wedge k \leq -74.8746249105567$$
$$k \geq -72.7802298081635 \wedge k \leq -68.5914396033772$$
$$k \geq -66.497044500984 \wedge k \leq -62.3082542961976$$
$$k \geq -60.2138591938044 \wedge k \leq -56.025068989018$$
$$k \geq -53.9306738866248 \wedge k \leq -49.7418836818384$$
$$k \geq -47.6474885794452 \wedge k \leq -43.4586983746588$$
$$k \geq -41.3643032722656 \wedge k \leq -37.1755130674792$$
$$k \geq -35.081117965086 \wedge k \leq -30.8923277602996$$
$$k \geq -28.7979326579064 \wedge k \leq -24.60914245312$$
$$k \geq -22.5147473507269 \wedge k \leq -18.3259571459405$$
$$k \geq -16.2315620435473 \wedge k \leq -12.0427718387609$$
$$k \geq -9.94837673636768 \wedge k \leq -5.75958653158129$$
$$k \geq -3.66519142918809 \wedge k \leq 0.523598775598299$$
$$k \geq 2.61799387799149 \wedge k \leq 6.80678408277789$$
$$k \geq 8.90117918517108 \wedge k \leq 13.0899693899575$$
$$k \geq 15.1843644923507 \wedge k \leq 19.3731546971371$$
$$k \geq 21.4675497995303 \wedge k \leq 25.6563400043166$$
$$k \geq 27.7507351067098 \wedge k \leq 31.9395253114962$$
$$k \geq 34.0339204138894 \wedge k \leq 38.2227106186758$$
$$k \geq 40.317105721069 \wedge k \leq 44.5058959258554$$
$$k \geq 46.6002910282486 \wedge k \leq 50.789081233035$$
$$k \geq 52.8834763354282 \wedge k \leq 57.0722665402146$$
$$k \geq 59.1666616426078 \wedge k \leq 63.3554518473942$$
$$k \geq 65.4498469497874 \wedge k \leq 69.6386371545737$$
$$k \geq 71.733032256967 \wedge k \leq 75.9218224617533$$
$$k \geq 78.0162175641465 \wedge k \leq 82.2050077689329$$
$$k \geq 84.2994028713261 \wedge k \leq 88.4881930761125$$
$$k \geq 90.5825881785057 \wedge k \leq 94.7713783832921$$
$$k \geq 96.8657734856853 \wedge k \leq 101.054563690472$$
$$k \geq 134.564885328763 \wedge k \leq 138.753675533549$$
$$k \geq 17438.4572213013$$
Rapid solution 2
[src]
pi 5*pi [--, ----] 6 6
$$k\ in\ \left[\frac{\pi}{6}, \frac{5 \pi}{6}\right]$$
k in Interval(pi/6, 5*pi/6)
Rapid solution
[src]
/pi 5*pi\ And|-- <= t, t <= ----| \6 6 /
$$\frac{\pi}{6} \leq t \wedge t \leq \frac{5 \pi}{6}$$
(pi/6 <= t)∧(t <= 5*pi/6)