Mister Exam
5sinx+cos2x≥-2 inequation
The solution
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5*sin(x) + cos(2*x) >= -2
$$5 \sin{\left(x \right)} + \cos{\left(2 x \right)} \geq -2$$
5*sin(x) + cos(2*x) >= -2
Detail solution
Given the inequality:
$$5 \sin{\left(x \right)} + \cos{\left(2 x \right)} \geq -2$$
To solve this inequality, we must first solve the corresponding equation:
$$5 \sin{\left(x \right)} + \cos{\left(2 x \right)} = -2$$
Solve:
Given the equation
$$5 \sin{\left(x \right)} + \cos{\left(2 x \right)} = -2$$
transform
$$5 \sin{\left(x \right)} + \cos{\left(2 x \right)} + 2 = 0$$
$$- 2 \sin^{2}{\left(x \right)} + 5 \sin{\left(x \right)} + 3 = 0$$
Do replacement
$$w = \sin{\left(x \right)}$$
This equation is of the form
A quadratic equation can be solved
using the discriminant.
The roots of the quadratic equation:
$$w_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$w_{2} = \frac{- \sqrt{D} - b}{2 a}$$
where D = b^2 - 4*a*c - it is the discriminant.
Because
$$a = -2$$
$$b = 5$$
$$c = 3$$
, then
Because D > 0, then the equation has two roots.
or
$$w_{1} = - \frac{1}{2}$$
$$w_{2} = 3$$
do backward replacement
$$\sin{\left(x \right)} = w$$
Given the equation
$$\sin{\left(x \right)} = w$$
- this is the simplest trigonometric equation
This equation is transformed to
$$x = 2 \pi n + \operatorname{asin}{\left(w \right)}$$
$$x = 2 \pi n - \operatorname{asin}{\left(w \right)} + \pi$$
Or
$$x = 2 \pi n + \operatorname{asin}{\left(w \right)}$$
$$x = 2 \pi n - \operatorname{asin}{\left(w \right)} + \pi$$
, where n - is a integer
substitute w:
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(w_{1} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(- \frac{1}{2} \right)}$$
$$x_{1} = 2 \pi n - \frac{\pi}{6}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(w_{2} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(3 \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(3 \right)}$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(- \frac{1}{2} \right)} + \pi$$
$$x_{3} = 2 \pi n + \frac{7 \pi}{6}$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(w_{2} \right)} + \pi$$
$$x_{4} = 2 \pi n + \pi - \operatorname{asin}{\left(3 \right)}$$
$$x_{4} = 2 \pi n + \pi - \operatorname{asin}{\left(3 \right)}$$
$$x_{1} = -45867.7763411866$$
$$x_{2} = 60.2138591938044$$
$$x_{3} = -75.9218224617533$$
$$x_{4} = 30.8923277602996$$
$$x_{5} = -6.80678408277789$$
$$x_{6} = 43.4586983746588$$
$$x_{7} = 18.3259571459405$$
$$x_{8} = 93.7241808320955$$
$$x_{9} = -78.0162175641465$$
$$x_{10} = -8.90117918517108$$
$$x_{11} = -643.502895210309$$
$$x_{12} = -96.8657734856853$$
$$x_{13} = 91.6297857297023$$
$$x_{14} = 791.15774992903$$
$$x_{15} = 53.9306738866248$$
$$x_{16} = -19.3731546971371$$
$$x_{17} = -71.733032256967$$
$$x_{18} = -50.789081233035$$
$$x_{19} = -40.317105721069$$
$$x_{20} = 37.1755130674792$$
$$x_{21} = 41.3643032722656$$
$$x_{22} = -21.4675497995303$$
$$x_{23} = -31.9395253114962$$
$$x_{24} = 24.60914245312$$
$$x_{25} = 16.2315620435473$$
$$x_{26} = -239.284640448423$$
$$x_{27} = -82.2050077689329$$
$$x_{28} = 12.0427718387609$$
$$x_{29} = 74.8746249105567$$
$$x_{30} = -15.1843644923507$$
$$x_{31} = 72.7802298081635$$
$$x_{32} = 22.5147473507269$$
$$x_{33} = 79.0634151153431$$
$$x_{34} = -52.8834763354282$$
$$x_{35} = -46.6002910282486$$
$$x_{36} = -34.0339204138894$$
$$x_{37} = -59.1666616426078$$
$$x_{38} = -69.6386371545737$$
$$x_{39} = 3.66519142918809$$
$$x_{40} = -27.7507351067098$$
$$x_{41} = 35.081117965086$$
$$x_{42} = -65.4498469497874$$
$$x_{43} = 87.4409955249159$$
$$x_{44} = -38.2227106186758$$
$$x_{45} = 66.497044500984$$
$$x_{46} = -101.054563690472$$
$$x_{47} = 62.3082542961976$$
$$x_{48} = 100.007366139275$$
$$x_{49} = -13.0899693899575$$
$$x_{50} = 97.9129710368819$$
$$x_{51} = -90.5825881785057$$
$$x_{52} = 5.75958653158129$$
$$x_{53} = 49.7418836818384$$
$$x_{54} = 85.3466004225227$$
$$x_{55} = 9.94837673636768$$
$$x_{56} = -2.61799387799149$$
$$x_{57} = 28.7979326579064$$
$$x_{58} = 68.5914396033772$$
$$x_{59} = 112.573736753634$$
$$x_{60} = -88.4881930761125$$
$$x_{61} = -25.6563400043166$$
$$x_{62} = -63.3554518473942$$
$$x_{63} = 110.479341651241$$
$$x_{64} = -94.7713783832921$$
$$x_{65} = 56.025068989018$$
$$x_{66} = -84.2994028713261$$
$$x_{67} = -44.5058959258554$$
$$x_{68} = -0.523598775598299$$
$$x_{69} = 47.6474885794452$$
$$x_{70} = 81.1578102177363$$
$$x_{71} = -57.0722665402146$$
$$x_{1} = -45867.7763411866$$
$$x_{2} = 60.2138591938044$$
$$x_{3} = -75.9218224617533$$
$$x_{4} = 30.8923277602996$$
$$x_{5} = -6.80678408277789$$
$$x_{6} = 43.4586983746588$$
$$x_{7} = 18.3259571459405$$
$$x_{8} = 93.7241808320955$$
$$x_{9} = -78.0162175641465$$
$$x_{10} = -8.90117918517108$$
$$x_{11} = -643.502895210309$$
$$x_{12} = -96.8657734856853$$
$$x_{13} = 91.6297857297023$$
$$x_{14} = 791.15774992903$$
$$x_{15} = 53.9306738866248$$
$$x_{16} = -19.3731546971371$$
$$x_{17} = -71.733032256967$$
$$x_{18} = -50.789081233035$$
$$x_{19} = -40.317105721069$$
$$x_{20} = 37.1755130674792$$
$$x_{21} = 41.3643032722656$$
$$x_{22} = -21.4675497995303$$
$$x_{23} = -31.9395253114962$$
$$x_{24} = 24.60914245312$$
$$x_{25} = 16.2315620435473$$
$$x_{26} = -239.284640448423$$
$$x_{27} = -82.2050077689329$$
$$x_{28} = 12.0427718387609$$
$$x_{29} = 74.8746249105567$$
$$x_{30} = -15.1843644923507$$
$$x_{31} = 72.7802298081635$$
$$x_{32} = 22.5147473507269$$
$$x_{33} = 79.0634151153431$$
$$x_{34} = -52.8834763354282$$
$$x_{35} = -46.6002910282486$$
$$x_{36} = -34.0339204138894$$
$$x_{37} = -59.1666616426078$$
$$x_{38} = -69.6386371545737$$
$$x_{39} = 3.66519142918809$$
$$x_{40} = -27.7507351067098$$
$$x_{41} = 35.081117965086$$
$$x_{42} = -65.4498469497874$$
$$x_{43} = 87.4409955249159$$
$$x_{44} = -38.2227106186758$$
$$x_{45} = 66.497044500984$$
$$x_{46} = -101.054563690472$$
$$x_{47} = 62.3082542961976$$
$$x_{48} = 100.007366139275$$
$$x_{49} = -13.0899693899575$$
$$x_{50} = 97.9129710368819$$
$$x_{51} = -90.5825881785057$$
$$x_{52} = 5.75958653158129$$
$$x_{53} = 49.7418836818384$$
$$x_{54} = 85.3466004225227$$
$$x_{55} = 9.94837673636768$$
$$x_{56} = -2.61799387799149$$
$$x_{57} = 28.7979326579064$$
$$x_{58} = 68.5914396033772$$
$$x_{59} = 112.573736753634$$
$$x_{60} = -88.4881930761125$$
$$x_{61} = -25.6563400043166$$
$$x_{62} = -63.3554518473942$$
$$x_{63} = 110.479341651241$$
$$x_{64} = -94.7713783832921$$
$$x_{65} = 56.025068989018$$
$$x_{66} = -84.2994028713261$$
$$x_{67} = -44.5058959258554$$
$$x_{68} = -0.523598775598299$$
$$x_{69} = 47.6474885794452$$
$$x_{70} = 81.1578102177363$$
$$x_{71} = -57.0722665402146$$
This roots
$$x_{1} = -45867.7763411866$$
$$x_{11} = -643.502895210309$$
$$x_{26} = -239.284640448423$$
$$x_{46} = -101.054563690472$$
$$x_{12} = -96.8657734856853$$
$$x_{64} = -94.7713783832921$$
$$x_{51} = -90.5825881785057$$
$$x_{60} = -88.4881930761125$$
$$x_{66} = -84.2994028713261$$
$$x_{27} = -82.2050077689329$$
$$x_{9} = -78.0162175641465$$
$$x_{3} = -75.9218224617533$$
$$x_{17} = -71.733032256967$$
$$x_{38} = -69.6386371545737$$
$$x_{42} = -65.4498469497874$$
$$x_{62} = -63.3554518473942$$
$$x_{37} = -59.1666616426078$$
$$x_{71} = -57.0722665402146$$
$$x_{34} = -52.8834763354282$$
$$x_{18} = -50.789081233035$$
$$x_{35} = -46.6002910282486$$
$$x_{67} = -44.5058959258554$$
$$x_{19} = -40.317105721069$$
$$x_{44} = -38.2227106186758$$
$$x_{36} = -34.0339204138894$$
$$x_{23} = -31.9395253114962$$
$$x_{40} = -27.7507351067098$$
$$x_{61} = -25.6563400043166$$
$$x_{22} = -21.4675497995303$$
$$x_{16} = -19.3731546971371$$
$$x_{30} = -15.1843644923507$$
$$x_{49} = -13.0899693899575$$
$$x_{10} = -8.90117918517108$$
$$x_{5} = -6.80678408277789$$
$$x_{56} = -2.61799387799149$$
$$x_{68} = -0.523598775598299$$
$$x_{39} = 3.66519142918809$$
$$x_{52} = 5.75958653158129$$
$$x_{55} = 9.94837673636768$$
$$x_{28} = 12.0427718387609$$
$$x_{25} = 16.2315620435473$$
$$x_{7} = 18.3259571459405$$
$$x_{32} = 22.5147473507269$$
$$x_{24} = 24.60914245312$$
$$x_{57} = 28.7979326579064$$
$$x_{4} = 30.8923277602996$$
$$x_{41} = 35.081117965086$$
$$x_{20} = 37.1755130674792$$
$$x_{21} = 41.3643032722656$$
$$x_{6} = 43.4586983746588$$
$$x_{69} = 47.6474885794452$$
$$x_{53} = 49.7418836818384$$
$$x_{15} = 53.9306738866248$$
$$x_{65} = 56.025068989018$$
$$x_{2} = 60.2138591938044$$
$$x_{47} = 62.3082542961976$$
$$x_{45} = 66.497044500984$$
$$x_{58} = 68.5914396033772$$
$$x_{31} = 72.7802298081635$$
$$x_{29} = 74.8746249105567$$
$$x_{33} = 79.0634151153431$$
$$x_{70} = 81.1578102177363$$
$$x_{54} = 85.3466004225227$$
$$x_{43} = 87.4409955249159$$
$$x_{13} = 91.6297857297023$$
$$x_{8} = 93.7241808320955$$
$$x_{50} = 97.9129710368819$$
$$x_{48} = 100.007366139275$$
$$x_{63} = 110.479341651241$$
$$x_{59} = 112.573736753634$$
$$x_{14} = 791.15774992903$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} \leq x_{1}$$
For example, let's take the point
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$-45867.7763411866 + - \frac{1}{10}$$
=
$$-45867.8763411866$$
substitute to the expression
$$5 \sin{\left(x \right)} + \cos{\left(2 x \right)} \geq -2$$
$$5 \sin{\left(-45867.8763411866 \right)} + \cos{\left(\left(-45867.8763411866\right) 2 \right)} \geq -2$$
but
Then
$$x \leq -45867.7763411866$$
no execute
one of the solutions of our inequality is:
$$x \geq -45867.7763411866 \wedge x \leq -643.502895210309$$
Other solutions will get with the changeover to the next point
etc.
The answer:
$$x \geq -45867.7763411866 \wedge x \leq -643.502895210309$$
$$x \geq -239.284640448423 \wedge x \leq -101.054563690472$$
$$x \geq -96.8657734856853 \wedge x \leq -94.7713783832921$$
$$x \geq -90.5825881785057 \wedge x \leq -88.4881930761125$$
$$x \geq -84.2994028713261 \wedge x \leq -82.2050077689329$$
$$x \geq -78.0162175641465 \wedge x \leq -75.9218224617533$$
$$x \geq -71.733032256967 \wedge x \leq -69.6386371545737$$
$$x \geq -65.4498469497874 \wedge x \leq -63.3554518473942$$
$$x \geq -59.1666616426078 \wedge x \leq -57.0722665402146$$
$$x \geq -52.8834763354282 \wedge x \leq -50.789081233035$$
$$x \geq -46.6002910282486 \wedge x \leq -44.5058959258554$$
$$x \geq -40.317105721069 \wedge x \leq -38.2227106186758$$
$$x \geq -34.0339204138894 \wedge x \leq -31.9395253114962$$
$$x \geq -27.7507351067098 \wedge x \leq -25.6563400043166$$
$$x \geq -21.4675497995303 \wedge x \leq -19.3731546971371$$
$$x \geq -15.1843644923507 \wedge x \leq -13.0899693899575$$
$$x \geq -8.90117918517108 \wedge x \leq -6.80678408277789$$
$$x \geq -2.61799387799149 \wedge x \leq -0.523598775598299$$
$$x \geq 3.66519142918809 \wedge x \leq 5.75958653158129$$
$$x \geq 9.94837673636768 \wedge x \leq 12.0427718387609$$
$$x \geq 16.2315620435473 \wedge x \leq 18.3259571459405$$
$$x \geq 22.5147473507269 \wedge x \leq 24.60914245312$$
$$x \geq 28.7979326579064 \wedge x \leq 30.8923277602996$$
$$x \geq 35.081117965086 \wedge x \leq 37.1755130674792$$
$$x \geq 41.3643032722656 \wedge x \leq 43.4586983746588$$
$$x \geq 47.6474885794452 \wedge x \leq 49.7418836818384$$
$$x \geq 53.9306738866248 \wedge x \leq 56.025068989018$$
$$x \geq 60.2138591938044 \wedge x \leq 62.3082542961976$$
$$x \geq 66.497044500984 \wedge x \leq 68.5914396033772$$
$$x \geq 72.7802298081635 \wedge x \leq 74.8746249105567$$
$$x \geq 79.0634151153431 \wedge x \leq 81.1578102177363$$
$$x \geq 85.3466004225227 \wedge x \leq 87.4409955249159$$
$$x \geq 91.6297857297023 \wedge x \leq 93.7241808320955$$
$$x \geq 97.9129710368819 \wedge x \leq 100.007366139275$$
$$x \geq 110.479341651241 \wedge x \leq 112.573736753634$$
$$x \geq 791.15774992903$$
$$5 \sin{\left(x \right)} + \cos{\left(2 x \right)} \geq -2$$
To solve this inequality, we must first solve the corresponding equation:
$$5 \sin{\left(x \right)} + \cos{\left(2 x \right)} = -2$$
Solve:
Given the equation
$$5 \sin{\left(x \right)} + \cos{\left(2 x \right)} = -2$$
transform
$$5 \sin{\left(x \right)} + \cos{\left(2 x \right)} + 2 = 0$$
$$- 2 \sin^{2}{\left(x \right)} + 5 \sin{\left(x \right)} + 3 = 0$$
Do replacement
$$w = \sin{\left(x \right)}$$
This equation is of the form
a*w^2 + b*w + c = 0
A quadratic equation can be solved
using the discriminant.
The roots of the quadratic equation:
$$w_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$w_{2} = \frac{- \sqrt{D} - b}{2 a}$$
where D = b^2 - 4*a*c - it is the discriminant.
Because
$$a = -2$$
$$b = 5$$
$$c = 3$$
, then
D = b^2 - 4 * a * c =
(5)^2 - 4 * (-2) * (3) = 49
Because D > 0, then the equation has two roots.
w1 = (-b + sqrt(D)) / (2*a)
w2 = (-b - sqrt(D)) / (2*a)
or
$$w_{1} = - \frac{1}{2}$$
$$w_{2} = 3$$
do backward replacement
$$\sin{\left(x \right)} = w$$
Given the equation
$$\sin{\left(x \right)} = w$$
- this is the simplest trigonometric equation
This equation is transformed to
$$x = 2 \pi n + \operatorname{asin}{\left(w \right)}$$
$$x = 2 \pi n - \operatorname{asin}{\left(w \right)} + \pi$$
Or
$$x = 2 \pi n + \operatorname{asin}{\left(w \right)}$$
$$x = 2 \pi n - \operatorname{asin}{\left(w \right)} + \pi$$
, where n - is a integer
substitute w:
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(w_{1} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(- \frac{1}{2} \right)}$$
$$x_{1} = 2 \pi n - \frac{\pi}{6}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(w_{2} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(3 \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(3 \right)}$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(- \frac{1}{2} \right)} + \pi$$
$$x_{3} = 2 \pi n + \frac{7 \pi}{6}$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(w_{2} \right)} + \pi$$
$$x_{4} = 2 \pi n + \pi - \operatorname{asin}{\left(3 \right)}$$
$$x_{4} = 2 \pi n + \pi - \operatorname{asin}{\left(3 \right)}$$
$$x_{1} = -45867.7763411866$$
$$x_{2} = 60.2138591938044$$
$$x_{3} = -75.9218224617533$$
$$x_{4} = 30.8923277602996$$
$$x_{5} = -6.80678408277789$$
$$x_{6} = 43.4586983746588$$
$$x_{7} = 18.3259571459405$$
$$x_{8} = 93.7241808320955$$
$$x_{9} = -78.0162175641465$$
$$x_{10} = -8.90117918517108$$
$$x_{11} = -643.502895210309$$
$$x_{12} = -96.8657734856853$$
$$x_{13} = 91.6297857297023$$
$$x_{14} = 791.15774992903$$
$$x_{15} = 53.9306738866248$$
$$x_{16} = -19.3731546971371$$
$$x_{17} = -71.733032256967$$
$$x_{18} = -50.789081233035$$
$$x_{19} = -40.317105721069$$
$$x_{20} = 37.1755130674792$$
$$x_{21} = 41.3643032722656$$
$$x_{22} = -21.4675497995303$$
$$x_{23} = -31.9395253114962$$
$$x_{24} = 24.60914245312$$
$$x_{25} = 16.2315620435473$$
$$x_{26} = -239.284640448423$$
$$x_{27} = -82.2050077689329$$
$$x_{28} = 12.0427718387609$$
$$x_{29} = 74.8746249105567$$
$$x_{30} = -15.1843644923507$$
$$x_{31} = 72.7802298081635$$
$$x_{32} = 22.5147473507269$$
$$x_{33} = 79.0634151153431$$
$$x_{34} = -52.8834763354282$$
$$x_{35} = -46.6002910282486$$
$$x_{36} = -34.0339204138894$$
$$x_{37} = -59.1666616426078$$
$$x_{38} = -69.6386371545737$$
$$x_{39} = 3.66519142918809$$
$$x_{40} = -27.7507351067098$$
$$x_{41} = 35.081117965086$$
$$x_{42} = -65.4498469497874$$
$$x_{43} = 87.4409955249159$$
$$x_{44} = -38.2227106186758$$
$$x_{45} = 66.497044500984$$
$$x_{46} = -101.054563690472$$
$$x_{47} = 62.3082542961976$$
$$x_{48} = 100.007366139275$$
$$x_{49} = -13.0899693899575$$
$$x_{50} = 97.9129710368819$$
$$x_{51} = -90.5825881785057$$
$$x_{52} = 5.75958653158129$$
$$x_{53} = 49.7418836818384$$
$$x_{54} = 85.3466004225227$$
$$x_{55} = 9.94837673636768$$
$$x_{56} = -2.61799387799149$$
$$x_{57} = 28.7979326579064$$
$$x_{58} = 68.5914396033772$$
$$x_{59} = 112.573736753634$$
$$x_{60} = -88.4881930761125$$
$$x_{61} = -25.6563400043166$$
$$x_{62} = -63.3554518473942$$
$$x_{63} = 110.479341651241$$
$$x_{64} = -94.7713783832921$$
$$x_{65} = 56.025068989018$$
$$x_{66} = -84.2994028713261$$
$$x_{67} = -44.5058959258554$$
$$x_{68} = -0.523598775598299$$
$$x_{69} = 47.6474885794452$$
$$x_{70} = 81.1578102177363$$
$$x_{71} = -57.0722665402146$$
$$x_{1} = -45867.7763411866$$
$$x_{2} = 60.2138591938044$$
$$x_{3} = -75.9218224617533$$
$$x_{4} = 30.8923277602996$$
$$x_{5} = -6.80678408277789$$
$$x_{6} = 43.4586983746588$$
$$x_{7} = 18.3259571459405$$
$$x_{8} = 93.7241808320955$$
$$x_{9} = -78.0162175641465$$
$$x_{10} = -8.90117918517108$$
$$x_{11} = -643.502895210309$$
$$x_{12} = -96.8657734856853$$
$$x_{13} = 91.6297857297023$$
$$x_{14} = 791.15774992903$$
$$x_{15} = 53.9306738866248$$
$$x_{16} = -19.3731546971371$$
$$x_{17} = -71.733032256967$$
$$x_{18} = -50.789081233035$$
$$x_{19} = -40.317105721069$$
$$x_{20} = 37.1755130674792$$
$$x_{21} = 41.3643032722656$$
$$x_{22} = -21.4675497995303$$
$$x_{23} = -31.9395253114962$$
$$x_{24} = 24.60914245312$$
$$x_{25} = 16.2315620435473$$
$$x_{26} = -239.284640448423$$
$$x_{27} = -82.2050077689329$$
$$x_{28} = 12.0427718387609$$
$$x_{29} = 74.8746249105567$$
$$x_{30} = -15.1843644923507$$
$$x_{31} = 72.7802298081635$$
$$x_{32} = 22.5147473507269$$
$$x_{33} = 79.0634151153431$$
$$x_{34} = -52.8834763354282$$
$$x_{35} = -46.6002910282486$$
$$x_{36} = -34.0339204138894$$
$$x_{37} = -59.1666616426078$$
$$x_{38} = -69.6386371545737$$
$$x_{39} = 3.66519142918809$$
$$x_{40} = -27.7507351067098$$
$$x_{41} = 35.081117965086$$
$$x_{42} = -65.4498469497874$$
$$x_{43} = 87.4409955249159$$
$$x_{44} = -38.2227106186758$$
$$x_{45} = 66.497044500984$$
$$x_{46} = -101.054563690472$$
$$x_{47} = 62.3082542961976$$
$$x_{48} = 100.007366139275$$
$$x_{49} = -13.0899693899575$$
$$x_{50} = 97.9129710368819$$
$$x_{51} = -90.5825881785057$$
$$x_{52} = 5.75958653158129$$
$$x_{53} = 49.7418836818384$$
$$x_{54} = 85.3466004225227$$
$$x_{55} = 9.94837673636768$$
$$x_{56} = -2.61799387799149$$
$$x_{57} = 28.7979326579064$$
$$x_{58} = 68.5914396033772$$
$$x_{59} = 112.573736753634$$
$$x_{60} = -88.4881930761125$$
$$x_{61} = -25.6563400043166$$
$$x_{62} = -63.3554518473942$$
$$x_{63} = 110.479341651241$$
$$x_{64} = -94.7713783832921$$
$$x_{65} = 56.025068989018$$
$$x_{66} = -84.2994028713261$$
$$x_{67} = -44.5058959258554$$
$$x_{68} = -0.523598775598299$$
$$x_{69} = 47.6474885794452$$
$$x_{70} = 81.1578102177363$$
$$x_{71} = -57.0722665402146$$
This roots
$$x_{1} = -45867.7763411866$$
$$x_{11} = -643.502895210309$$
$$x_{26} = -239.284640448423$$
$$x_{46} = -101.054563690472$$
$$x_{12} = -96.8657734856853$$
$$x_{64} = -94.7713783832921$$
$$x_{51} = -90.5825881785057$$
$$x_{60} = -88.4881930761125$$
$$x_{66} = -84.2994028713261$$
$$x_{27} = -82.2050077689329$$
$$x_{9} = -78.0162175641465$$
$$x_{3} = -75.9218224617533$$
$$x_{17} = -71.733032256967$$
$$x_{38} = -69.6386371545737$$
$$x_{42} = -65.4498469497874$$
$$x_{62} = -63.3554518473942$$
$$x_{37} = -59.1666616426078$$
$$x_{71} = -57.0722665402146$$
$$x_{34} = -52.8834763354282$$
$$x_{18} = -50.789081233035$$
$$x_{35} = -46.6002910282486$$
$$x_{67} = -44.5058959258554$$
$$x_{19} = -40.317105721069$$
$$x_{44} = -38.2227106186758$$
$$x_{36} = -34.0339204138894$$
$$x_{23} = -31.9395253114962$$
$$x_{40} = -27.7507351067098$$
$$x_{61} = -25.6563400043166$$
$$x_{22} = -21.4675497995303$$
$$x_{16} = -19.3731546971371$$
$$x_{30} = -15.1843644923507$$
$$x_{49} = -13.0899693899575$$
$$x_{10} = -8.90117918517108$$
$$x_{5} = -6.80678408277789$$
$$x_{56} = -2.61799387799149$$
$$x_{68} = -0.523598775598299$$
$$x_{39} = 3.66519142918809$$
$$x_{52} = 5.75958653158129$$
$$x_{55} = 9.94837673636768$$
$$x_{28} = 12.0427718387609$$
$$x_{25} = 16.2315620435473$$
$$x_{7} = 18.3259571459405$$
$$x_{32} = 22.5147473507269$$
$$x_{24} = 24.60914245312$$
$$x_{57} = 28.7979326579064$$
$$x_{4} = 30.8923277602996$$
$$x_{41} = 35.081117965086$$
$$x_{20} = 37.1755130674792$$
$$x_{21} = 41.3643032722656$$
$$x_{6} = 43.4586983746588$$
$$x_{69} = 47.6474885794452$$
$$x_{53} = 49.7418836818384$$
$$x_{15} = 53.9306738866248$$
$$x_{65} = 56.025068989018$$
$$x_{2} = 60.2138591938044$$
$$x_{47} = 62.3082542961976$$
$$x_{45} = 66.497044500984$$
$$x_{58} = 68.5914396033772$$
$$x_{31} = 72.7802298081635$$
$$x_{29} = 74.8746249105567$$
$$x_{33} = 79.0634151153431$$
$$x_{70} = 81.1578102177363$$
$$x_{54} = 85.3466004225227$$
$$x_{43} = 87.4409955249159$$
$$x_{13} = 91.6297857297023$$
$$x_{8} = 93.7241808320955$$
$$x_{50} = 97.9129710368819$$
$$x_{48} = 100.007366139275$$
$$x_{63} = 110.479341651241$$
$$x_{59} = 112.573736753634$$
$$x_{14} = 791.15774992903$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} \leq x_{1}$$
For example, let's take the point
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$-45867.7763411866 + - \frac{1}{10}$$
=
$$-45867.8763411866$$
substitute to the expression
$$5 \sin{\left(x \right)} + \cos{\left(2 x \right)} \geq -2$$
$$5 \sin{\left(-45867.8763411866 \right)} + \cos{\left(\left(-45867.8763411866\right) 2 \right)} \geq -2$$
-2.60182118651354 >= -2
but
-2.60182118651354 < -2
Then
$$x \leq -45867.7763411866$$
no execute
one of the solutions of our inequality is:
$$x \geq -45867.7763411866 \wedge x \leq -643.502895210309$$
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x1 x11 x26 x46 x12 x64 x51 x60 x66 x27 x9 x3 x17 x38 x42 x62 x37 x71 x34 x18 x35 x67 x19 x44 x36 x23 x40 x61 x22 x16 x30 x49 x10 x5 x56 x68 x39 x52 x55 x28 x25 x7 x32 x24 x57 x4 x41 x20 x21 x6 x69 x53 x15 x65 x2 x47 x45 x58 x31 x29 x33 x70 x54 x43 x13 x8 x50 x48 x63 x59 x14Other solutions will get with the changeover to the next point
etc.
The answer:
$$x \geq -45867.7763411866 \wedge x \leq -643.502895210309$$
$$x \geq -239.284640448423 \wedge x \leq -101.054563690472$$
$$x \geq -96.8657734856853 \wedge x \leq -94.7713783832921$$
$$x \geq -90.5825881785057 \wedge x \leq -88.4881930761125$$
$$x \geq -84.2994028713261 \wedge x \leq -82.2050077689329$$
$$x \geq -78.0162175641465 \wedge x \leq -75.9218224617533$$
$$x \geq -71.733032256967 \wedge x \leq -69.6386371545737$$
$$x \geq -65.4498469497874 \wedge x \leq -63.3554518473942$$
$$x \geq -59.1666616426078 \wedge x \leq -57.0722665402146$$
$$x \geq -52.8834763354282 \wedge x \leq -50.789081233035$$
$$x \geq -46.6002910282486 \wedge x \leq -44.5058959258554$$
$$x \geq -40.317105721069 \wedge x \leq -38.2227106186758$$
$$x \geq -34.0339204138894 \wedge x \leq -31.9395253114962$$
$$x \geq -27.7507351067098 \wedge x \leq -25.6563400043166$$
$$x \geq -21.4675497995303 \wedge x \leq -19.3731546971371$$
$$x \geq -15.1843644923507 \wedge x \leq -13.0899693899575$$
$$x \geq -8.90117918517108 \wedge x \leq -6.80678408277789$$
$$x \geq -2.61799387799149 \wedge x \leq -0.523598775598299$$
$$x \geq 3.66519142918809 \wedge x \leq 5.75958653158129$$
$$x \geq 9.94837673636768 \wedge x \leq 12.0427718387609$$
$$x \geq 16.2315620435473 \wedge x \leq 18.3259571459405$$
$$x \geq 22.5147473507269 \wedge x \leq 24.60914245312$$
$$x \geq 28.7979326579064 \wedge x \leq 30.8923277602996$$
$$x \geq 35.081117965086 \wedge x \leq 37.1755130674792$$
$$x \geq 41.3643032722656 \wedge x \leq 43.4586983746588$$
$$x \geq 47.6474885794452 \wedge x \leq 49.7418836818384$$
$$x \geq 53.9306738866248 \wedge x \leq 56.025068989018$$
$$x \geq 60.2138591938044 \wedge x \leq 62.3082542961976$$
$$x \geq 66.497044500984 \wedge x \leq 68.5914396033772$$
$$x \geq 72.7802298081635 \wedge x \leq 74.8746249105567$$
$$x \geq 79.0634151153431 \wedge x \leq 81.1578102177363$$
$$x \geq 85.3466004225227 \wedge x \leq 87.4409955249159$$
$$x \geq 91.6297857297023 \wedge x \leq 93.7241808320955$$
$$x \geq 97.9129710368819 \wedge x \leq 100.007366139275$$
$$x \geq 110.479341651241 \wedge x \leq 112.573736753634$$
$$x \geq 791.15774992903$$
Solving inequality on a graph
Rapid solution
[src]
/ / 7*pi\ /11*pi \\ Or|And|0 <= x, x <= ----|, And|----- <= x, x <= 2*pi|| \ \ 6 / \ 6 //
$$\left(0 \leq x \wedge x \leq \frac{7 \pi}{6}\right) \vee \left(\frac{11 \pi}{6} \leq x \wedge x \leq 2 \pi\right)$$
((0 <= x)∧(x <= 7*pi/6))∨((11*pi/6 <= x)∧(x <= 2*pi))
Rapid solution 2
[src]
7*pi 11*pi
[0, ----] U [-----, 2*pi]
6 6
$$x\ in\ \left[0, \frac{7 \pi}{6}\right] \cup \left[\frac{11 \pi}{6}, 2 \pi\right]$$
x in Union(Interval(0, 7*pi/6), Interval(11*pi/6, 2*pi))