Mister Exam
6sin3xcos3x+(sin6x/2)+1=>0 inequation
The solution
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sin(6*x)
6*sin(3*x)*cos(3*x) + -------- + 1 >= 0
2
$$\left(6 \sin{\left(3 x \right)} \cos{\left(3 x \right)} + \frac{\sin{\left(6 x \right)}}{2}\right) + 1 \geq 0$$
(6*sin(3*x))*cos(3*x) + sin(6*x)/2 + 1 >= 0
Detail solution
Given the inequality:
$$\left(6 \sin{\left(3 x \right)} \cos{\left(3 x \right)} + \frac{\sin{\left(6 x \right)}}{2}\right) + 1 \geq 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\left(6 \sin{\left(3 x \right)} \cos{\left(3 x \right)} + \frac{\sin{\left(6 x \right)}}{2}\right) + 1 = 0$$
Solve:
$$x_{1} = -15.7562552181883$$
$$x_{2} = 35.1294099153254$$
$$x_{3} = -41.9361939981032$$
$$x_{4} = 68.0195488775395$$
$$x_{5} = 87.9163023502749$$
$$x_{6} = 96.2938827598476$$
$$x_{7} = -71.6847403067276$$
$$x_{8} = -27.7024431564705$$
$$x_{9} = 80.1589046167791$$
$$x_{10} = -97.8646790866426$$
$$x_{11} = -58.6913548172488$$
$$x_{12} = -19.9450454229747$$
$$x_{13} = 2.04610315215385$$
$$x_{14} = 46.028400302411$$
$$x_{15} = 12.0910637890002$$
$$x_{16} = 50.2171905071974$$
$$x_{17} = -11.9944798885215$$
$$x_{18} = 24.0372517272824$$
$$x_{19} = 94.1994876574545$$
$$x_{20} = 28.2260419320688$$
$$x_{21} = -63.9273425732318$$
$$x_{22} = 90.0106974526681$$
$$x_{23} = 92.1050925550613$$
$$x_{24} = 65.9251537751463$$
$$x_{25} = -31.8912333612569$$
$$x_{26} = 100.055658089514$$
$$x_{27} = 26.1316468296756$$
$$x_{28} = -39.8417988957101$$
$$x_{29} = 4.14049825454705$$
$$x_{30} = -77.9679256139072$$
$$x_{31} = 18.3742490961798$$
$$x_{32} = -24.5608505028807$$
$$x_{33} = 9.99666868660702$$
$$x_{34} = -81.729700943574$$
$$x_{35} = -93.6758888818562$$
$$x_{36} = 21.9428566248892$$
$$x_{37} = 31.9878172617356$$
$$x_{38} = -59.7385523684454$$
$$x_{39} = 84.3476948215655$$
$$x_{40} = 60.2621511440437$$
$$x_{41} = 38.2710025689152$$
$$x_{42} = -7.80568968373514$$
$$x_{43} = -37.7474037933169$$
$$x_{44} = 36.176607466522$$
$$x_{45} = -68.1161327780182$$
$$x_{46} = -49.693591731599$$
$$x_{47} = 14.1854588913934$$
$$x_{48} = -118.808630110574$$
$$x_{49} = 48.1227954048042$$
$$x_{50} = -44.0305891004964$$
$$x_{51} = 16.2798539937866$$
$$x_{52} = -61.8329474708386$$
$$x_{53} = -99.9590741890357$$
$$x_{54} = 97.9612629871212$$
$$x_{55} = 75.9701144119927$$
$$x_{56} = 53.9789658368641$$
$$x_{57} = 40.3653976713084$$
$$x_{58} = 130.424387074216$$
$$x_{59} = -3.18988460382913$$
$$x_{60} = -80.0623207163004$$
$$x_{61} = -55.9767770387786$$
$$x_{62} = -2.56970192775215$$
$$x_{63} = -0.0482919502393412$$
$$x_{64} = -66.021737675625$$
$$x_{65} = 78.0645095143859$$
$$x_{66} = -17.8506503205815$$
$$x_{67} = -46.1249842028896$$
$$x_{68} = -73.7791354091208$$
$$x_{69} = -22.0394405253679$$
$$x_{70} = 62.3565462464369$$
$$x_{71} = -95.7702839842493$$
$$x_{72} = 74.3027341847191$$
$$x_{73} = 82.2532997191723$$
$$x_{74} = -90.1072813531467$$
$$x_{75} = -75.873530511514$$
$$x_{76} = -33.9856284636501$$
$$x_{77} = 30.940619710539$$
$$x_{78} = 72.2083390823259$$
$$x_{79} = -48.2193793052828$$
$$x_{80} = -53.8823819363854$$
$$x_{81} = -29.7968382588637$$
$$x_{82} = -88.0128862507536$$
$$x_{83} = -25.1810331789577$$
$$x_{84} = -5.71129458134195$$
$$x_{85} = -51.7879868339922$$
$$x_{86} = 70.1139439799327$$
$$x_{87} = 56.0733609392573$$
$$x_{88} = -9.90008478612834$$
$$x_{89} = 6.23489335694024$$
$$x_{90} = -83.8240960459672$$
$$x_{91} = -85.9184911483604$$
$$x_{92} = 58.1677560416505$$
$$x_{93} = 43.9340052000178$$
$$x_{94} = 52.3115856095905$$
$$x_{95} = 34.0822123641288$$
$$x_{1} = -15.7562552181883$$
$$x_{2} = 35.1294099153254$$
$$x_{3} = -41.9361939981032$$
$$x_{4} = 68.0195488775395$$
$$x_{5} = 87.9163023502749$$
$$x_{6} = 96.2938827598476$$
$$x_{7} = -71.6847403067276$$
$$x_{8} = -27.7024431564705$$
$$x_{9} = 80.1589046167791$$
$$x_{10} = -97.8646790866426$$
$$x_{11} = -58.6913548172488$$
$$x_{12} = -19.9450454229747$$
$$x_{13} = 2.04610315215385$$
$$x_{14} = 46.028400302411$$
$$x_{15} = 12.0910637890002$$
$$x_{16} = 50.2171905071974$$
$$x_{17} = -11.9944798885215$$
$$x_{18} = 24.0372517272824$$
$$x_{19} = 94.1994876574545$$
$$x_{20} = 28.2260419320688$$
$$x_{21} = -63.9273425732318$$
$$x_{22} = 90.0106974526681$$
$$x_{23} = 92.1050925550613$$
$$x_{24} = 65.9251537751463$$
$$x_{25} = -31.8912333612569$$
$$x_{26} = 100.055658089514$$
$$x_{27} = 26.1316468296756$$
$$x_{28} = -39.8417988957101$$
$$x_{29} = 4.14049825454705$$
$$x_{30} = -77.9679256139072$$
$$x_{31} = 18.3742490961798$$
$$x_{32} = -24.5608505028807$$
$$x_{33} = 9.99666868660702$$
$$x_{34} = -81.729700943574$$
$$x_{35} = -93.6758888818562$$
$$x_{36} = 21.9428566248892$$
$$x_{37} = 31.9878172617356$$
$$x_{38} = -59.7385523684454$$
$$x_{39} = 84.3476948215655$$
$$x_{40} = 60.2621511440437$$
$$x_{41} = 38.2710025689152$$
$$x_{42} = -7.80568968373514$$
$$x_{43} = -37.7474037933169$$
$$x_{44} = 36.176607466522$$
$$x_{45} = -68.1161327780182$$
$$x_{46} = -49.693591731599$$
$$x_{47} = 14.1854588913934$$
$$x_{48} = -118.808630110574$$
$$x_{49} = 48.1227954048042$$
$$x_{50} = -44.0305891004964$$
$$x_{51} = 16.2798539937866$$
$$x_{52} = -61.8329474708386$$
$$x_{53} = -99.9590741890357$$
$$x_{54} = 97.9612629871212$$
$$x_{55} = 75.9701144119927$$
$$x_{56} = 53.9789658368641$$
$$x_{57} = 40.3653976713084$$
$$x_{58} = 130.424387074216$$
$$x_{59} = -3.18988460382913$$
$$x_{60} = -80.0623207163004$$
$$x_{61} = -55.9767770387786$$
$$x_{62} = -2.56970192775215$$
$$x_{63} = -0.0482919502393412$$
$$x_{64} = -66.021737675625$$
$$x_{65} = 78.0645095143859$$
$$x_{66} = -17.8506503205815$$
$$x_{67} = -46.1249842028896$$
$$x_{68} = -73.7791354091208$$
$$x_{69} = -22.0394405253679$$
$$x_{70} = 62.3565462464369$$
$$x_{71} = -95.7702839842493$$
$$x_{72} = 74.3027341847191$$
$$x_{73} = 82.2532997191723$$
$$x_{74} = -90.1072813531467$$
$$x_{75} = -75.873530511514$$
$$x_{76} = -33.9856284636501$$
$$x_{77} = 30.940619710539$$
$$x_{78} = 72.2083390823259$$
$$x_{79} = -48.2193793052828$$
$$x_{80} = -53.8823819363854$$
$$x_{81} = -29.7968382588637$$
$$x_{82} = -88.0128862507536$$
$$x_{83} = -25.1810331789577$$
$$x_{84} = -5.71129458134195$$
$$x_{85} = -51.7879868339922$$
$$x_{86} = 70.1139439799327$$
$$x_{87} = 56.0733609392573$$
$$x_{88} = -9.90008478612834$$
$$x_{89} = 6.23489335694024$$
$$x_{90} = -83.8240960459672$$
$$x_{91} = -85.9184911483604$$
$$x_{92} = 58.1677560416505$$
$$x_{93} = 43.9340052000178$$
$$x_{94} = 52.3115856095905$$
$$x_{95} = 34.0822123641288$$
This roots
$$x_{48} = -118.808630110574$$
$$x_{53} = -99.9590741890357$$
$$x_{10} = -97.8646790866426$$
$$x_{71} = -95.7702839842493$$
$$x_{35} = -93.6758888818562$$
$$x_{74} = -90.1072813531467$$
$$x_{82} = -88.0128862507536$$
$$x_{91} = -85.9184911483604$$
$$x_{90} = -83.8240960459672$$
$$x_{34} = -81.729700943574$$
$$x_{60} = -80.0623207163004$$
$$x_{30} = -77.9679256139072$$
$$x_{75} = -75.873530511514$$
$$x_{68} = -73.7791354091208$$
$$x_{7} = -71.6847403067276$$
$$x_{45} = -68.1161327780182$$
$$x_{64} = -66.021737675625$$
$$x_{21} = -63.9273425732318$$
$$x_{52} = -61.8329474708386$$
$$x_{38} = -59.7385523684454$$
$$x_{11} = -58.6913548172488$$
$$x_{61} = -55.9767770387786$$
$$x_{80} = -53.8823819363854$$
$$x_{85} = -51.7879868339922$$
$$x_{46} = -49.693591731599$$
$$x_{79} = -48.2193793052828$$
$$x_{67} = -46.1249842028896$$
$$x_{50} = -44.0305891004964$$
$$x_{3} = -41.9361939981032$$
$$x_{28} = -39.8417988957101$$
$$x_{43} = -37.7474037933169$$
$$x_{76} = -33.9856284636501$$
$$x_{25} = -31.8912333612569$$
$$x_{81} = -29.7968382588637$$
$$x_{8} = -27.7024431564705$$
$$x_{83} = -25.1810331789577$$
$$x_{32} = -24.5608505028807$$
$$x_{69} = -22.0394405253679$$
$$x_{12} = -19.9450454229747$$
$$x_{66} = -17.8506503205815$$
$$x_{1} = -15.7562552181883$$
$$x_{17} = -11.9944798885215$$
$$x_{88} = -9.90008478612834$$
$$x_{42} = -7.80568968373514$$
$$x_{84} = -5.71129458134195$$
$$x_{59} = -3.18988460382913$$
$$x_{62} = -2.56970192775215$$
$$x_{63} = -0.0482919502393412$$
$$x_{13} = 2.04610315215385$$
$$x_{29} = 4.14049825454705$$
$$x_{89} = 6.23489335694024$$
$$x_{33} = 9.99666868660702$$
$$x_{15} = 12.0910637890002$$
$$x_{47} = 14.1854588913934$$
$$x_{51} = 16.2798539937866$$
$$x_{31} = 18.3742490961798$$
$$x_{36} = 21.9428566248892$$
$$x_{18} = 24.0372517272824$$
$$x_{27} = 26.1316468296756$$
$$x_{20} = 28.2260419320688$$
$$x_{77} = 30.940619710539$$
$$x_{37} = 31.9878172617356$$
$$x_{95} = 34.0822123641288$$
$$x_{2} = 35.1294099153254$$
$$x_{44} = 36.176607466522$$
$$x_{41} = 38.2710025689152$$
$$x_{57} = 40.3653976713084$$
$$x_{93} = 43.9340052000178$$
$$x_{14} = 46.028400302411$$
$$x_{49} = 48.1227954048042$$
$$x_{16} = 50.2171905071974$$
$$x_{94} = 52.3115856095905$$
$$x_{56} = 53.9789658368641$$
$$x_{87} = 56.0733609392573$$
$$x_{92} = 58.1677560416505$$
$$x_{40} = 60.2621511440437$$
$$x_{70} = 62.3565462464369$$
$$x_{24} = 65.9251537751463$$
$$x_{4} = 68.0195488775395$$
$$x_{86} = 70.1139439799327$$
$$x_{78} = 72.2083390823259$$
$$x_{72} = 74.3027341847191$$
$$x_{55} = 75.9701144119927$$
$$x_{65} = 78.0645095143859$$
$$x_{9} = 80.1589046167791$$
$$x_{73} = 82.2532997191723$$
$$x_{39} = 84.3476948215655$$
$$x_{5} = 87.9163023502749$$
$$x_{22} = 90.0106974526681$$
$$x_{23} = 92.1050925550613$$
$$x_{19} = 94.1994876574545$$
$$x_{6} = 96.2938827598476$$
$$x_{54} = 97.9612629871212$$
$$x_{26} = 100.055658089514$$
$$x_{58} = 130.424387074216$$
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
$$x_{0} = x_{48} - \frac{1}{10}$$
=
$$-118.808630110574 + - \frac{1}{10}$$
=
$$-118.908630110574$$
substitute to the expression
$$\left(6 \sin{\left(3 x \right)} \cos{\left(3 x \right)} + \frac{\sin{\left(6 x \right)}}{2}\right) + 1 \geq 0$$
$$1 + \left(\frac{\sin{\left(\left(-118.908630110574\right) 6 \right)}}{2} + 6 \sin{\left(\left(-118.908630110574\right) 3 \right)} \cos{\left(\left(-118.908630110574\right) 3 \right)}\right) \geq 0$$
one of the solutions of our inequality is:
$$x \leq -118.808630110574$$
Other solutions will get with the changeover to the next point
etc.
The answer:
$$x \leq -118.808630110574$$
$$x \geq -99.9590741890357 \wedge x \leq -97.8646790866426$$
$$x \geq -95.7702839842493 \wedge x \leq -93.6758888818562$$
$$x \geq -90.1072813531467 \wedge x \leq -88.0128862507536$$
$$x \geq -85.9184911483604 \wedge x \leq -83.8240960459672$$
$$x \geq -81.729700943574 \wedge x \leq -80.0623207163004$$
$$x \geq -77.9679256139072 \wedge x \leq -75.873530511514$$
$$x \geq -73.7791354091208 \wedge x \leq -71.6847403067276$$
$$x \geq -68.1161327780182 \wedge x \leq -66.021737675625$$
$$x \geq -63.9273425732318 \wedge x \leq -61.8329474708386$$
$$x \geq -59.7385523684454 \wedge x \leq -58.6913548172488$$
$$x \geq -55.9767770387786 \wedge x \leq -53.8823819363854$$
$$x \geq -51.7879868339922 \wedge x \leq -49.693591731599$$
$$x \geq -48.2193793052828 \wedge x \leq -46.1249842028896$$
$$x \geq -44.0305891004964 \wedge x \leq -41.9361939981032$$
$$x \geq -39.8417988957101 \wedge x \leq -37.7474037933169$$
$$x \geq -33.9856284636501 \wedge x \leq -31.8912333612569$$
$$x \geq -29.7968382588637 \wedge x \leq -27.7024431564705$$
$$x \geq -25.1810331789577 \wedge x \leq -24.5608505028807$$
$$x \geq -22.0394405253679 \wedge x \leq -19.9450454229747$$
$$x \geq -17.8506503205815 \wedge x \leq -15.7562552181883$$
$$x \geq -11.9944798885215 \wedge x \leq -9.90008478612834$$
$$x \geq -7.80568968373514 \wedge x \leq -5.71129458134195$$
$$x \geq -3.18988460382913 \wedge x \leq -2.56970192775215$$
$$x \geq -0.0482919502393412 \wedge x \leq 2.04610315215385$$
$$x \geq 4.14049825454705 \wedge x \leq 6.23489335694024$$
$$x \geq 9.99666868660702 \wedge x \leq 12.0910637890002$$
$$x \geq 14.1854588913934 \wedge x \leq 16.2798539937866$$
$$x \geq 18.3742490961798 \wedge x \leq 21.9428566248892$$
$$x \geq 24.0372517272824 \wedge x \leq 26.1316468296756$$
$$x \geq 28.2260419320688 \wedge x \leq 30.940619710539$$
$$x \geq 31.9878172617356 \wedge x \leq 34.0822123641288$$
$$x \geq 35.1294099153254 \wedge x \leq 36.176607466522$$
$$x \geq 38.2710025689152 \wedge x \leq 40.3653976713084$$
$$x \geq 43.9340052000178 \wedge x \leq 46.028400302411$$
$$x \geq 48.1227954048042 \wedge x \leq 50.2171905071974$$
$$x \geq 52.3115856095905 \wedge x \leq 53.9789658368641$$
$$x \geq 56.0733609392573 \wedge x \leq 58.1677560416505$$
$$x \geq 60.2621511440437 \wedge x \leq 62.3565462464369$$
$$x \geq 65.9251537751463 \wedge x \leq 68.0195488775395$$
$$x \geq 70.1139439799327 \wedge x \leq 72.2083390823259$$
$$x \geq 74.3027341847191 \wedge x \leq 75.9701144119927$$
$$x \geq 78.0645095143859 \wedge x \leq 80.1589046167791$$
$$x \geq 82.2532997191723 \wedge x \leq 84.3476948215655$$
$$x \geq 87.9163023502749 \wedge x \leq 90.0106974526681$$
$$x \geq 92.1050925550613 \wedge x \leq 94.1994876574545$$
$$x \geq 96.2938827598476 \wedge x \leq 97.9612629871212$$
$$x \geq 100.055658089514 \wedge x \leq 130.424387074216$$
$$\left(6 \sin{\left(3 x \right)} \cos{\left(3 x \right)} + \frac{\sin{\left(6 x \right)}}{2}\right) + 1 \geq 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\left(6 \sin{\left(3 x \right)} \cos{\left(3 x \right)} + \frac{\sin{\left(6 x \right)}}{2}\right) + 1 = 0$$
Solve:
$$x_{1} = -15.7562552181883$$
$$x_{2} = 35.1294099153254$$
$$x_{3} = -41.9361939981032$$
$$x_{4} = 68.0195488775395$$
$$x_{5} = 87.9163023502749$$
$$x_{6} = 96.2938827598476$$
$$x_{7} = -71.6847403067276$$
$$x_{8} = -27.7024431564705$$
$$x_{9} = 80.1589046167791$$
$$x_{10} = -97.8646790866426$$
$$x_{11} = -58.6913548172488$$
$$x_{12} = -19.9450454229747$$
$$x_{13} = 2.04610315215385$$
$$x_{14} = 46.028400302411$$
$$x_{15} = 12.0910637890002$$
$$x_{16} = 50.2171905071974$$
$$x_{17} = -11.9944798885215$$
$$x_{18} = 24.0372517272824$$
$$x_{19} = 94.1994876574545$$
$$x_{20} = 28.2260419320688$$
$$x_{21} = -63.9273425732318$$
$$x_{22} = 90.0106974526681$$
$$x_{23} = 92.1050925550613$$
$$x_{24} = 65.9251537751463$$
$$x_{25} = -31.8912333612569$$
$$x_{26} = 100.055658089514$$
$$x_{27} = 26.1316468296756$$
$$x_{28} = -39.8417988957101$$
$$x_{29} = 4.14049825454705$$
$$x_{30} = -77.9679256139072$$
$$x_{31} = 18.3742490961798$$
$$x_{32} = -24.5608505028807$$
$$x_{33} = 9.99666868660702$$
$$x_{34} = -81.729700943574$$
$$x_{35} = -93.6758888818562$$
$$x_{36} = 21.9428566248892$$
$$x_{37} = 31.9878172617356$$
$$x_{38} = -59.7385523684454$$
$$x_{39} = 84.3476948215655$$
$$x_{40} = 60.2621511440437$$
$$x_{41} = 38.2710025689152$$
$$x_{42} = -7.80568968373514$$
$$x_{43} = -37.7474037933169$$
$$x_{44} = 36.176607466522$$
$$x_{45} = -68.1161327780182$$
$$x_{46} = -49.693591731599$$
$$x_{47} = 14.1854588913934$$
$$x_{48} = -118.808630110574$$
$$x_{49} = 48.1227954048042$$
$$x_{50} = -44.0305891004964$$
$$x_{51} = 16.2798539937866$$
$$x_{52} = -61.8329474708386$$
$$x_{53} = -99.9590741890357$$
$$x_{54} = 97.9612629871212$$
$$x_{55} = 75.9701144119927$$
$$x_{56} = 53.9789658368641$$
$$x_{57} = 40.3653976713084$$
$$x_{58} = 130.424387074216$$
$$x_{59} = -3.18988460382913$$
$$x_{60} = -80.0623207163004$$
$$x_{61} = -55.9767770387786$$
$$x_{62} = -2.56970192775215$$
$$x_{63} = -0.0482919502393412$$
$$x_{64} = -66.021737675625$$
$$x_{65} = 78.0645095143859$$
$$x_{66} = -17.8506503205815$$
$$x_{67} = -46.1249842028896$$
$$x_{68} = -73.7791354091208$$
$$x_{69} = -22.0394405253679$$
$$x_{70} = 62.3565462464369$$
$$x_{71} = -95.7702839842493$$
$$x_{72} = 74.3027341847191$$
$$x_{73} = 82.2532997191723$$
$$x_{74} = -90.1072813531467$$
$$x_{75} = -75.873530511514$$
$$x_{76} = -33.9856284636501$$
$$x_{77} = 30.940619710539$$
$$x_{78} = 72.2083390823259$$
$$x_{79} = -48.2193793052828$$
$$x_{80} = -53.8823819363854$$
$$x_{81} = -29.7968382588637$$
$$x_{82} = -88.0128862507536$$
$$x_{83} = -25.1810331789577$$
$$x_{84} = -5.71129458134195$$
$$x_{85} = -51.7879868339922$$
$$x_{86} = 70.1139439799327$$
$$x_{87} = 56.0733609392573$$
$$x_{88} = -9.90008478612834$$
$$x_{89} = 6.23489335694024$$
$$x_{90} = -83.8240960459672$$
$$x_{91} = -85.9184911483604$$
$$x_{92} = 58.1677560416505$$
$$x_{93} = 43.9340052000178$$
$$x_{94} = 52.3115856095905$$
$$x_{95} = 34.0822123641288$$
$$x_{1} = -15.7562552181883$$
$$x_{2} = 35.1294099153254$$
$$x_{3} = -41.9361939981032$$
$$x_{4} = 68.0195488775395$$
$$x_{5} = 87.9163023502749$$
$$x_{6} = 96.2938827598476$$
$$x_{7} = -71.6847403067276$$
$$x_{8} = -27.7024431564705$$
$$x_{9} = 80.1589046167791$$
$$x_{10} = -97.8646790866426$$
$$x_{11} = -58.6913548172488$$
$$x_{12} = -19.9450454229747$$
$$x_{13} = 2.04610315215385$$
$$x_{14} = 46.028400302411$$
$$x_{15} = 12.0910637890002$$
$$x_{16} = 50.2171905071974$$
$$x_{17} = -11.9944798885215$$
$$x_{18} = 24.0372517272824$$
$$x_{19} = 94.1994876574545$$
$$x_{20} = 28.2260419320688$$
$$x_{21} = -63.9273425732318$$
$$x_{22} = 90.0106974526681$$
$$x_{23} = 92.1050925550613$$
$$x_{24} = 65.9251537751463$$
$$x_{25} = -31.8912333612569$$
$$x_{26} = 100.055658089514$$
$$x_{27} = 26.1316468296756$$
$$x_{28} = -39.8417988957101$$
$$x_{29} = 4.14049825454705$$
$$x_{30} = -77.9679256139072$$
$$x_{31} = 18.3742490961798$$
$$x_{32} = -24.5608505028807$$
$$x_{33} = 9.99666868660702$$
$$x_{34} = -81.729700943574$$
$$x_{35} = -93.6758888818562$$
$$x_{36} = 21.9428566248892$$
$$x_{37} = 31.9878172617356$$
$$x_{38} = -59.7385523684454$$
$$x_{39} = 84.3476948215655$$
$$x_{40} = 60.2621511440437$$
$$x_{41} = 38.2710025689152$$
$$x_{42} = -7.80568968373514$$
$$x_{43} = -37.7474037933169$$
$$x_{44} = 36.176607466522$$
$$x_{45} = -68.1161327780182$$
$$x_{46} = -49.693591731599$$
$$x_{47} = 14.1854588913934$$
$$x_{48} = -118.808630110574$$
$$x_{49} = 48.1227954048042$$
$$x_{50} = -44.0305891004964$$
$$x_{51} = 16.2798539937866$$
$$x_{52} = -61.8329474708386$$
$$x_{53} = -99.9590741890357$$
$$x_{54} = 97.9612629871212$$
$$x_{55} = 75.9701144119927$$
$$x_{56} = 53.9789658368641$$
$$x_{57} = 40.3653976713084$$
$$x_{58} = 130.424387074216$$
$$x_{59} = -3.18988460382913$$
$$x_{60} = -80.0623207163004$$
$$x_{61} = -55.9767770387786$$
$$x_{62} = -2.56970192775215$$
$$x_{63} = -0.0482919502393412$$
$$x_{64} = -66.021737675625$$
$$x_{65} = 78.0645095143859$$
$$x_{66} = -17.8506503205815$$
$$x_{67} = -46.1249842028896$$
$$x_{68} = -73.7791354091208$$
$$x_{69} = -22.0394405253679$$
$$x_{70} = 62.3565462464369$$
$$x_{71} = -95.7702839842493$$
$$x_{72} = 74.3027341847191$$
$$x_{73} = 82.2532997191723$$
$$x_{74} = -90.1072813531467$$
$$x_{75} = -75.873530511514$$
$$x_{76} = -33.9856284636501$$
$$x_{77} = 30.940619710539$$
$$x_{78} = 72.2083390823259$$
$$x_{79} = -48.2193793052828$$
$$x_{80} = -53.8823819363854$$
$$x_{81} = -29.7968382588637$$
$$x_{82} = -88.0128862507536$$
$$x_{83} = -25.1810331789577$$
$$x_{84} = -5.71129458134195$$
$$x_{85} = -51.7879868339922$$
$$x_{86} = 70.1139439799327$$
$$x_{87} = 56.0733609392573$$
$$x_{88} = -9.90008478612834$$
$$x_{89} = 6.23489335694024$$
$$x_{90} = -83.8240960459672$$
$$x_{91} = -85.9184911483604$$
$$x_{92} = 58.1677560416505$$
$$x_{93} = 43.9340052000178$$
$$x_{94} = 52.3115856095905$$
$$x_{95} = 34.0822123641288$$
This roots
$$x_{48} = -118.808630110574$$
$$x_{53} = -99.9590741890357$$
$$x_{10} = -97.8646790866426$$
$$x_{71} = -95.7702839842493$$
$$x_{35} = -93.6758888818562$$
$$x_{74} = -90.1072813531467$$
$$x_{82} = -88.0128862507536$$
$$x_{91} = -85.9184911483604$$
$$x_{90} = -83.8240960459672$$
$$x_{34} = -81.729700943574$$
$$x_{60} = -80.0623207163004$$
$$x_{30} = -77.9679256139072$$
$$x_{75} = -75.873530511514$$
$$x_{68} = -73.7791354091208$$
$$x_{7} = -71.6847403067276$$
$$x_{45} = -68.1161327780182$$
$$x_{64} = -66.021737675625$$
$$x_{21} = -63.9273425732318$$
$$x_{52} = -61.8329474708386$$
$$x_{38} = -59.7385523684454$$
$$x_{11} = -58.6913548172488$$
$$x_{61} = -55.9767770387786$$
$$x_{80} = -53.8823819363854$$
$$x_{85} = -51.7879868339922$$
$$x_{46} = -49.693591731599$$
$$x_{79} = -48.2193793052828$$
$$x_{67} = -46.1249842028896$$
$$x_{50} = -44.0305891004964$$
$$x_{3} = -41.9361939981032$$
$$x_{28} = -39.8417988957101$$
$$x_{43} = -37.7474037933169$$
$$x_{76} = -33.9856284636501$$
$$x_{25} = -31.8912333612569$$
$$x_{81} = -29.7968382588637$$
$$x_{8} = -27.7024431564705$$
$$x_{83} = -25.1810331789577$$
$$x_{32} = -24.5608505028807$$
$$x_{69} = -22.0394405253679$$
$$x_{12} = -19.9450454229747$$
$$x_{66} = -17.8506503205815$$
$$x_{1} = -15.7562552181883$$
$$x_{17} = -11.9944798885215$$
$$x_{88} = -9.90008478612834$$
$$x_{42} = -7.80568968373514$$
$$x_{84} = -5.71129458134195$$
$$x_{59} = -3.18988460382913$$
$$x_{62} = -2.56970192775215$$
$$x_{63} = -0.0482919502393412$$
$$x_{13} = 2.04610315215385$$
$$x_{29} = 4.14049825454705$$
$$x_{89} = 6.23489335694024$$
$$x_{33} = 9.99666868660702$$
$$x_{15} = 12.0910637890002$$
$$x_{47} = 14.1854588913934$$
$$x_{51} = 16.2798539937866$$
$$x_{31} = 18.3742490961798$$
$$x_{36} = 21.9428566248892$$
$$x_{18} = 24.0372517272824$$
$$x_{27} = 26.1316468296756$$
$$x_{20} = 28.2260419320688$$
$$x_{77} = 30.940619710539$$
$$x_{37} = 31.9878172617356$$
$$x_{95} = 34.0822123641288$$
$$x_{2} = 35.1294099153254$$
$$x_{44} = 36.176607466522$$
$$x_{41} = 38.2710025689152$$
$$x_{57} = 40.3653976713084$$
$$x_{93} = 43.9340052000178$$
$$x_{14} = 46.028400302411$$
$$x_{49} = 48.1227954048042$$
$$x_{16} = 50.2171905071974$$
$$x_{94} = 52.3115856095905$$
$$x_{56} = 53.9789658368641$$
$$x_{87} = 56.0733609392573$$
$$x_{92} = 58.1677560416505$$
$$x_{40} = 60.2621511440437$$
$$x_{70} = 62.3565462464369$$
$$x_{24} = 65.9251537751463$$
$$x_{4} = 68.0195488775395$$
$$x_{86} = 70.1139439799327$$
$$x_{78} = 72.2083390823259$$
$$x_{72} = 74.3027341847191$$
$$x_{55} = 75.9701144119927$$
$$x_{65} = 78.0645095143859$$
$$x_{9} = 80.1589046167791$$
$$x_{73} = 82.2532997191723$$
$$x_{39} = 84.3476948215655$$
$$x_{5} = 87.9163023502749$$
$$x_{22} = 90.0106974526681$$
$$x_{23} = 92.1050925550613$$
$$x_{19} = 94.1994876574545$$
$$x_{6} = 96.2938827598476$$
$$x_{54} = 97.9612629871212$$
$$x_{26} = 100.055658089514$$
$$x_{58} = 130.424387074216$$
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
$$x_{0} = x_{48} - \frac{1}{10}$$
=
$$-118.808630110574 + - \frac{1}{10}$$
=
$$-118.908630110574$$
substitute to the expression
$$\left(6 \sin{\left(3 x \right)} \cos{\left(3 x \right)} + \frac{\sin{\left(6 x \right)}}{2}\right) + 1 \geq 0$$
$$1 + \left(\frac{\sin{\left(\left(-118.908630110574\right) 6 \right)}}{2} + 6 \sin{\left(\left(-118.908630110574\right) 3 \right)} \cos{\left(\left(-118.908630110574\right) 3 \right)}\right) \geq 0$$
2.06853281533239 >= 0
one of the solutions of our inequality is:
$$x \leq -118.808630110574$$
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x48 x53 x10 x71 x35 x74 x82 x91 x90 x34 x60 x30 x75 x68 x7 x45 x64 x21 x52 x38 x11 x61 x80 x85 x46 x79 x67 x50 x3 x28 x43 x76 x25 x81 x8 x83 x32 x69 x12 x66 x1 x17 x88 x42 x84 x59 x62 x63 x13 x29 x89 x33 x15 x47 x51 x31 x36 x18 x27 x20 x77 x37 x95 x2 x44 x41 x57 x93 x14 x49 x16 x94 x56 x87 x92 x40 x70 x24 x4 x86 x78 x72 x55 x65 x9 x73 x39 x5 x22 x23 x19 x6 x54 x26 x58Other solutions will get with the changeover to the next point
etc.
The answer:
$$x \leq -118.808630110574$$
$$x \geq -99.9590741890357 \wedge x \leq -97.8646790866426$$
$$x \geq -95.7702839842493 \wedge x \leq -93.6758888818562$$
$$x \geq -90.1072813531467 \wedge x \leq -88.0128862507536$$
$$x \geq -85.9184911483604 \wedge x \leq -83.8240960459672$$
$$x \geq -81.729700943574 \wedge x \leq -80.0623207163004$$
$$x \geq -77.9679256139072 \wedge x \leq -75.873530511514$$
$$x \geq -73.7791354091208 \wedge x \leq -71.6847403067276$$
$$x \geq -68.1161327780182 \wedge x \leq -66.021737675625$$
$$x \geq -63.9273425732318 \wedge x \leq -61.8329474708386$$
$$x \geq -59.7385523684454 \wedge x \leq -58.6913548172488$$
$$x \geq -55.9767770387786 \wedge x \leq -53.8823819363854$$
$$x \geq -51.7879868339922 \wedge x \leq -49.693591731599$$
$$x \geq -48.2193793052828 \wedge x \leq -46.1249842028896$$
$$x \geq -44.0305891004964 \wedge x \leq -41.9361939981032$$
$$x \geq -39.8417988957101 \wedge x \leq -37.7474037933169$$
$$x \geq -33.9856284636501 \wedge x \leq -31.8912333612569$$
$$x \geq -29.7968382588637 \wedge x \leq -27.7024431564705$$
$$x \geq -25.1810331789577 \wedge x \leq -24.5608505028807$$
$$x \geq -22.0394405253679 \wedge x \leq -19.9450454229747$$
$$x \geq -17.8506503205815 \wedge x \leq -15.7562552181883$$
$$x \geq -11.9944798885215 \wedge x \leq -9.90008478612834$$
$$x \geq -7.80568968373514 \wedge x \leq -5.71129458134195$$
$$x \geq -3.18988460382913 \wedge x \leq -2.56970192775215$$
$$x \geq -0.0482919502393412 \wedge x \leq 2.04610315215385$$
$$x \geq 4.14049825454705 \wedge x \leq 6.23489335694024$$
$$x \geq 9.99666868660702 \wedge x \leq 12.0910637890002$$
$$x \geq 14.1854588913934 \wedge x \leq 16.2798539937866$$
$$x \geq 18.3742490961798 \wedge x \leq 21.9428566248892$$
$$x \geq 24.0372517272824 \wedge x \leq 26.1316468296756$$
$$x \geq 28.2260419320688 \wedge x \leq 30.940619710539$$
$$x \geq 31.9878172617356 \wedge x \leq 34.0822123641288$$
$$x \geq 35.1294099153254 \wedge x \leq 36.176607466522$$
$$x \geq 38.2710025689152 \wedge x \leq 40.3653976713084$$
$$x \geq 43.9340052000178 \wedge x \leq 46.028400302411$$
$$x \geq 48.1227954048042 \wedge x \leq 50.2171905071974$$
$$x \geq 52.3115856095905 \wedge x \leq 53.9789658368641$$
$$x \geq 56.0733609392573 \wedge x \leq 58.1677560416505$$
$$x \geq 60.2621511440437 \wedge x \leq 62.3565462464369$$
$$x \geq 65.9251537751463 \wedge x \leq 68.0195488775395$$
$$x \geq 70.1139439799327 \wedge x \leq 72.2083390823259$$
$$x \geq 74.3027341847191 \wedge x \leq 75.9701144119927$$
$$x \geq 78.0645095143859 \wedge x \leq 80.1589046167791$$
$$x \geq 82.2532997191723 \wedge x \leq 84.3476948215655$$
$$x \geq 87.9163023502749 \wedge x \leq 90.0106974526681$$
$$x \geq 92.1050925550613 \wedge x \leq 94.1994876574545$$
$$x \geq 96.2938827598476 \wedge x \leq 97.9612629871212$$
$$x \geq 100.055658089514 \wedge x \leq 130.424387074216$$
Solving inequality on a graph
Rapid solution 2
[src]
/ ___\ / ___\
|7 3*\/ 5 | |7 3*\/ 5 |
atan|- + -------| atan|- - -------|
\2 2 / pi \2 2 / pi pi
[0, - ----------------- + --] U [- ----------------- + --, --]
3 3 3 3 3
$$x\ in\ \left[0, - \frac{\operatorname{atan}{\left(\frac{3 \sqrt{5}}{2} + \frac{7}{2} \right)}}{3} + \frac{\pi}{3}\right] \cup \left[- \frac{\operatorname{atan}{\left(\frac{7}{2} - \frac{3 \sqrt{5}}{2} \right)}}{3} + \frac{\pi}{3}, \frac{\pi}{3}\right]$$
x in Union(Interval(0, -atan(3*sqrt(5)/2 + 7/2)/3 + pi/3), Interval(-atan(7/2 - 3*sqrt(5)/2)/3 + pi/3, pi/3))
Rapid solution
[src]
/ / / ___\ \ / / ___\ \\ | | |7 3*\/ 5 | | | |7 3*\/ 5 | || | | atan|- + -------| | | atan|- - -------| || | | \2 2 / pi| | pi \2 2 / pi || Or|And|0 <= x, x <= - ----------------- + --|, And|x <= --, - ----------------- + -- <= x|| \ \ 3 3 / \ 3 3 3 //
$$\left(0 \leq x \wedge x \leq - \frac{\operatorname{atan}{\left(\frac{3 \sqrt{5}}{2} + \frac{7}{2} \right)}}{3} + \frac{\pi}{3}\right) \vee \left(x \leq \frac{\pi}{3} \wedge - \frac{\operatorname{atan}{\left(\frac{7}{2} - \frac{3 \sqrt{5}}{2} \right)}}{3} + \frac{\pi}{3} \leq x\right)$$
((0 <= x)∧(x <= -atan(7/2 + 3*sqrt(5)/2)/3 + pi/3))∨((x <= pi/3)∧(-atan(7/2 - 3*sqrt(5)/2)/3 + pi/3 <= x))