Sr Examen
Otras calculadoras
- Sistemas de desigualdades paso a paso
- Ecuaciones básicas paso a paso
- Sistemas de ecuaciones paso a paso
- Solución de desigualdades trigonométricas en línea
- Resolución de desigualdades exponenciales en línea
- Resolución de desigualdades con un módulo en línea
-
¿Cómo usar?
- Desigualdades:
-
-3-5x<=x+3
-
2-7x>0
-
5x^2+4x>0
-
(x+4)*(x-2)<0
- Forma canónica:
- =0
-
Expresiones idénticas
- (ctg(dos *x))^ dos -3ctg(x)+ dos >= cero
- (ctg(2 multiplicar por x)) al cuadrado menos 3ctg(x) más 2 más o igual a 0
- (ctg(dos multiplicar por x)) en el grado dos menos 3ctg(x) más dos más o igual a cero
- (ctg(2*x))2-3ctg(x)+2>=0
- ctg2*x2-3ctgx+2>=0
- (ctg(2*x))²-3ctg(x)+2>=0
- (ctg(2*x)) en el grado 2-3ctg(x)+2>=0
- (ctg(2x))^2-3ctg(x)+2>=0
- (ctg(2x))2-3ctg(x)+2>=0
- ctg2x2-3ctgx+2>=0
- ctg2x^2-3ctgx+2>=0
- (ctg(2*x))^2-3ctg(x)+2>=O
-
Expresiones semejantes
- (ctg(2*x))^2+3ctg(x)+2>=0
- (ctg(2*x))^2-3ctg(x)-2>=0
-
Expresiones con funciones
- ctg
- ctg(2x)>=0
- ctg(2x)<=(-sqrt(3))*1/2
- ctg(x)>=y
(ctg(2*x))^2-3ctg(x)+2>=0 desigualdades
Solución
Ha introducido
[src]
2 cot (2*x) - 3*cot(x) + 2 >= 0
$$\left(- 3 \cot{\left(x \right)} + \cot^{2}{\left(2 x \right)}\right) + 2 \geq 0$$
-3*cot(x) + cot(2*x)^2 + 2 >= 0
Solución detallada
Se da la desigualdad:
$$\left(- 3 \cot{\left(x \right)} + \cot^{2}{\left(2 x \right)}\right) + 2 \geq 0$$
Para resolver esta desigualdad primero hay que resolver la ecuación correspondiente:
$$\left(- 3 \cot{\left(x \right)} + \cot^{2}{\left(2 x \right)}\right) + 2 = 0$$
Resolvemos:
Tenemos la ecuación
$$\left(- 3 \cot{\left(x \right)} + \cot^{2}{\left(2 x \right)}\right) + 2 = 0$$
cambiamos
$$- 3 \cot{\left(x \right)} + \cot^{2}{\left(2 x \right)} + 1 = 0$$
$$- 3 \cot{\left(x \right)} + \cot^{2}{\left(2 x \right)} + 1 = 0$$
Sustituimos
$$w = \cot{\left(2 x \right)}$$
Es la ecuación de la forma
La ecuación cuadrática puede ser resuelta
con la ayuda del discriminante.
Las raíces de la ecuación cuadrática:
$$w_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$w_{2} = \frac{- \sqrt{D} - b}{2 a}$$
donde D = b^2 - 4*a*c es el discriminante.
Como
$$a = 1$$
$$b = 0$$
$$c = 1 - 3 \cot{\left(x \right)}$$
, entonces
La ecuación tiene dos raíces.
o
$$w_{1} = \frac{\sqrt{12 \cot{\left(x \right)} - 4}}{2}$$
$$w_{2} = - \frac{\sqrt{12 \cot{\left(x \right)} - 4}}{2}$$
hacemos cambio inverso
$$\cot{\left(2 x \right)} = w$$
sustituimos w:
$$x_{1} = -2.1868803298028$$
$$x_{2} = 7.23789763096658$$
$$x_{3} = 66.9281580491727$$
$$x_{4} = 82.6361213171216$$
$$x_{5} = -14.753250944162$$
$$x_{6} = 79.4945286635318$$
$$x_{7} = -27.3196215585211$$
$$x_{8} = -99.5762525910864$$
$$x_{9} = 98.3440845850706$$
$$x_{10} = 76.352936009942$$
$$x_{11} = 13.5210829381462$$
$$x_{12} = 0.954712323786992$$
$$x_{13} = 54.3617874348135$$
$$x_{14} = 22.9458608989155$$
$$x_{15} = -93.2930672839068$$
$$x_{16} = 2646.17572664639$$
$$x_{17} = -77.5851040159578$$
$$x_{18} = -65.0187334015987$$
$$x_{19} = 48.0786021276339$$
$$x_{20} = 70.0697507027624$$
$$x_{21} = -83.8682893231374$$
$$x_{22} = -71.3019187087783$$
$$x_{23} = -87.0098819767272$$
$$x_{24} = -33.6028068657007$$
$$x_{25} = -61.8771407480089$$
$$x_{26} = 95.2024919314808$$
$$x_{27} = -43.0275848264701$$
$$x_{28} = 10.3794902845564$$
$$x_{29} = -30.4612142121109$$
$$x_{30} = 26.0874535525053$$
$$x_{31} = -39.8859921728803$$
$$x_{32} = 41.7954168204543$$
$$x_{33} = 19.8042682453258$$
$$x_{34} = 60.6449727419931$$
$$x_{35} = 62.9187678846351$$
$$x_{36} = -68.1603260551885$$
$$x_{37} = 16.662675591736$$
$$x_{38} = -52.4523627872395$$
$$x_{39} = 38.6538241668645$$
$$x_{40} = 32.3706388596849$$
$$x_{41} = -24.1780289049314$$
$$x_{42} = -49.3107701336497$$
$$x_{43} = -5.32847298339259$$
$$x_{44} = -21.0364362513416$$
$$x_{45} = -96.4346599374966$$
$$x_{46} = 44.9370094740441$$
$$x_{47} = -36.7443995192905$$
$$x_{48} = -8.47006563698239$$
$$x_{49} = -90.151474630317$$
$$x_{50} = 63.7865653955829$$
$$x_{51} = 92.060899277891$$
$$x_{52} = -17.8948435977518$$
$$x_{53} = -74.443511362368$$
$$x_{54} = 88.9193066243012$$
$$x_{55} = -46.1691774800599$$
$$x_{56} = 4.09630497737678$$
$$x_{57} = -11.6116582905722$$
$$x_{58} = 85.7777139707114$$
$$x_{59} = -55.5939554408293$$
$$x_{1} = -2.1868803298028$$
$$x_{2} = 7.23789763096658$$
$$x_{3} = 66.9281580491727$$
$$x_{4} = 82.6361213171216$$
$$x_{5} = -14.753250944162$$
$$x_{6} = 79.4945286635318$$
$$x_{7} = -27.3196215585211$$
$$x_{8} = -99.5762525910864$$
$$x_{9} = 98.3440845850706$$
$$x_{10} = 76.352936009942$$
$$x_{11} = 13.5210829381462$$
$$x_{12} = 0.954712323786992$$
$$x_{13} = 54.3617874348135$$
$$x_{14} = 22.9458608989155$$
$$x_{15} = -93.2930672839068$$
$$x_{16} = 2646.17572664639$$
$$x_{17} = -77.5851040159578$$
$$x_{18} = -65.0187334015987$$
$$x_{19} = 48.0786021276339$$
$$x_{20} = 70.0697507027624$$
$$x_{21} = -83.8682893231374$$
$$x_{22} = -71.3019187087783$$
$$x_{23} = -87.0098819767272$$
$$x_{24} = -33.6028068657007$$
$$x_{25} = -61.8771407480089$$
$$x_{26} = 95.2024919314808$$
$$x_{27} = -43.0275848264701$$
$$x_{28} = 10.3794902845564$$
$$x_{29} = -30.4612142121109$$
$$x_{30} = 26.0874535525053$$
$$x_{31} = -39.8859921728803$$
$$x_{32} = 41.7954168204543$$
$$x_{33} = 19.8042682453258$$
$$x_{34} = 60.6449727419931$$
$$x_{35} = 62.9187678846351$$
$$x_{36} = -68.1603260551885$$
$$x_{37} = 16.662675591736$$
$$x_{38} = -52.4523627872395$$
$$x_{39} = 38.6538241668645$$
$$x_{40} = 32.3706388596849$$
$$x_{41} = -24.1780289049314$$
$$x_{42} = -49.3107701336497$$
$$x_{43} = -5.32847298339259$$
$$x_{44} = -21.0364362513416$$
$$x_{45} = -96.4346599374966$$
$$x_{46} = 44.9370094740441$$
$$x_{47} = -36.7443995192905$$
$$x_{48} = -8.47006563698239$$
$$x_{49} = -90.151474630317$$
$$x_{50} = 63.7865653955829$$
$$x_{51} = 92.060899277891$$
$$x_{52} = -17.8948435977518$$
$$x_{53} = -74.443511362368$$
$$x_{54} = 88.9193066243012$$
$$x_{55} = -46.1691774800599$$
$$x_{56} = 4.09630497737678$$
$$x_{57} = -11.6116582905722$$
$$x_{58} = 85.7777139707114$$
$$x_{59} = -55.5939554408293$$
Las raíces dadas
$$x_{8} = -99.5762525910864$$
$$x_{45} = -96.4346599374966$$
$$x_{15} = -93.2930672839068$$
$$x_{49} = -90.151474630317$$
$$x_{23} = -87.0098819767272$$
$$x_{21} = -83.8682893231374$$
$$x_{17} = -77.5851040159578$$
$$x_{53} = -74.443511362368$$
$$x_{22} = -71.3019187087783$$
$$x_{36} = -68.1603260551885$$
$$x_{18} = -65.0187334015987$$
$$x_{25} = -61.8771407480089$$
$$x_{59} = -55.5939554408293$$
$$x_{38} = -52.4523627872395$$
$$x_{42} = -49.3107701336497$$
$$x_{55} = -46.1691774800599$$
$$x_{27} = -43.0275848264701$$
$$x_{31} = -39.8859921728803$$
$$x_{47} = -36.7443995192905$$
$$x_{24} = -33.6028068657007$$
$$x_{29} = -30.4612142121109$$
$$x_{7} = -27.3196215585211$$
$$x_{41} = -24.1780289049314$$
$$x_{44} = -21.0364362513416$$
$$x_{52} = -17.8948435977518$$
$$x_{5} = -14.753250944162$$
$$x_{57} = -11.6116582905722$$
$$x_{48} = -8.47006563698239$$
$$x_{43} = -5.32847298339259$$
$$x_{1} = -2.1868803298028$$
$$x_{12} = 0.954712323786992$$
$$x_{56} = 4.09630497737678$$
$$x_{2} = 7.23789763096658$$
$$x_{28} = 10.3794902845564$$
$$x_{11} = 13.5210829381462$$
$$x_{37} = 16.662675591736$$
$$x_{33} = 19.8042682453258$$
$$x_{14} = 22.9458608989155$$
$$x_{30} = 26.0874535525053$$
$$x_{40} = 32.3706388596849$$
$$x_{39} = 38.6538241668645$$
$$x_{32} = 41.7954168204543$$
$$x_{46} = 44.9370094740441$$
$$x_{19} = 48.0786021276339$$
$$x_{13} = 54.3617874348135$$
$$x_{34} = 60.6449727419931$$
$$x_{35} = 62.9187678846351$$
$$x_{50} = 63.7865653955829$$
$$x_{3} = 66.9281580491727$$
$$x_{20} = 70.0697507027624$$
$$x_{10} = 76.352936009942$$
$$x_{6} = 79.4945286635318$$
$$x_{4} = 82.6361213171216$$
$$x_{58} = 85.7777139707114$$
$$x_{54} = 88.9193066243012$$
$$x_{51} = 92.060899277891$$
$$x_{26} = 95.2024919314808$$
$$x_{9} = 98.3440845850706$$
$$x_{16} = 2646.17572664639$$
son puntos de cambio del signo de desigualdad en las soluciones.
Primero definámonos con el signo hasta el punto extremo izquierdo:
$$x_{0} \leq x_{8}$$
Consideremos, por ejemplo, el punto
$$x_{0} = x_{8} - \frac{1}{10}$$
=
$$-99.5762525910864 + - \frac{1}{10}$$
=
$$-99.6762525910864$$
lo sustituimos en la expresión
$$\left(- 3 \cot{\left(x \right)} + \cot^{2}{\left(2 x \right)}\right) + 2 \geq 0$$
$$\left(- 3 \cot{\left(-99.6762525910864 \right)} + \cot^{2}{\left(\left(-99.6762525910864\right) 2 \right)}\right) + 2 \geq 0$$
pero
Entonces
$$x \leq -99.5762525910864$$
no se cumple
significa que una de las soluciones de nuestra ecuación será con:
$$x \geq -99.5762525910864 \wedge x \leq -96.4346599374966$$
Recibiremos otras soluciones de la desigualdad pasando al polo siguiente etc.
etc.
Respuesta:
$$x \geq -99.5762525910864 \wedge x \leq -96.4346599374966$$
$$x \geq -93.2930672839068 \wedge x \leq -90.151474630317$$
$$x \geq -87.0098819767272 \wedge x \leq -83.8682893231374$$
$$x \geq -77.5851040159578 \wedge x \leq -74.443511362368$$
$$x \geq -71.3019187087783 \wedge x \leq -68.1603260551885$$
$$x \geq -65.0187334015987 \wedge x \leq -61.8771407480089$$
$$x \geq -55.5939554408293 \wedge x \leq -52.4523627872395$$
$$x \geq -49.3107701336497 \wedge x \leq -46.1691774800599$$
$$x \geq -43.0275848264701 \wedge x \leq -39.8859921728803$$
$$x \geq -36.7443995192905 \wedge x \leq -33.6028068657007$$
$$x \geq -30.4612142121109 \wedge x \leq -27.3196215585211$$
$$x \geq -24.1780289049314 \wedge x \leq -21.0364362513416$$
$$x \geq -17.8948435977518 \wedge x \leq -14.753250944162$$
$$x \geq -11.6116582905722 \wedge x \leq -8.47006563698239$$
$$x \geq -5.32847298339259 \wedge x \leq -2.1868803298028$$
$$x \geq 0.954712323786992 \wedge x \leq 4.09630497737678$$
$$x \geq 7.23789763096658 \wedge x \leq 10.3794902845564$$
$$x \geq 13.5210829381462 \wedge x \leq 16.662675591736$$
$$x \geq 19.8042682453258 \wedge x \leq 22.9458608989155$$
$$x \geq 26.0874535525053 \wedge x \leq 32.3706388596849$$
$$x \geq 38.6538241668645 \wedge x \leq 41.7954168204543$$
$$x \geq 44.9370094740441 \wedge x \leq 48.0786021276339$$
$$x \geq 54.3617874348135 \wedge x \leq 60.6449727419931$$
$$x \geq 62.9187678846351 \wedge x \leq 63.7865653955829$$
$$x \geq 66.9281580491727 \wedge x \leq 70.0697507027624$$
$$x \geq 76.352936009942 \wedge x \leq 79.4945286635318$$
$$x \geq 82.6361213171216 \wedge x \leq 85.7777139707114$$
$$x \geq 88.9193066243012 \wedge x \leq 92.060899277891$$
$$x \geq 95.2024919314808 \wedge x \leq 98.3440845850706$$
$$x \geq 2646.17572664639$$
$$\left(- 3 \cot{\left(x \right)} + \cot^{2}{\left(2 x \right)}\right) + 2 \geq 0$$
Para resolver esta desigualdad primero hay que resolver la ecuación correspondiente:
$$\left(- 3 \cot{\left(x \right)} + \cot^{2}{\left(2 x \right)}\right) + 2 = 0$$
Resolvemos:
Tenemos la ecuación
$$\left(- 3 \cot{\left(x \right)} + \cot^{2}{\left(2 x \right)}\right) + 2 = 0$$
cambiamos
$$- 3 \cot{\left(x \right)} + \cot^{2}{\left(2 x \right)} + 1 = 0$$
$$- 3 \cot{\left(x \right)} + \cot^{2}{\left(2 x \right)} + 1 = 0$$
Sustituimos
$$w = \cot{\left(2 x \right)}$$
Es la ecuación de la forma
a*w^2 + b*w + c = 0
La ecuación cuadrática puede ser resuelta
con la ayuda del discriminante.
Las raíces de la ecuación cuadrática:
$$w_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$w_{2} = \frac{- \sqrt{D} - b}{2 a}$$
donde D = b^2 - 4*a*c es el discriminante.
Como
$$a = 1$$
$$b = 0$$
$$c = 1 - 3 \cot{\left(x \right)}$$
, entonces
D = b^2 - 4 * a * c =
(0)^2 - 4 * (1) * (1 - 3*cot(x)) = -4 + 12*cot(x)
La ecuación tiene dos raíces.
w1 = (-b + sqrt(D)) / (2*a)
w2 = (-b - sqrt(D)) / (2*a)
o
$$w_{1} = \frac{\sqrt{12 \cot{\left(x \right)} - 4}}{2}$$
$$w_{2} = - \frac{\sqrt{12 \cot{\left(x \right)} - 4}}{2}$$
hacemos cambio inverso
$$\cot{\left(2 x \right)} = w$$
sustituimos w:
$$x_{1} = -2.1868803298028$$
$$x_{2} = 7.23789763096658$$
$$x_{3} = 66.9281580491727$$
$$x_{4} = 82.6361213171216$$
$$x_{5} = -14.753250944162$$
$$x_{6} = 79.4945286635318$$
$$x_{7} = -27.3196215585211$$
$$x_{8} = -99.5762525910864$$
$$x_{9} = 98.3440845850706$$
$$x_{10} = 76.352936009942$$
$$x_{11} = 13.5210829381462$$
$$x_{12} = 0.954712323786992$$
$$x_{13} = 54.3617874348135$$
$$x_{14} = 22.9458608989155$$
$$x_{15} = -93.2930672839068$$
$$x_{16} = 2646.17572664639$$
$$x_{17} = -77.5851040159578$$
$$x_{18} = -65.0187334015987$$
$$x_{19} = 48.0786021276339$$
$$x_{20} = 70.0697507027624$$
$$x_{21} = -83.8682893231374$$
$$x_{22} = -71.3019187087783$$
$$x_{23} = -87.0098819767272$$
$$x_{24} = -33.6028068657007$$
$$x_{25} = -61.8771407480089$$
$$x_{26} = 95.2024919314808$$
$$x_{27} = -43.0275848264701$$
$$x_{28} = 10.3794902845564$$
$$x_{29} = -30.4612142121109$$
$$x_{30} = 26.0874535525053$$
$$x_{31} = -39.8859921728803$$
$$x_{32} = 41.7954168204543$$
$$x_{33} = 19.8042682453258$$
$$x_{34} = 60.6449727419931$$
$$x_{35} = 62.9187678846351$$
$$x_{36} = -68.1603260551885$$
$$x_{37} = 16.662675591736$$
$$x_{38} = -52.4523627872395$$
$$x_{39} = 38.6538241668645$$
$$x_{40} = 32.3706388596849$$
$$x_{41} = -24.1780289049314$$
$$x_{42} = -49.3107701336497$$
$$x_{43} = -5.32847298339259$$
$$x_{44} = -21.0364362513416$$
$$x_{45} = -96.4346599374966$$
$$x_{46} = 44.9370094740441$$
$$x_{47} = -36.7443995192905$$
$$x_{48} = -8.47006563698239$$
$$x_{49} = -90.151474630317$$
$$x_{50} = 63.7865653955829$$
$$x_{51} = 92.060899277891$$
$$x_{52} = -17.8948435977518$$
$$x_{53} = -74.443511362368$$
$$x_{54} = 88.9193066243012$$
$$x_{55} = -46.1691774800599$$
$$x_{56} = 4.09630497737678$$
$$x_{57} = -11.6116582905722$$
$$x_{58} = 85.7777139707114$$
$$x_{59} = -55.5939554408293$$
$$x_{1} = -2.1868803298028$$
$$x_{2} = 7.23789763096658$$
$$x_{3} = 66.9281580491727$$
$$x_{4} = 82.6361213171216$$
$$x_{5} = -14.753250944162$$
$$x_{6} = 79.4945286635318$$
$$x_{7} = -27.3196215585211$$
$$x_{8} = -99.5762525910864$$
$$x_{9} = 98.3440845850706$$
$$x_{10} = 76.352936009942$$
$$x_{11} = 13.5210829381462$$
$$x_{12} = 0.954712323786992$$
$$x_{13} = 54.3617874348135$$
$$x_{14} = 22.9458608989155$$
$$x_{15} = -93.2930672839068$$
$$x_{16} = 2646.17572664639$$
$$x_{17} = -77.5851040159578$$
$$x_{18} = -65.0187334015987$$
$$x_{19} = 48.0786021276339$$
$$x_{20} = 70.0697507027624$$
$$x_{21} = -83.8682893231374$$
$$x_{22} = -71.3019187087783$$
$$x_{23} = -87.0098819767272$$
$$x_{24} = -33.6028068657007$$
$$x_{25} = -61.8771407480089$$
$$x_{26} = 95.2024919314808$$
$$x_{27} = -43.0275848264701$$
$$x_{28} = 10.3794902845564$$
$$x_{29} = -30.4612142121109$$
$$x_{30} = 26.0874535525053$$
$$x_{31} = -39.8859921728803$$
$$x_{32} = 41.7954168204543$$
$$x_{33} = 19.8042682453258$$
$$x_{34} = 60.6449727419931$$
$$x_{35} = 62.9187678846351$$
$$x_{36} = -68.1603260551885$$
$$x_{37} = 16.662675591736$$
$$x_{38} = -52.4523627872395$$
$$x_{39} = 38.6538241668645$$
$$x_{40} = 32.3706388596849$$
$$x_{41} = -24.1780289049314$$
$$x_{42} = -49.3107701336497$$
$$x_{43} = -5.32847298339259$$
$$x_{44} = -21.0364362513416$$
$$x_{45} = -96.4346599374966$$
$$x_{46} = 44.9370094740441$$
$$x_{47} = -36.7443995192905$$
$$x_{48} = -8.47006563698239$$
$$x_{49} = -90.151474630317$$
$$x_{50} = 63.7865653955829$$
$$x_{51} = 92.060899277891$$
$$x_{52} = -17.8948435977518$$
$$x_{53} = -74.443511362368$$
$$x_{54} = 88.9193066243012$$
$$x_{55} = -46.1691774800599$$
$$x_{56} = 4.09630497737678$$
$$x_{57} = -11.6116582905722$$
$$x_{58} = 85.7777139707114$$
$$x_{59} = -55.5939554408293$$
Las raíces dadas
$$x_{8} = -99.5762525910864$$
$$x_{45} = -96.4346599374966$$
$$x_{15} = -93.2930672839068$$
$$x_{49} = -90.151474630317$$
$$x_{23} = -87.0098819767272$$
$$x_{21} = -83.8682893231374$$
$$x_{17} = -77.5851040159578$$
$$x_{53} = -74.443511362368$$
$$x_{22} = -71.3019187087783$$
$$x_{36} = -68.1603260551885$$
$$x_{18} = -65.0187334015987$$
$$x_{25} = -61.8771407480089$$
$$x_{59} = -55.5939554408293$$
$$x_{38} = -52.4523627872395$$
$$x_{42} = -49.3107701336497$$
$$x_{55} = -46.1691774800599$$
$$x_{27} = -43.0275848264701$$
$$x_{31} = -39.8859921728803$$
$$x_{47} = -36.7443995192905$$
$$x_{24} = -33.6028068657007$$
$$x_{29} = -30.4612142121109$$
$$x_{7} = -27.3196215585211$$
$$x_{41} = -24.1780289049314$$
$$x_{44} = -21.0364362513416$$
$$x_{52} = -17.8948435977518$$
$$x_{5} = -14.753250944162$$
$$x_{57} = -11.6116582905722$$
$$x_{48} = -8.47006563698239$$
$$x_{43} = -5.32847298339259$$
$$x_{1} = -2.1868803298028$$
$$x_{12} = 0.954712323786992$$
$$x_{56} = 4.09630497737678$$
$$x_{2} = 7.23789763096658$$
$$x_{28} = 10.3794902845564$$
$$x_{11} = 13.5210829381462$$
$$x_{37} = 16.662675591736$$
$$x_{33} = 19.8042682453258$$
$$x_{14} = 22.9458608989155$$
$$x_{30} = 26.0874535525053$$
$$x_{40} = 32.3706388596849$$
$$x_{39} = 38.6538241668645$$
$$x_{32} = 41.7954168204543$$
$$x_{46} = 44.9370094740441$$
$$x_{19} = 48.0786021276339$$
$$x_{13} = 54.3617874348135$$
$$x_{34} = 60.6449727419931$$
$$x_{35} = 62.9187678846351$$
$$x_{50} = 63.7865653955829$$
$$x_{3} = 66.9281580491727$$
$$x_{20} = 70.0697507027624$$
$$x_{10} = 76.352936009942$$
$$x_{6} = 79.4945286635318$$
$$x_{4} = 82.6361213171216$$
$$x_{58} = 85.7777139707114$$
$$x_{54} = 88.9193066243012$$
$$x_{51} = 92.060899277891$$
$$x_{26} = 95.2024919314808$$
$$x_{9} = 98.3440845850706$$
$$x_{16} = 2646.17572664639$$
son puntos de cambio del signo de desigualdad en las soluciones.
Primero definámonos con el signo hasta el punto extremo izquierdo:
$$x_{0} \leq x_{8}$$
Consideremos, por ejemplo, el punto
$$x_{0} = x_{8} - \frac{1}{10}$$
=
$$-99.5762525910864 + - \frac{1}{10}$$
=
$$-99.6762525910864$$
lo sustituimos en la expresión
$$\left(- 3 \cot{\left(x \right)} + \cot^{2}{\left(2 x \right)}\right) + 2 \geq 0$$
$$\left(- 3 \cot{\left(-99.6762525910864 \right)} + \cot^{2}{\left(\left(-99.6762525910864\right) 2 \right)}\right) + 2 \geq 0$$
-0.591022891543910 >= 0
pero
-0.591022891543910 < 0
Entonces
$$x \leq -99.5762525910864$$
no se cumple
significa que una de las soluciones de nuestra ecuación será con:
$$x \geq -99.5762525910864 \wedge x \leq -96.4346599374966$$
_____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____
/ \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------
x8 x45 x15 x49 x23 x21 x17 x53 x22 x36 x18 x25 x59 x38 x42 x55 x27 x31 x47 x24 x29 x7 x41 x44 x52 x5 x57 x48 x43 x1 x12 x56 x2 x28 x11 x37 x33 x14 x30 x40 x39 x32 x46 x19 x13 x34 x35 x50 x3 x20 x10 x6 x4 x58 x54 x51 x26 x9 x16Recibiremos otras soluciones de la desigualdad pasando al polo siguiente etc.
etc.
Respuesta:
$$x \geq -99.5762525910864 \wedge x \leq -96.4346599374966$$
$$x \geq -93.2930672839068 \wedge x \leq -90.151474630317$$
$$x \geq -87.0098819767272 \wedge x \leq -83.8682893231374$$
$$x \geq -77.5851040159578 \wedge x \leq -74.443511362368$$
$$x \geq -71.3019187087783 \wedge x \leq -68.1603260551885$$
$$x \geq -65.0187334015987 \wedge x \leq -61.8771407480089$$
$$x \geq -55.5939554408293 \wedge x \leq -52.4523627872395$$
$$x \geq -49.3107701336497 \wedge x \leq -46.1691774800599$$
$$x \geq -43.0275848264701 \wedge x \leq -39.8859921728803$$
$$x \geq -36.7443995192905 \wedge x \leq -33.6028068657007$$
$$x \geq -30.4612142121109 \wedge x \leq -27.3196215585211$$
$$x \geq -24.1780289049314 \wedge x \leq -21.0364362513416$$
$$x \geq -17.8948435977518 \wedge x \leq -14.753250944162$$
$$x \geq -11.6116582905722 \wedge x \leq -8.47006563698239$$
$$x \geq -5.32847298339259 \wedge x \leq -2.1868803298028$$
$$x \geq 0.954712323786992 \wedge x \leq 4.09630497737678$$
$$x \geq 7.23789763096658 \wedge x \leq 10.3794902845564$$
$$x \geq 13.5210829381462 \wedge x \leq 16.662675591736$$
$$x \geq 19.8042682453258 \wedge x \leq 22.9458608989155$$
$$x \geq 26.0874535525053 \wedge x \leq 32.3706388596849$$
$$x \geq 38.6538241668645 \wedge x \leq 41.7954168204543$$
$$x \geq 44.9370094740441 \wedge x \leq 48.0786021276339$$
$$x \geq 54.3617874348135 \wedge x \leq 60.6449727419931$$
$$x \geq 62.9187678846351 \wedge x \leq 63.7865653955829$$
$$x \geq 66.9281580491727 \wedge x \leq 70.0697507027624$$
$$x \geq 76.352936009942 \wedge x \leq 79.4945286635318$$
$$x \geq 82.6361213171216 \wedge x \leq 85.7777139707114$$
$$x \geq 88.9193066243012 \wedge x \leq 92.060899277891$$
$$x \geq 95.2024919314808 \wedge x \leq 98.3440845850706$$
$$x \geq 2646.17572664639$$