((cos^2)(x/3))>3/4 desigualdades

Se da la desigualdad:
$$\frac{x}{3} \cos^{2}{\left(x \right)} > \frac{3}{4}$$
Para resolver esta desigualdad primero hay que resolver la ecuación correspondiente:
$$\frac{x}{3} \cos^{2}{\left(x \right)} = \frac{3}{4}$$
Resolvemos:
$$x_{1} = 2.7149394618688$$
$$x_{2} = 33.2498832685612$$
$$x_{3} = 64.2143518632908$$
$$x_{4} = 55.1811971114491$$
$$x_{5} = 3.84073890468295$$
$$x_{6} = 20.7558575201056$$
$$x_{7} = 26.4073283652303$$
$$x_{8} = 3.84073890468315$$
$$x_{9} = 86.5557338591619$$
$$x_{10} = 2.71493946186879$$
$$x_{11} = 10.5146631440493$$
$$x_{12} = 95.9722958186243$$
$$x_{13} = 89.3760524205387$$
$$x_{14} = 80.2788183378609$$
$$x_{15} = 36.379645236609$$
$$x_{16} = 11.4547383396048$$
$$x_{17} = 20.0790127496302$$
$$x_{18} = 32.7213949076838$$
$$x_{19} = 48.4775464839108$$
$$x_{20} = 39.0274312679191$$
$$x_{21} = 61.4535892593694$$
$$x_{22} = 16.905319283048$$
$$x_{23} = 76.7970063550568$$
$$x_{24} = 70.506231164574$$
$$x_{25} = 83.4171868972646$$
$$x_{26} = 98.8086883883327$$
$$x_{27} = 86.2315552891753$$
$$x_{28} = 45.7766528451696$$
$$x_{29} = 74.0026915500082$$
$$x_{30} = 92.5203988110945$$
$$x_{31} = 52.0457279878448$$
$$x_{32} = 99.1114154863499$$
$$x_{33} = 89.6944433654572$$
$$x_{34} = 42.1784315096021$$
$$x_{35} = 39.5108675097475$$
$$x_{36} = 77.1406462915622$$
$$x_{37} = 83.0868931300394$$
$$x_{38} = 58.317172920454$$
$$x_{39} = 8.39803652194002$$
$$x_{40} = 54.7737803428729$$
$$x_{41} = 5.41304458387641$$
$$x_{42} = 17.6439242925576$$
$$x_{43} = 61.0679095593369$$
$$x_{44} = 30.1219587411443$$
$$x_{45} = 67.7275338829031$$
$$x_{46} = 23.8739770445748$$
$$x_{1} = 2.7149394618688$$
$$x_{2} = 33.2498832685612$$
$$x_{3} = 64.2143518632908$$
$$x_{4} = 55.1811971114491$$
$$x_{5} = 3.84073890468295$$
$$x_{6} = 20.7558575201056$$
$$x_{7} = 26.4073283652303$$
$$x_{8} = 3.84073890468315$$
$$x_{9} = 86.5557338591619$$
$$x_{10} = 2.71493946186879$$
$$x_{11} = 10.5146631440493$$
$$x_{12} = 95.9722958186243$$
$$x_{13} = 89.3760524205387$$
$$x_{14} = 80.2788183378609$$
$$x_{15} = 36.379645236609$$
$$x_{16} = 11.4547383396048$$
$$x_{17} = 20.0790127496302$$
$$x_{18} = 32.7213949076838$$
$$x_{19} = 48.4775464839108$$
$$x_{20} = 39.0274312679191$$
$$x_{21} = 61.4535892593694$$
$$x_{22} = 16.905319283048$$
$$x_{23} = 76.7970063550568$$
$$x_{24} = 70.506231164574$$
$$x_{25} = 83.4171868972646$$
$$x_{26} = 98.8086883883327$$
$$x_{27} = 86.2315552891753$$
$$x_{28} = 45.7766528451696$$
$$x_{29} = 74.0026915500082$$
$$x_{30} = 92.5203988110945$$
$$x_{31} = 52.0457279878448$$
$$x_{32} = 99.1114154863499$$
$$x_{33} = 89.6944433654572$$
$$x_{34} = 42.1784315096021$$
$$x_{35} = 39.5108675097475$$
$$x_{36} = 77.1406462915622$$
$$x_{37} = 83.0868931300394$$
$$x_{38} = 58.317172920454$$
$$x_{39} = 8.39803652194002$$
$$x_{40} = 54.7737803428729$$
$$x_{41} = 5.41304458387641$$
$$x_{42} = 17.6439242925576$$
$$x_{43} = 61.0679095593369$$
$$x_{44} = 30.1219587411443$$
$$x_{45} = 67.7275338829031$$
$$x_{46} = 23.8739770445748$$
Las raíces dadas
$$x_{10} = 2.71493946186879$$
$$x_{1} = 2.7149394618688$$
$$x_{5} = 3.84073890468295$$
$$x_{8} = 3.84073890468315$$
$$x_{41} = 5.41304458387641$$
$$x_{39} = 8.39803652194002$$
$$x_{11} = 10.5146631440493$$
$$x_{16} = 11.4547383396048$$
$$x_{22} = 16.905319283048$$
$$x_{42} = 17.6439242925576$$
$$x_{17} = 20.0790127496302$$
$$x_{6} = 20.7558575201056$$
$$x_{46} = 23.8739770445748$$
$$x_{7} = 26.4073283652303$$
$$x_{44} = 30.1219587411443$$
$$x_{18} = 32.7213949076838$$
$$x_{2} = 33.2498832685612$$
$$x_{15} = 36.379645236609$$
$$x_{20} = 39.0274312679191$$
$$x_{35} = 39.5108675097475$$
$$x_{34} = 42.1784315096021$$
$$x_{28} = 45.7766528451696$$
$$x_{19} = 48.4775464839108$$
$$x_{31} = 52.0457279878448$$
$$x_{40} = 54.7737803428729$$
$$x_{4} = 55.1811971114491$$
$$x_{38} = 58.317172920454$$
$$x_{43} = 61.0679095593369$$
$$x_{21} = 61.4535892593694$$
$$x_{3} = 64.2143518632908$$
$$x_{45} = 67.7275338829031$$
$$x_{24} = 70.506231164574$$
$$x_{29} = 74.0026915500082$$
$$x_{23} = 76.7970063550568$$
$$x_{36} = 77.1406462915622$$
$$x_{14} = 80.2788183378609$$
$$x_{37} = 83.0868931300394$$
$$x_{25} = 83.4171868972646$$
$$x_{27} = 86.2315552891753$$
$$x_{9} = 86.5557338591619$$
$$x_{13} = 89.3760524205387$$
$$x_{33} = 89.6944433654572$$
$$x_{30} = 92.5203988110945$$
$$x_{12} = 95.9722958186243$$
$$x_{26} = 98.8086883883327$$
$$x_{32} = 99.1114154863499$$
son puntos de cambio del signo de desigualdad en las soluciones.
Primero definámonos con el signo hasta el punto extremo izquierdo:
$$x_{0} < x_{10}$$
Consideremos, por ejemplo, el punto
$$x_{0} = x_{10} - \frac{1}{10}$$
=
$$- \frac{1}{10} + 2.71493946186879$$
=
$$2.61493946186879$$
lo sustituimos en la expresión
$$\frac{x}{3} \cos^{2}{\left(x \right)} > \frac{3}{4}$$
$$\frac{2.61493946186879}{3} \cos^{2}{\left(2.61493946186879 \right)} > \frac{3}{4}$$

0.65142513283262 > 3/4

Entonces
$$x < 2.71493946186879$$
no se cumple
significa que una de las soluciones de nuestra ecuación será con:
$$x > 2.71493946186879 \wedge x < 2.7149394618688$$

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Recibiremos otras soluciones de la desigualdad pasando al polo siguiente etc.
etc.
Respuesta:
$$x > 2.71493946186879 \wedge x < 2.7149394618688$$
$$x > 3.84073890468295 \wedge x < 3.84073890468315$$
$$x > 5.41304458387641 \wedge x < 8.39803652194002$$
$$x > 10.5146631440493 \wedge x < 11.4547383396048$$
$$x > 16.905319283048 \wedge x < 17.6439242925576$$
$$x > 20.0790127496302 \wedge x < 20.7558575201056$$
$$x > 23.8739770445748 \wedge x < 26.4073283652303$$
$$x > 30.1219587411443 \wedge x < 32.7213949076838$$
$$x > 33.2498832685612 \wedge x < 36.379645236609$$
$$x > 39.0274312679191 \wedge x < 39.5108675097475$$
$$x > 42.1784315096021 \wedge x < 45.7766528451696$$
$$x > 48.4775464839108 \wedge x < 52.0457279878448$$
$$x > 54.7737803428729 \wedge x < 55.1811971114491$$
$$x > 58.317172920454 \wedge x < 61.0679095593369$$
$$x > 61.4535892593694 \wedge x < 64.2143518632908$$
$$x > 67.7275338829031 \wedge x < 70.506231164574$$
$$x > 74.0026915500082 \wedge x < 76.7970063550568$$
$$x > 77.1406462915622 \wedge x < 80.2788183378609$$
$$x > 83.0868931300394 \wedge x < 83.4171868972646$$
$$x > 86.2315552891753 \wedge x < 86.5557338591619$$
$$x > 89.3760524205387 \wedge x < 89.6944433654572$$
$$x > 92.5203988110945 \wedge x < 95.9722958186243$$
$$x > 98.8086883883327 \wedge x < 99.1114154863499$$