Se da la desigualdad:
$$\sin^{4}{\left(x \right)} + \cos^{4}{\left(x \right)} \geq \frac{\sqrt{3}}{2}$$
Para resolver esta desigualdad primero hay que resolver la ecuación correspondiente:
$$\sin^{4}{\left(x \right)} + \cos^{4}{\left(x \right)} = \frac{\sqrt{3}}{2}$$
Resolvemos:
$$x_{1} = - 2 \operatorname{atan}{\left(\sqrt{4 \sqrt{3} + 7 + 4 \sqrt{5 + 3 \sqrt{3}} + 2 \sqrt{2} \sqrt{22 + 4 \sqrt{3} \sqrt{5 + 3 \sqrt{3}} + 7 \sqrt{5 + 3 \sqrt{3}} + 13 \sqrt{3}}} \right)}$$
$$x_{2} = 2 \operatorname{atan}{\left(\sqrt{4 \sqrt{3} + 7 + 4 \sqrt{5 + 3 \sqrt{3}} + 2 \sqrt{2} \sqrt{22 + 4 \sqrt{3} \sqrt{5 + 3 \sqrt{3}} + 7 \sqrt{5 + 3 \sqrt{3}} + 13 \sqrt{3}}} \right)}$$
$$x_{3} = - 2 \operatorname{atan}{\left(\sqrt{- 2 \sqrt{2} \sqrt{22 + 4 \sqrt{3} \sqrt{5 + 3 \sqrt{3}} + 7 \sqrt{5 + 3 \sqrt{3}} + 13 \sqrt{3}} + 4 \sqrt{3} + 7 + 4 \sqrt{5 + 3 \sqrt{3}}} \right)}$$
$$x_{4} = 2 \operatorname{atan}{\left(\sqrt{- 2 \sqrt{2} \sqrt{22 + 4 \sqrt{3} \sqrt{5 + 3 \sqrt{3}} + 7 \sqrt{5 + 3 \sqrt{3}} + 13 \sqrt{3}} + 4 \sqrt{3} + 7 + 4 \sqrt{5 + 3 \sqrt{3}}} \right)}$$
$$x_{5} = - 2 \operatorname{atan}{\left(\sqrt{- 4 \sqrt{5 + 3 \sqrt{3}} - 2 \sqrt{2} \sqrt{- 7 \sqrt{5 + 3 \sqrt{3}} - 4 \sqrt{3} \sqrt{5 + 3 \sqrt{3}} + 22 + 13 \sqrt{3}} + 4 \sqrt{3} + 7} \right)}$$
$$x_{6} = 2 \operatorname{atan}{\left(\sqrt{- 4 \sqrt{5 + 3 \sqrt{3}} - 2 \sqrt{2} \sqrt{- 7 \sqrt{5 + 3 \sqrt{3}} - 4 \sqrt{3} \sqrt{5 + 3 \sqrt{3}} + 22 + 13 \sqrt{3}} + 4 \sqrt{3} + 7} \right)}$$
$$x_{7} = - 2 \operatorname{atan}{\left(\sqrt{- 4 \sqrt{5 + 3 \sqrt{3}} + 2 \sqrt{2} \sqrt{- 7 \sqrt{5 + 3 \sqrt{3}} - 4 \sqrt{3} \sqrt{5 + 3 \sqrt{3}} + 22 + 13 \sqrt{3}} + 4 \sqrt{3} + 7} \right)}$$
$$x_{8} = 2 \operatorname{atan}{\left(\sqrt{- 4 \sqrt{5 + 3 \sqrt{3}} + 2 \sqrt{2} \sqrt{- 7 \sqrt{5 + 3 \sqrt{3}} - 4 \sqrt{3} \sqrt{5 + 3 \sqrt{3}} + 22 + 13 \sqrt{3}} + 4 \sqrt{3} + 7} \right)}$$
$$x_{1} = - 2 \operatorname{atan}{\left(\sqrt{4 \sqrt{3} + 7 + 4 \sqrt{5 + 3 \sqrt{3}} + 2 \sqrt{2} \sqrt{22 + 4 \sqrt{3} \sqrt{5 + 3 \sqrt{3}} + 7 \sqrt{5 + 3 \sqrt{3}} + 13 \sqrt{3}}} \right)}$$
$$x_{2} = 2 \operatorname{atan}{\left(\sqrt{4 \sqrt{3} + 7 + 4 \sqrt{5 + 3 \sqrt{3}} + 2 \sqrt{2} \sqrt{22 + 4 \sqrt{3} \sqrt{5 + 3 \sqrt{3}} + 7 \sqrt{5 + 3 \sqrt{3}} + 13 \sqrt{3}}} \right)}$$
$$x_{3} = - 2 \operatorname{atan}{\left(\sqrt{- 2 \sqrt{2} \sqrt{22 + 4 \sqrt{3} \sqrt{5 + 3 \sqrt{3}} + 7 \sqrt{5 + 3 \sqrt{3}} + 13 \sqrt{3}} + 4 \sqrt{3} + 7 + 4 \sqrt{5 + 3 \sqrt{3}}} \right)}$$
$$x_{4} = 2 \operatorname{atan}{\left(\sqrt{- 2 \sqrt{2} \sqrt{22 + 4 \sqrt{3} \sqrt{5 + 3 \sqrt{3}} + 7 \sqrt{5 + 3 \sqrt{3}} + 13 \sqrt{3}} + 4 \sqrt{3} + 7 + 4 \sqrt{5 + 3 \sqrt{3}}} \right)}$$
$$x_{5} = - 2 \operatorname{atan}{\left(\sqrt{- 4 \sqrt{5 + 3 \sqrt{3}} - 2 \sqrt{2} \sqrt{- 7 \sqrt{5 + 3 \sqrt{3}} - 4 \sqrt{3} \sqrt{5 + 3 \sqrt{3}} + 22 + 13 \sqrt{3}} + 4 \sqrt{3} + 7} \right)}$$
$$x_{6} = 2 \operatorname{atan}{\left(\sqrt{- 4 \sqrt{5 + 3 \sqrt{3}} - 2 \sqrt{2} \sqrt{- 7 \sqrt{5 + 3 \sqrt{3}} - 4 \sqrt{3} \sqrt{5 + 3 \sqrt{3}} + 22 + 13 \sqrt{3}} + 4 \sqrt{3} + 7} \right)}$$
$$x_{7} = - 2 \operatorname{atan}{\left(\sqrt{- 4 \sqrt{5 + 3 \sqrt{3}} + 2 \sqrt{2} \sqrt{- 7 \sqrt{5 + 3 \sqrt{3}} - 4 \sqrt{3} \sqrt{5 + 3 \sqrt{3}} + 22 + 13 \sqrt{3}} + 4 \sqrt{3} + 7} \right)}$$
$$x_{8} = 2 \operatorname{atan}{\left(\sqrt{- 4 \sqrt{5 + 3 \sqrt{3}} + 2 \sqrt{2} \sqrt{- 7 \sqrt{5 + 3 \sqrt{3}} - 4 \sqrt{3} \sqrt{5 + 3 \sqrt{3}} + 22 + 13 \sqrt{3}} + 4 \sqrt{3} + 7} \right)}$$
Las raíces dadas
$$x_{1} = - 2 \operatorname{atan}{\left(\sqrt{4 \sqrt{3} + 7 + 4 \sqrt{5 + 3 \sqrt{3}} + 2 \sqrt{2} \sqrt{22 + 4 \sqrt{3} \sqrt{5 + 3 \sqrt{3}} + 7 \sqrt{5 + 3 \sqrt{3}} + 13 \sqrt{3}}} \right)}$$
$$x_{7} = - 2 \operatorname{atan}{\left(\sqrt{- 4 \sqrt{5 + 3 \sqrt{3}} + 2 \sqrt{2} \sqrt{- 7 \sqrt{5 + 3 \sqrt{3}} - 4 \sqrt{3} \sqrt{5 + 3 \sqrt{3}} + 22 + 13 \sqrt{3}} + 4 \sqrt{3} + 7} \right)}$$
$$x_{5} = - 2 \operatorname{atan}{\left(\sqrt{- 4 \sqrt{5 + 3 \sqrt{3}} - 2 \sqrt{2} \sqrt{- 7 \sqrt{5 + 3 \sqrt{3}} - 4 \sqrt{3} \sqrt{5 + 3 \sqrt{3}} + 22 + 13 \sqrt{3}} + 4 \sqrt{3} + 7} \right)}$$
$$x_{3} = - 2 \operatorname{atan}{\left(\sqrt{- 2 \sqrt{2} \sqrt{22 + 4 \sqrt{3} \sqrt{5 + 3 \sqrt{3}} + 7 \sqrt{5 + 3 \sqrt{3}} + 13 \sqrt{3}} + 4 \sqrt{3} + 7 + 4 \sqrt{5 + 3 \sqrt{3}}} \right)}$$
$$x_{4} = 2 \operatorname{atan}{\left(\sqrt{- 2 \sqrt{2} \sqrt{22 + 4 \sqrt{3} \sqrt{5 + 3 \sqrt{3}} + 7 \sqrt{5 + 3 \sqrt{3}} + 13 \sqrt{3}} + 4 \sqrt{3} + 7 + 4 \sqrt{5 + 3 \sqrt{3}}} \right)}$$
$$x_{6} = 2 \operatorname{atan}{\left(\sqrt{- 4 \sqrt{5 + 3 \sqrt{3}} - 2 \sqrt{2} \sqrt{- 7 \sqrt{5 + 3 \sqrt{3}} - 4 \sqrt{3} \sqrt{5 + 3 \sqrt{3}} + 22 + 13 \sqrt{3}} + 4 \sqrt{3} + 7} \right)}$$
$$x_{8} = 2 \operatorname{atan}{\left(\sqrt{- 4 \sqrt{5 + 3 \sqrt{3}} + 2 \sqrt{2} \sqrt{- 7 \sqrt{5 + 3 \sqrt{3}} - 4 \sqrt{3} \sqrt{5 + 3 \sqrt{3}} + 22 + 13 \sqrt{3}} + 4 \sqrt{3} + 7} \right)}$$
$$x_{2} = 2 \operatorname{atan}{\left(\sqrt{4 \sqrt{3} + 7 + 4 \sqrt{5 + 3 \sqrt{3}} + 2 \sqrt{2} \sqrt{22 + 4 \sqrt{3} \sqrt{5 + 3 \sqrt{3}} + 7 \sqrt{5 + 3 \sqrt{3}} + 13 \sqrt{3}}} \right)}$$
son puntos de cambio del signo de desigualdad en las soluciones.
Primero definámonos con el signo hasta el punto extremo izquierdo:
$$x_{0} \leq x_{1}$$
Consideremos, por ejemplo, el punto
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- 2 \operatorname{atan}{\left(\sqrt{4 \sqrt{3} + 7 + 4 \sqrt{5 + 3 \sqrt{3}} + 2 \sqrt{2} \sqrt{22 + 4 \sqrt{3} \sqrt{5 + 3 \sqrt{3}} + 7 \sqrt{5 + 3 \sqrt{3}} + 13 \sqrt{3}}} \right)} - \frac{1}{10}$$
=
$$- 2 \operatorname{atan}{\left(\sqrt{4 \sqrt{3} + 7 + 4 \sqrt{5 + 3 \sqrt{3}} + 2 \sqrt{2} \sqrt{22 + 4 \sqrt{3} \sqrt{5 + 3 \sqrt{3}} + 7 \sqrt{5 + 3 \sqrt{3}} + 13 \sqrt{3}}} \right)} - \frac{1}{10}$$
lo sustituimos en la expresión
$$\sin^{4}{\left(x \right)} + \cos^{4}{\left(x \right)} \geq \frac{\sqrt{3}}{2}$$
$$\sin^{4}{\left(- 2 \operatorname{atan}{\left(\sqrt{4 \sqrt{3} + 7 + 4 \sqrt{5 + 3 \sqrt{3}} + 2 \sqrt{2} \sqrt{22 + 4 \sqrt{3} \sqrt{5 + 3 \sqrt{3}} + 7 \sqrt{5 + 3 \sqrt{3}} + 13 \sqrt{3}}} \right)} - \frac{1}{10} \right)} + \cos^{4}{\left(- 2 \operatorname{atan}{\left(\sqrt{4 \sqrt{3} + 7 + 4 \sqrt{5 + 3 \sqrt{3}} + 2 \sqrt{2} \sqrt{22 + 4 \sqrt{3} \sqrt{5 + 3 \sqrt{3}} + 7 \sqrt{5 + 3 \sqrt{3}} + 13 \sqrt{3}}} \right)} - \frac{1}{10} \right)} \geq \frac{\sqrt{3}}{2}$$
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| | / _____________ / _____________ _____________ || | | / _____________ / _____________ _____________ || \/ 3
4|1 | / ___ / ___ ___ / / ___ ___ ___ / ___ || 4|1 | / ___ / ___ ___ / / ___ ___ ___ / ___ || >= -----
cos |-- + 2*atan\\/ 7 + 4*\/ 3 + 4*\/ 5 + 3*\/ 3 + 2*\/ 2 *\/ 22 + 7*\/ 5 + 3*\/ 3 + 13*\/ 3 + 4*\/ 3 *\/ 5 + 3*\/ 3 /| + sin |-- + 2*atan\\/ 7 + 4*\/ 3 + 4*\/ 5 + 3*\/ 3 + 2*\/ 2 *\/ 22 + 7*\/ 5 + 3*\/ 3 + 13*\/ 3 + 4*\/ 3 *\/ 5 + 3*\/ 3 /| 2
\10 / \10 /
significa que una de las soluciones de nuestra ecuación será con:
$$x \leq - 2 \operatorname{atan}{\left(\sqrt{4 \sqrt{3} + 7 + 4 \sqrt{5 + 3 \sqrt{3}} + 2 \sqrt{2} \sqrt{22 + 4 \sqrt{3} \sqrt{5 + 3 \sqrt{3}} + 7 \sqrt{5 + 3 \sqrt{3}} + 13 \sqrt{3}}} \right)}$$
_____ _____ _____ _____ _____
\ / \ / \ / \ /
-------•-------•-------•-------•-------•-------•-------•-------•-------
x1 x7 x5 x3 x4 x6 x8 x2
Recibiremos otras soluciones de la desigualdad pasando al polo siguiente etc.
etc.
Respuesta:
$$x \leq - 2 \operatorname{atan}{\left(\sqrt{4 \sqrt{3} + 7 + 4 \sqrt{5 + 3 \sqrt{3}} + 2 \sqrt{2} \sqrt{22 + 4 \sqrt{3} \sqrt{5 + 3 \sqrt{3}} + 7 \sqrt{5 + 3 \sqrt{3}} + 13 \sqrt{3}}} \right)}$$
$$x \geq - 2 \operatorname{atan}{\left(\sqrt{- 4 \sqrt{5 + 3 \sqrt{3}} + 2 \sqrt{2} \sqrt{- 7 \sqrt{5 + 3 \sqrt{3}} - 4 \sqrt{3} \sqrt{5 + 3 \sqrt{3}} + 22 + 13 \sqrt{3}} + 4 \sqrt{3} + 7} \right)} \wedge x \leq - 2 \operatorname{atan}{\left(\sqrt{- 4 \sqrt{5 + 3 \sqrt{3}} - 2 \sqrt{2} \sqrt{- 7 \sqrt{5 + 3 \sqrt{3}} - 4 \sqrt{3} \sqrt{5 + 3 \sqrt{3}} + 22 + 13 \sqrt{3}} + 4 \sqrt{3} + 7} \right)}$$
$$x \geq - 2 \operatorname{atan}{\left(\sqrt{- 2 \sqrt{2} \sqrt{22 + 4 \sqrt{3} \sqrt{5 + 3 \sqrt{3}} + 7 \sqrt{5 + 3 \sqrt{3}} + 13 \sqrt{3}} + 4 \sqrt{3} + 7 + 4 \sqrt{5 + 3 \sqrt{3}}} \right)} \wedge x \leq 2 \operatorname{atan}{\left(\sqrt{- 2 \sqrt{2} \sqrt{22 + 4 \sqrt{3} \sqrt{5 + 3 \sqrt{3}} + 7 \sqrt{5 + 3 \sqrt{3}} + 13 \sqrt{3}} + 4 \sqrt{3} + 7 + 4 \sqrt{5 + 3 \sqrt{3}}} \right)}$$
$$x \geq 2 \operatorname{atan}{\left(\sqrt{- 4 \sqrt{5 + 3 \sqrt{3}} - 2 \sqrt{2} \sqrt{- 7 \sqrt{5 + 3 \sqrt{3}} - 4 \sqrt{3} \sqrt{5 + 3 \sqrt{3}} + 22 + 13 \sqrt{3}} + 4 \sqrt{3} + 7} \right)} \wedge x \leq 2 \operatorname{atan}{\left(\sqrt{- 4 \sqrt{5 + 3 \sqrt{3}} + 2 \sqrt{2} \sqrt{- 7 \sqrt{5 + 3 \sqrt{3}} - 4 \sqrt{3} \sqrt{5 + 3 \sqrt{3}} + 22 + 13 \sqrt{3}} + 4 \sqrt{3} + 7} \right)}$$
$$x \geq 2 \operatorname{atan}{\left(\sqrt{4 \sqrt{3} + 7 + 4 \sqrt{5 + 3 \sqrt{3}} + 2 \sqrt{2} \sqrt{22 + 4 \sqrt{3} \sqrt{5 + 3 \sqrt{3}} + 7 \sqrt{5 + 3 \sqrt{3}} + 13 \sqrt{3}}} \right)}$$