\(4{\sin ^3}x + 4{\cos ^2}x \ge 1 + 3\sin x\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,4{\sin ^3}x + 4-4{\sin ^2}x \ge 1 + 3\sin x\,\,\,\,\,\,\, \Leftrightarrow \)
\( \Leftrightarrow \,\,\,\,\,\,\,4{\sin ^3}x-4{\sin ^2}x-3\sin x + 3 \ge 0.\)
Пусть \(\sin x = t.\) Тогда:
\(4{t^3}-4{t^2}-3t + 3 \ge 0\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,4{t^2}\left( {t-1} \right)-3\left( {t-1} \right) \ge 0\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\,\left( {t-1} \right)\left( {4{t^2}-3} \right) \ge 0.\)
\(\left[ {\begin{array}{*{20}{c}}{-\dfrac{{\sqrt 3 }}{2} \le t \le \dfrac{{\sqrt 3 }}{2},}\\{t \ge 1\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,}\end{array}} \right.\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\left[ {\begin{array}{*{20}{c}}{-\dfrac{{\sqrt 3 }}{2} \le \sin x \le \dfrac{{\sqrt 3 }}{2},}\\{\sin x \ge 1\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,}\end{array}\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\left[ {\begin{array}{*{20}{c}}{x\, \in \,\left[ {-\dfrac{\pi }{3} + \pi k;\dfrac{\pi }{3} + \pi k} \right],}\\{x = \dfrac{\pi }{2} + 2\pi k,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,}\end{array}} \right.} \right.\,\,\,\,\,\,k\, \in \,Z.\,\,\,\,\)
Ответ: \(\left[ {-\dfrac{\pi }{3} + \pi k;\dfrac{\pi }{3} + \pi k} \right] \cup \left\{ {\dfrac{\pi }{2} + 2\pi k} \right\},\,\,\,\,k\, \in \,Z.\)