\[\begin{array}{l}
1 + 2co{s^2}x + 5sinx = 0\\
\Leftrightarrow 1 + 2\left( {1 - {{\sin }^2}x} \right) + 5\sin x = 0\\
\Leftrightarrow 1 + 2 - 2{\sin ^2}x + 5\sin x = 0\\
\Leftrightarrow - 2{\sin ^2}x + 6\sin x - \sin x + 3 = 0\\
\Leftrightarrow - 2\sin x\left( {\sin x - 3} \right) - \left( {\sin x - 3} \right) = 0\\
\Leftrightarrow - \left( {\sin x - 3} \right)\left( {2\sin x + 1} \right) = 0\\
\Leftrightarrow \left[ \begin{array}{l}
\sin x - 3 = 0\\
2\sin x + 1 = 0
\end{array} \right.\\
\Leftrightarrow \left[ \begin{array}{l}
\sin x = 3\left( {ktm} \right)\\
\sin x = \frac{{ - 1}}{2}
\end{array} \right.\\
\Leftrightarrow \sin x = \sin \frac{{ - \pi }}{6}\\
\Leftrightarrow \left[ \begin{array}{l}
x = \frac{{ - \pi }}{6} + k2\pi \\
x = \pi - \left( {\frac{{ - \pi }}{6}} \right) + k2\pi
\end{array} \right.\\
\Leftrightarrow \left[ \begin{array}{l}
x = \frac{{ - \pi }}{6} + k2\pi \\
x = \frac{{7\pi }}{6} + k2\pi
\end{array} \right.\left( {k \in Z} \right)\\
vay:S = \{ \frac{{ - \pi }}{6} + k2\pi ;\frac{{7\pi }}{6} + k2\}
\end{array}\]