Giải thích các bước giải:
Ta có:
\(\begin{array}{l}
\frac{\pi }{2} < a < \pi \Rightarrow \left\{ \begin{array}{l}
\sin a > 0\\
\cos a < 0
\end{array} \right.\\
\sin a + 2\cos a = \frac{1}{2} \Leftrightarrow \sin a = \frac{1}{2} - 2\cos a\\
{\sin ^2}a + {\cos ^2}a = 1\\
\Leftrightarrow {\left( {\frac{1}{2} - 2\cos a} \right)^2} + {\cos ^2}a = 1\\
\Leftrightarrow \frac{1}{4} - 2.\frac{1}{2}.2\cos a + 4{\cos ^2}a + {\cos ^2}a = 1\\
\Leftrightarrow 5{\cos ^2}a - 2\cos a - \frac{3}{4} = 0\\
\Leftrightarrow \cos a = \frac{{2 \pm \sqrt {19} }}{{10}}\\
\cos a < 0 \Rightarrow \cos a = \frac{{2 - \sqrt {19} }}{{10}}\\
\Rightarrow \sin a = \frac{1}{2} - 2\cos a = \frac{{1 + 2\sqrt {19} }}{{10}}\\
\sin 2a = 2\sin a.\cos a = \frac{{ - 36 + 3\sqrt {19} }}{{50}}\\
\cos 2a = 2{\cos ^2}a - 1 = - 0.8887....
\end{array}\)
