Đáp án:
\[\cos \frac{a}{2} = \frac{{\sqrt 5 }}{5}\]
Giải thích các bước giải:
Ta có:
\(\begin{array}{l}
\frac{\pi }{2} < a < \pi \Rightarrow \left\{ \begin{array}{l}
\sin a > 0\\
\cos a < 0
\end{array} \right.\\
{\sin ^2}a + {\cos ^2}a = 1\\
\cos a < 0 \Rightarrow \cos a = - \sqrt {1 - {{\sin }^2}a} = - \sqrt {1 - {{\left( {\frac{4}{5}} \right)}^2}} = - \frac{3}{5}\\
\cos a = 2{\cos ^2}\frac{a}{2} - 1 \Leftrightarrow - \frac{3}{5} = 2{\cos ^2}\frac{a}{2} - 1 \Rightarrow {\cos ^2}\frac{a}{2} = \frac{1}{5}\\
\frac{\pi }{2} < a < \pi \Rightarrow \frac{\pi }{4} < \frac{a}{2} < \frac{\pi }{2} \Rightarrow \cos \frac{a}{2} > 0\\
\Rightarrow \cos \frac{a}{2} = \frac{{\sqrt 5 }}{5}
\end{array}\)
Vậy \(\cos \frac{a}{2} = \frac{{\sqrt 5 }}{5}\)
