$$\eqalign{
& \cos a - {\mathop{\rm sina}\nolimits} = {1 \over 2} \Leftrightarrow {\mathop{\rm cosa}\nolimits} = \sin a + {1 \over 2} \cr
& Ta\,\,co:\,\,{\sin ^2}a + {\cos ^2}a = 1\,\,\forall a \cr
& \Rightarrow {\sin ^2}a + {\left( {\sin a + {1 \over 2}} \right)^2} = 1 \cr
& \Leftrightarrow {\sin ^2}a + {\sin ^2}a + \sin a + {1 \over 4} = 1 \cr
& \Leftrightarrow 2{\sin ^2}a + {\mathop{\rm sina}\nolimits} - {3 \over 4} = 0 \cr
& \Leftrightarrow \left[ \matrix{
\sin a = {{ - 1 + \sqrt 7 } \over 2}\,\,\left( {tm} \right) \Rightarrow \cos a = {{1 + \sqrt 7 } \over 4} \hfill \cr
\sin a = {{ - 1 - \sqrt 7 } \over 4}\,\,\left( {tm} \right) \Rightarrow \cos a = {{1 - \sqrt 7 } \over 4} \hfill \cr} \right. \cr} $$
