Đáp án:
\(\dfrac{{ - 32 + 255\sqrt 3 }}{{514}}\)
Giải thích các bước giải:
\(\begin{array}{l}
{\cos ^2}a + {\sin ^2}a = 1\\
\to {\left( {\dfrac{{255}}{{257}}} \right)^2} + {\cos ^2}a = 1\\
\to {\cos ^2}a = \dfrac{{1024}}{{66049}}\\
Do:a \in \left( {\dfrac{\pi }{2};\pi } \right)\\
\to \cos a < 0\\
\to \cos a = - \dfrac{{32}}{{257}}\\
\to \cos \left( {a - \dfrac{\pi }{3}} \right) = \cos a.\cos \dfrac{\pi }{3} + \sin a.\sin \dfrac{\pi }{3}\\
= - \dfrac{{32}}{{257}}.\dfrac{1}{2} + \dfrac{{255}}{{257}}.\dfrac{{\sqrt 3 }}{2}\\
= \dfrac{{ - 32 + 255\sqrt 3 }}{{514}}
\end{array}\)
