Đáp án:
\[\cos \left( {\frac{\pi }{3} - a} \right) = \frac{{13\sqrt 3 - 12}}{{2\sqrt {313} }}\]
Giải thích các bước giải:
Ta có:
\(\begin{array}{l}
\frac{{3\pi }}{2} < a < 2\pi \Rightarrow \left\{ \begin{array}{l}
\sin a < 0\\
\cos a > 0
\end{array} \right.\\
\tan a = \frac{{ - 12}}{{13}} \Leftrightarrow \frac{{\sin a}}{{\cos a}} = - \frac{{12}}{{13}} \Leftrightarrow \sin a = - \frac{{12}}{{13}}\cos a\\
{\sin ^2}a + {\cos ^2}a = 1\\
\Leftrightarrow {\left( { - \frac{{12}}{{13}}\cos a} \right)^2} + {\cos ^2}a = 1\\
\Leftrightarrow \frac{{313}}{{169}}{\cos ^2}a = 1\\
\Leftrightarrow {\cos ^2}a = \frac{{169}}{{313}}\\
\cos a > 0 \Rightarrow \cos a = \frac{{13}}{{\sqrt {313} }}\\
\sin a = \frac{{ - 12}}{{13}}\cos a = - \frac{{12}}{{\sqrt {313} }}\\
\cos \left( {\frac{\pi }{3} - a} \right) = \cos \frac{\pi }{3}.\cos a + \sin \frac{\pi }{3}.\sin a = \frac{1}{2}.\frac{{ - 12}}{{\sqrt {313} }} + \frac{{\sqrt 3 }}{2}.\frac{{13}}{{\sqrt {313} }} = \frac{{13\sqrt 3 - 12}}{{2\sqrt {313} }}
\end{array}\)
Vậy \(\cos \left( {\frac{\pi }{3} - a} \right) = \frac{{13\sqrt 3 - 12}}{{2\sqrt {313} }}\)
