Giải thích các bước giải:
Do:
$\sin a = \dfrac{3}{5}$; $\dfrac{\pi }{2} < a < \pi $
$ \Rightarrow \cos a < 0$
$ \Rightarrow \cos a = - \sqrt {1 - {{\sin }^2}a} = \dfrac{{ - 4}}{5}$
Khi đó:
$\begin{array}{l}
+ )\sin 2a = 2\sin a\cos a = 2.\dfrac{3}{5}.\left( {\dfrac{{ - 4}}{5}} \right) = \dfrac{{ - 24}}{{25}}\\
+ )\cos 2a = {\cos ^2}a - {\sin ^2}a = {\left( {\dfrac{{ - 4}}{5}} \right)^2} - {\left( {\dfrac{3}{5}} \right)^2} = \dfrac{7}{{25}}\\
+ )\tan 2a = \dfrac{{\sin 2a}}{{\cos 2a}} = \dfrac{{\dfrac{{ - 24}}{{25}}}}{{\dfrac{7}{{25}}}} = \dfrac{{ - 24}}{7}\\
+ )\cot \left( {a - \dfrac{\pi }{4}} \right)\\
= \dfrac{1}{{\tan \left( {a - \dfrac{\pi }{4}} \right)}}\\
= \dfrac{{1 + \tan a.\tan \dfrac{\pi }{4}}}{{\tan a - \tan \dfrac{\pi }{4}}}\\
= \dfrac{{1 + \dfrac{{\sin a}}{{\cos a}}.1}}{{\dfrac{{\sin a}}{{\cos a}} - 1}}\\
= \dfrac{{1 + \dfrac{{3/5}}{{ - 4/5}}}}{{\dfrac{{3/5}}{{ - 4/5}}}}\\
= \dfrac{{1 - \dfrac{3}{4}}}{{\dfrac{{ - 3}}{4}}}\\
= \dfrac{1}{4}.\dfrac{4}{{ - 3}}\\
= \dfrac{{ - 1}}{3}\\
+ )\dfrac{\pi }{2} < a < \pi \Rightarrow \dfrac{\pi }{4} < \dfrac{a}{2} < \dfrac{\pi }{2}\\
\Rightarrow \sin \dfrac{a}{2} > 0\\
\Rightarrow \sin \dfrac{a}{2} = \sqrt {\dfrac{{1 - \cos a}}{2}} = \dfrac{3}{{\sqrt {10} }}\\
\Rightarrow \cos \dfrac{a}{2} = \dfrac{{\sin a}}{{2\sin \dfrac{a}{2}}} = \dfrac{1}{{\sqrt {10} }}
\end{array}$
