Cho sin a = 3/5 với π/2 < a < π Tính sin 2a , cos 2a , tan 2a , cot ( a - π/4 ) , sin a/2 , cos a/2 Cảm ơn trc❤

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1. 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:

   $\begin{array}{l}
    + )\sin 2a = 2\sin a\cos a = 2.\dfrac{3}{5}.\left( {\dfrac{{ – 4}}{5}} \right) = \dfrac{{ – 24}}{{25}}\\
    + )\cos 2a = {\cos ^2}a – {\sin ^2}a = {\left( {\dfrac{{ – 4}}{5}} \right)^2} – {\left( {\dfrac{3}{5}} \right)^2} = \dfrac{7}{{25}}\\
    + )\tan 2a = \dfrac{{\sin 2a}}{{\cos 2a}} = \dfrac{{\dfrac{{ – 24}}{{25}}}}{{\dfrac{7}{{25}}}} = \dfrac{{ – 24}}{7}\\
    + )\cot \left( {a – \dfrac{\pi }{4}} \right)\\
    = \dfrac{1}{{\tan \left( {a – \dfrac{\pi }{4}} \right)}}\\
    = \dfrac{{1 + \tan a.\tan \dfrac{\pi }{4}}}{{\tan a – \tan \dfrac{\pi }{4}}}\\
    = \dfrac{{1 + \dfrac{{\sin a}}{{\cos a}}.1}}{{\dfrac{{\sin a}}{{\cos a}} – 1}}\\
    = \dfrac{{1 + \dfrac{{3/5}}{{ – 4/5}}}}{{\dfrac{{3/5}}{{ – 4/5}}}}\\
    = \dfrac{{1 – \dfrac{3}{4}}}{{\dfrac{{ – 3}}{4}}}\\
    = \dfrac{1}{4}.\dfrac{4}{{ – 3}}\\
    = \dfrac{{ – 1}}{3}\\
    + )\dfrac{\pi }{2} < a < \pi  \Rightarrow \dfrac{\pi }{4} < \dfrac{a}{2} < \dfrac{\pi }{2}\\
    \Rightarrow \sin \dfrac{a}{2} > 0\\
    \Rightarrow \sin \dfrac{a}{2} = \sqrt {\dfrac{{1 – \cos a}}{2}}  = \dfrac{3}{{\sqrt {10} }}\\
    \Rightarrow \cos \dfrac{a}{2} = \dfrac{{\sin a}}{{2\sin \dfrac{a}{2}}} = \dfrac{1}{{\sqrt {10} }}
   \end{array}$

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