Cho (cos a = frac{3}{4};sin a > 0;sin b = frac{3}{5};cos b

Câu hỏi:

Cho \(\cos a = \frac{3}{4};\sin a > 0;\sin b = \frac{3}{5};\cos b < 0\). Giá trị của \(\cos \left( {a + b} \right).\) bằng :

A.
\(\frac{3}{5}\left( {1 + \frac{{\sqrt 7 }}{4}} \right).\)

B.
\(-\frac{3}{5}\left( {1 + \frac{{\sqrt 7 }}{4}} \right).\)

C.
\(\frac{3}{5}\left( {1 – \frac{{\sqrt 7 }}{4}} \right).\)

D.
\(-\frac{3}{5}\left( {1 – \frac{{\sqrt 7 }}{4}} \right).\)

Đáp án đúng: A

Ta có 

\(\begin{array}{l}
\left\{ \begin{array}{l}
\cos a = \frac{3}{4}\\
\sin a > 0
\end{array} \right. \Rightarrow \sin a = \sqrt {1 – {{\cos }^2}a}  = \frac{{\sqrt 7 }}{4}\\
\left\{ \begin{array}{l}
\sin b = \frac{3}{5}\\
\cos b < 0
\end{array} \right. \Rightarrow \cos b =  – \sqrt {1 – {{\sin }^2}b}  =  – \frac{4}{5}.\\
\cos \left( {a + b} \right) = \cos a\cos b – \sin a\sin b = \frac{3}{4}.\left( { – \frac{4}{5}} \right) – \frac{{\sqrt 7 }}{4}.\frac{3}{5} =  – \frac{3}{5}\left( {1 + \frac{{\sqrt 7 }}{4}} \right).
\end{array}\)