Cho \(\sin \left( \alpha \right) + \cos \left( \alpha \right) = \frac{5}{4}\), khi đó \(\sin \left( {2\alpha } \right)\) có giá trị bằng:
-
A.
\(\frac{{16}}{9}\)
-
B.
\(\frac{6}{9}\)
-
C.
\(\frac{9}{{16}}\)
-
D.
\(\frac{9}{6}\)
Sử dụng công thức \(\sin \left( {2\alpha } \right) = 2\sin \left( \alpha \right)\cos \left( \alpha \right)\)
Ta có: \(\sin \left( \alpha \right) + \cos \left( \alpha \right) = \frac{5}{4} \Leftrightarrow {\left[ {\sin \left( \alpha \right) + \cos \left( \alpha \right)} \right]^2} = {\left( {\frac{5}{4}} \right)^2} = \frac{{25}}{{16}}\)
\( \Rightarrow {\sin ^2}\left( \alpha \right) + 2\sin \left( \alpha \right)\cos \left( \alpha \right) + {\cos ^2}\left( \alpha \right) = \frac{{25}}{{16}}\)
\( \Rightarrow 1 + \sin \left( {2\alpha } \right) = \frac{{25}}{{16}}\)
\( \Rightarrow \sin \left( {2\alpha } \right) = \frac{{25}}{{16}} - 1 = \frac{9}{{16}}\)
Đáp án : C
Danh sách bình luận