Cho s i n ( α ) + c o s ( β ) = 5 4 , khi đó sin ( 2 α ) có giá trị bằng:

9/20

Cho \[{\rm{sin}}\left( {\rm{\alpha }} \right){\rm{ + cos}}\left( {\rm{\beta }} \right){\rm{ = }}\frac{{\rm{5}}}{{\rm{4}}}\], khi đó \(\sin \left( {2\alpha } \right)\)có giá trị bằng:

\[\frac{{16}}{9}\]

\[\frac{6}{9}\]

\[\frac{9}{{16}}\]

\(\frac{9}{6}\)

Giải thích

Ta có:\[{\rm{sin}}\left( {\rm{\alpha }} \right){\rm{ + cos}}\left( {\rm{\beta }} \right){\rm{ = }}\frac{{\rm{5}}}{{\rm{4}}} \Leftrightarrow {\left[ {{\rm{sin}}\left( {\rm{\alpha }} \right){\rm{ + cos}}\left( {\rm{\beta }} \right)} \right]^{\rm{2}}}{\rm{ = }}{\left( {\frac{{\rm{5}}}{{\rm{4}}}} \right)^{\rm{2}}}{\rm{ = }}\frac{{{\rm{25}}}}{{{\rm{16}}}}\]

\[ \Rightarrow {\rm{si}}{{\rm{n}}^{\rm{2}}}\left( {\rm{\alpha }} \right){\rm{ + 2sin}}\left( {\rm{\alpha }} \right){\rm{cos}}\left( {\rm{\alpha }} \right){\rm{ + co}}{{\rm{s}}^{\rm{2}}}\left( {\rm{\beta }} \right){\rm{ = }}\frac{{{\rm{25}}}}{{{\rm{16}}}}\]

\[ \Rightarrow {\rm{1 + sin}}\left( {{\rm{2\alpha }}} \right){\rm{ = }}\frac{{{\rm{25}}}}{{{\rm{16}}}}\]

\[ \Rightarrow {\rm{sin}}\left( {{\rm{2\alpha }}} \right){\rm{ = }}\frac{{{\rm{25}}}}{{{\rm{16}}}} - {\rm{1 = }}\frac{{\rm{9}}}{{{\rm{16}}}}\]

Đáp án cần chọn là: C