Cho [ sin left( alpha right) = frac{1}{{ sqrt 3 }} ] với (0 < alpha < frac{ pi }{2} ). Tính giá trị của [ sin left( { alpha + frac{ pi }{3}} right) ] A. [ frac{{ sqrt 3 }}{6} - frac{{ sqrt 2 }}{2} ] B...

Ta có [ sin left( alpha right) = frac{1}{{ sqrt 3 }} ], [{ rm{si}}{{ rm{n}}^{ rm{2}}} left( { rm{ alpha }} right){ rm{ + co}}{{ rm{s}}^{ rm{2}}} left( { rm{ alpha }} right){ rm{ = 1}} Rightarrow { rm{co}}{{ rm{s}}^{ rm{2}}} left( { rm{ alpha }} right){ rm{ = 1}} - frac{{ rm{1}}}{{ rm{3}}}{ rm{ = }} frac{{ rm{2}}}{{ rm{3}}} ] Vì (0 < alpha < frac{ pi }{2} )nên [ cos left( alpha right) > 0 Rightarrow cos left( alpha right) = sqrt { frac{2}{3}} ]</> [ Rightarrow { rm{sin}} left( {{ rm{ alpha + }} frac{{ rm{ pi }}}{{ rm{3}}}} right){ rm{ = sin}} left( { rm{ alpha }} right){ rm{cos}} left( { frac{{ rm{ pi }}}{{ rm{3}}}} right){ rm{ + cos}} left( { rm{ alpha }} right){ rm{sin}} left( { frac{{ rm{ pi }}}{{ rm{3}}}} right){ rm{ = }} frac{{ rm{1}}}{{ sqrt { rm{3}} }}{ rm{.}} frac{{ rm{1}}}{{ rm{2}}}{ rm{ + }} sqrt { frac{{ rm{2}}}{{ rm{3}}}} { rm{.}} frac{{ sqrt { rm{3}} }}{{ rm{2}}}{ rm{ = }} frac{{ sqrt { rm{3}} }}{{ rm{6}}}{ rm{ + }} frac{{ sqrt { rm{2}} }}{{ rm{2}}} ] Đáp án cần chọn là: D