Cho \(\sin \alpha  = \frac{1}{{\sqrt 3 }}\) với \(0 var DOM...

* Hướng dẫn giải

Ta có: \(\sin \alpha  = \frac{1}{{\sqrt 3 }}\) mà \({\sin ^2}\alpha  + {\cos ^2}\alpha  = 1 \Rightarrow {\cos ^2}\alpha  = \frac{2}{3}.\)

Lại có \(0 < \alpha  < \frac{\pi }{2}\) nên \(\cos \alpha  > 0 \Rightarrow \cos \alpha  = \sqrt {\frac{2}{3}} \)

\(\begin{array}{l} \Rightarrow \sin \left( {\alpha  + \frac{\pi }{3}} \right) = \sin \alpha \cos \frac{\pi }{3} + \cos \alpha \sin \frac{\pi }{3}\\\, = \frac{1}{{\sqrt 3 }}.\frac{1}{2} + \cos \alpha .\frac{{\sqrt 3 }}{2}\\ = \frac{1}{{\sqrt 3 }}.\frac{1}{2} + \sqrt {\frac{2}{3}} .\frac{{\sqrt 3 }}{2} = \frac{{\sqrt 3 }}{6} + \frac{{\sqrt 2 }}{2}.\end{array}\)