Cho \(\sin \alpha = \frac{2}{3}\) với \(0 < \alpha < \frac{\pi }{2}\).

a) \(\sin 2\alpha = \frac{{4\sqrt 5 }}{9}\).

b) \(\cos \left( {\alpha + \frac{{3\pi }}{2}} \right) = \frac{2}{3}\).

c) \(\sqrt 2 \cos \left( {\alpha + \frac{\pi }{4}} \right) = \frac{{2 + \sqrt 5 }}{3}\).

d) \(D = \frac{{\cot \alpha + \tan \alpha }}{{\cot \alpha - \tan \alpha }} = \frac{1}{9}\).

a) \(\sin \alpha = \frac{2}{3}\) với \(0 < \alpha < \frac{\pi }{2}\) suy ra \(\cos \alpha = \frac{{\sqrt 5 }}{3}\) \( \Rightarrow \sin 2\alpha = 2\sin \alpha .\cos \alpha = 2.\frac{2}{3}.\frac{{\sqrt 5 }}{3} = \frac{{4\sqrt 5 }}{9}\).

b) \(\cos \left( {\alpha + \frac{{3\pi }}{2}} \right) = \cos \left( {\alpha + 2\pi - \frac{\pi }{2}} \right) = \cos \left( {\alpha - \frac{\pi }{2}} \right) = \cos \left( {\frac{\pi }{2} - \alpha } \right) = \sin \alpha = \frac{2}{3}\).

c) \(\sqrt 2 \cos \left( {\alpha + \frac{\pi }{4}} \right) = \cos \alpha - \sin \alpha = \frac{{\sqrt 5 - 2}}{3}\).

d) Có \(\sin \alpha = \frac{2}{3};\cos \alpha = \frac{{\sqrt 5 }}{3}\) \( \Rightarrow \tan \alpha = \frac{2}{{\sqrt 5 }},\cot \alpha = \frac{{\sqrt 5 }}{2}\).

Do đó \(D = \frac{{\cot \alpha + \tan \alpha }}{{\cot \alpha - \tan \alpha }} = \frac{{\frac{{\sqrt 5 }}{2} + \frac{2}{{\sqrt 5 }}}}{{\frac{{\sqrt 5 }}{2} - \frac{2}{{\sqrt 5 }}}} = 9\).

Đáp án: a) Đúng;   b) Đúng;   c) Sai;   d) Sai.