Sin a+ cos a=-2/3 với 3π/4<a<π Tính cos 2a

Ta có $\displaystyle \sin \alpha +\cos \alpha =-\frac{2}{3} \Rightarrow (\sin \alpha +\cos \alpha )^{2} =\left(\frac{-2}{3}\right)^{2}$
$\displaystyle  \begin{array}{{>{\displaystyle}l}}
\Rightarrow \sin^{2} \alpha +\cos^{2} \alpha +2\sin \alpha \cos \alpha =\frac{4}{9}\\
\Leftrightarrow 1+\sin 2\alpha =\frac{4}{9}\\
\Leftrightarrow \sin 2\alpha =\frac{-5}{9}
\end{array}$
Ta có $\displaystyle \sin^{2} 2\alpha +\cos^{2} 2\alpha =1\Rightarrow \cos^{2} 2\alpha =1-\left(\frac{-5}{9}\right)^{2} =\frac{56}{81}$
Có $\displaystyle \frac{3\pi }{4} < \alpha < \pi \Rightarrow \frac{3\pi }{2} < 2\alpha < 2\pi \Rightarrow \cos 2\alpha =\frac{2\sqrt{14}}{9}$