[tex]\sin a+\cos a =\frac{1}{3} \ |^2\\ \\ \sin^2 a+\cos^2a+2\sin a \cos a=\frac{1}{9}\\ \\ 1+\sin2a=\frac{1}{9}\\ \\ \Rightarrow \sin2a=\frac{1}{9}-\frac{9}{9}\\ \\ \Rightarrow \sin2a=\frac{-8}{9}[/tex]
