Answer
$\frac{1-2\sqrt 6}{6}$
Work Step by Step
1. $f(\alpha+\beta)=sin(\alpha+\beta)=sin\alpha cos\beta +cos\alpha sin\beta $
2. From the given figures, we have $sin\alpha=\frac{1}{2}, cos\alpha=\frac{\sqrt 3}{2}, sin\beta=-\frac{2\sqrt 2}{3}, cos\beta=\frac{1}{3}$
3. Thus $f(\alpha+\beta)=(\frac{1}{2})(\frac{1}{3})+(\frac{\sqrt 3}{2})(-\frac{2\sqrt 2}{3})=\frac{1-2\sqrt 6}{6}$
