From Förberedande kurs i matematik 1

We can write the expression \displaystyle \sin (x+y) in terms of \displaystyle \sin x, \displaystyle \cos x, \displaystyle \sin y and \displaystyle \cos y if we use the addition formula for sine,

\displaystyle \sin (x+y) = \sin x\cdot \cos y + \cos x\cdot \sin y\,\textrm{.}

In turn, it is possible to express the factors \displaystyle \cos x and \displaystyle \cos y in terms of \displaystyle \sin x and \displaystyle \sin y by using the Pythagorean identity,

\displaystyle \begin{align} \cos x &= \pm \sqrt{1-\sin^2\!x} = \pm \sqrt{1-(2/3)^2} = \pm\frac{\sqrt{5}}{3}\,,\\[5pt] \cos y &= \pm \sqrt{1-\sin^2\!y} = \pm \sqrt{1-(1/3)^{2}} = \pm \frac{2\sqrt{2}}{3}\,\textrm{.} \end{align}

Because x and y are angles in the first quadrant, \displaystyle \cos x and \displaystyle \cos y are positive, so we in fact have

\displaystyle \cos x = \frac{\sqrt{5}}{3}\qquad\text{and}\qquad\cos y = \frac{2\sqrt{2}}{3}\,\textrm{.}

Finally, we obtain

\displaystyle \sin (x+y) = \frac{2}{3}\cdot \frac{2\sqrt{2}}{3} + \frac{\sqrt{5}}{3}\cdot \frac{1}{3} = \frac{4\sqrt{2} + \sqrt{5}}{9}\,\textrm{.}