Find the Exact Value Under the Given Conditions $\cos \alpha = \frac { 1 } { 3 } , 0 < \alpha < \frac { \pi } { 2 } ; \quad \sin \beta = - \frac { 1 } { 2 } , - \frac { \pi } { 2 } < \beta < 0 \quad$

Multiple Choice

Find the exact value under the given conditions.
- $\cos \alpha = \frac { 1 } { 3 } , 0 < \alpha < \frac { \pi } { 2 } ; \quad \sin \beta = - \frac { 1 } { 2 } , - \frac { \pi } { 2 } < \beta < 0 \quad$ Find $\tan ( \alpha + \beta )$ .

A) $\frac { 9 \sqrt { 3 } + 8 \sqrt { 2 } } { 3 }$
B) $\frac { 9 \sqrt { 3 } + 8 \sqrt { 2 } } { 5 }$
C) $\frac { 9 \sqrt { 3 } - 8 \sqrt { 2 } } { 5 }$
D) $\frac { 9 \sqrt { 3 } - 8 \sqrt { 2 } } { 3 }$