Answer
$$\frac{{\sqrt {10} - 3\sqrt {30} }}{{20}}$$
Work Step by Step
$$\eqalign{
& \sin \left( {{{\sin }^{ - 1}}\frac{1}{2} + {{\tan }^{ - 1}}\left( { - 3} \right)} \right) \approx - 0.6634699532 \cr
& = \sin \left( {{{\sin }^{ - 1}}\frac{1}{2} - {{\tan }^{ - 1}}\left( 3 \right)} \right) \cr
& {\text{Therefore,}} \cr
& \theta = {\sin ^{ - 1}}\frac{1}{2} \to \sin \theta = \frac{1}{2} = \frac{{{\text{opposite side}}}}{{{\text{hypotenuse}}}} \cr
& {\text{adjacent side}} = \sqrt {{2^2} - {1^2}} = \sqrt 3 \cr
& and \cr
& \beta = {\tan ^{ - 1}}\left( 3 \right) \to \tan \theta = \frac{3}{1} = \frac{{{\text{opposite side}}}}{{{\text{adjacent side}}}} \cr
& {\text{hypotenuse}} = \sqrt {{3^2} + {1^2}} = \sqrt {10} \cr
& \cr
& \sin \left( {{{\sin }^{ - 1}}\frac{3}{5} - {{\tan }^{ - 1}}\left( 3 \right)} \right) = \sin \left( {\theta - \beta } \right) \cr
& {\text{Use the sine of a difference}} \cr
& = \sin \theta \cos \beta - \cos \theta \sin \beta \cr
& = \left( {\frac{1}{2}} \right)\left( {\frac{1}{{\sqrt {10} }}} \right) - \left( {\frac{{\sqrt 3 }}{2}} \right)\left( {\frac{3}{{\sqrt {10} }}} \right) \cr
& {\text{Simplify}} \cr
& = \frac{{\sqrt {10} - 3\sqrt {30} }}{{20}} \cr} $$
