Answer
$$\frac{{\sqrt 6 + \sqrt 2 }}{4}$$
Work Step by Step
$$\eqalign{
& \sin \frac{{5\pi }}{{12}} \cr
& {\text{Write }}\frac{{5\pi }}{{12}}{\text{ as }}\frac{\pi }{4} + \frac{\pi }{6} \cr
& \sin \frac{{5\pi }}{{12}} = \sin \left( {\frac{\pi }{4} + \frac{\pi }{6}} \right) \cr
& {\text{Use sine sum identity}} \cr
& \sin \frac{{5\pi }}{{12}} = \sin \left( {\frac{\pi }{4}} \right)\cos \left( {\frac{\pi }{6}} \right) + \cos \left( {\frac{\pi }{4}} \right)\sin \left( {\frac{\pi }{6}} \right) \cr
& {\text{Substitute known values}} \cr
& \sin \frac{{5\pi }}{{12}} = \left( {\frac{{\sqrt 2 }}{2}} \right)\left( {\frac{{\sqrt 3 }}{2}} \right) + \left( {\frac{{\sqrt 2 }}{2}} \right)\left( {\frac{1}{2}} \right) \cr
& \sin \frac{{5\pi }}{{12}} = \frac{{\sqrt 6 }}{4} + \frac{{\sqrt 2 }}{4} \cr
& \sin \frac{{5\pi }}{{12}} = \frac{{\sqrt 6 + \sqrt 2 }}{4} \cr} $$
