[tex]\sin\dfrac{5\pi}{12}[/tex]
Notice that [tex]\dfrac{5\pi}{12}=\dfrac{3\pi}{12}+\dfrac{2\pi}{12}=\dfrac\pi4+\dfrac\pi6[/tex]. So,
[tex]\sin\dfrac{5\pi}{12}=\sin\left(\dfrac\pi4+\dfrac\pi6\right)[/tex]
By the angle sum identity for sine, this is equal to
[tex]\sin\left(\dfrac\pi4+\dfrac\pi6\right)=\sin\dfrac\pi4\cos\dfrac\pi6+\cos\dfrac\pi4\sin\dfrac\pi6=\dfrac1{\sqrt2}\times\dfrac{\sqrt3}2+\dfrac1{\sqrt2}\times\dfrac12=\dfrac{\sqrt3+1}{2\sqrt2}[/tex]
