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