Find the exact value of $\sin 150^\circ$.

$150^\circ$ lies in the second quadrant, where sine is positive, and $\sin 150^\circ = \sin(180^\circ - 30^\circ) = \sin 30^\circ = \dfrac{1}{2}$. The negative values arise from confusing the sign rule with that of cosine, and $\dfrac{\sqrt{3}}{2}$ is $\cos 30^\circ$.