Trigonometry 7th Edition

$-\dfrac{\sqrt3}{2}$
$\dfrac{4\pi}{3}$ is an angle in Quadrant III. The reference angle of an angle in Quadrant III can be found using the formula $\theta - \pi$. Thus, the reference angle of $\dfrac{4\pi}{3}$ is: $\dfrac{4\pi}{3} - \pi = \dfrac{4\pi}{3}-\dfrac{3\pi}{3} = \dfrac{\pi}{3}$ $\dfrac{\pi}{3}$ is a special angle whose sine value is $\dfrac{\sqrt3}{2}$. However, since $\dfrac{4\pi}{3}$ is in Quadrant III, then its sine value is negative. Thus, $\sin{(\frac{4\pi}{3})}=-\frac{\sqrt3}{2}$