Problem 59

Analysis

This question examines the method of calculating the radian measure of the central angle of a sector and the value of a trigonometric function, focusing on computational ability. By directly using the formula relating arc length, radius, and central angle, we can find the radian measure of the sector's central angle, and then determine the value of the trigonometric function for a special angle.

Solution

Given that the radius is $2$ and the arc length is $\frac{8\pi}{3}$, the central angle of the sector $\alpha$ can be calculated as $\alpha= \frac{ \frac{8\pi}{3}}{2}= \frac{4\pi}{3}$.

Therefore, $\sin \frac{4\pi}{3}=-\sin \frac{\pi}{3}=- \frac{ \sqrt{3}}{2}$.

Hence, the answer is $\boxed{- \frac{ \sqrt{3}}{2}}$.