### Instruction: Find $\sin \frac{4 \pi}{3}.$.### Response: I know that $\sin \frac{4 \pi}{3}$ is the same as $\sin (\pi + \frac{\pi}{3})$.
I also know that $\sin (\pi + \theta) = -\sin \theta$ for any angle $\theta$.
So I can simplify $\sin \frac{4 \pi}{3} = -\sin \frac{\pi}{3}$.
Now I need to recall the value of $\sin \frac{\pi}{3}$.
I can use the 30-60-90 right triangle, where the hypotenuse is 2, the shorter leg is 1, and the longer leg is $\sqrt{3}$.
The angle $\frac{\pi}{3}$ is opposite the longer leg, so $\sin \frac{\pi}{3} = \frac{\sqrt{3}}{2}$.
Therefore, $\sin \frac{4 \pi}{3} = -\sin \frac{\pi}{3} = -\frac{\sqrt{3}}{2}$.