find the exact value of $sin\frac{4pi}{3}$

Explanation:

Step1: Use the angle addition formula

$\sin\frac{4\pi}{3}=\sin(\pi + \frac{\pi}{3})$

Step2: Apply the sine addition formula $\sin(A + B)=\sin A\cos B+\cos A\sin B$

$\sin(\pi+\frac{\pi}{3})=\sin\pi\cos\frac{\pi}{3}+\cos\pi\sin\frac{\pi}{3}$ Since $\sin\pi = 0$ and $\cos\pi=- 1$ $\sin\pi\cos\frac{\pi}{3}+\cos\pi\sin\frac{\pi}{3}=0\times\cos\frac{\pi}{3}+(-1)\times\sin\frac{\pi}{3}$

Step3: Calculate the value

$0\times\cos\frac{\pi}{3}+(-1)\times\sin\frac{\pi}{3}=-\sin\frac{\pi}{3}$ And $\sin\frac{\pi}{3}=\frac{\sqrt{3}}{2}$

Answer:

$-\frac{\sqrt{3}}{2}$