find the exact value. sin (-4π/3) type + or - ? √/

Explanation:

Step1: Use the property of sine function

$\sin(-\alpha)=-\sin\alpha$. So, $\sin\frac{-4\pi}{3}=-\sin\frac{4\pi}{3}$.

Step2: Rewrite the angle

$\frac{4\pi}{3}=\pi+\frac{\pi}{3}$. Then, $-\sin\frac{4\pi}{3}=-\sin(\pi+\frac{\pi}{3})$.

Step3: Use the sine addition formula

$\sin(A + B)=\sin A\cos B+\cos A\sin B$. Here $A=\pi$, $B = \frac{\pi}{3}$. $\sin(\pi+\frac{\pi}{3})=\sin\pi\cos\frac{\pi}{3}+\cos\pi\sin\frac{\pi}{3}=0\times\frac{1}{2}+(- 1)\times\frac{\sqrt{3}}{2}=-\frac{\sqrt{3}}{2}$.

Step4: Calculate the final value

Since $\sin\frac{-4\pi}{3}=-\sin\frac{4\pi}{3}$ and $\sin\frac{4\pi}{3}=-\frac{\sqrt{3}}{2}$, then $\sin\frac{-4\pi}{3}=\frac{\sqrt{3}}{2}$.

Answer:

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