Question

find the exact value.

\\( \sin \frac { - 4 \pi } { 3 } \\)

type + or -

? \\( \frac { \sqrt { } } { } \\)

Explanation:

Step1: Use the property of sine function

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

Step2: Rewrite the angle

\(\frac{4\pi}{3}=\pi+\frac{\pi}{3}\). And we know that \(\sin(A + B)=\sin A\cos B+\cos A\sin B\). Here \(A=\pi\), \(B = \frac{\pi}{3}\). Since \(\sin\pi = 0\), \(\cos\pi=- 1\), \(\sin\frac{\pi}{3}=\frac{\sqrt{3}}{2}\), \(\cos\frac{\pi}{3}=\frac{1}{2}\). Then \(\sin(\pi+\frac{\pi}{3})=\sin\pi\cos\frac{\pi}{3}+\cos\pi\sin\frac{\pi}{3}=0\times\frac{1}{2}+(-1)\times\frac{\sqrt{3}}{2}=-\frac{\sqrt{3}}{2}\)

Step3: Calculate the original value

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

Answer:

\(+\frac{\sqrt{3}}{2}\)