find the exact value. \ncos\\frac{4\\pi}{3} \ntype + or - \n? \\frac{}{}

Explanation:

Step1: Determine the reference angle

The angle (\frac{4\pi}{3}) is in the third quadrant. The reference angle (\theta_{r}) for an angle (\theta=\frac{4\pi}{3}) is (\theta_{r}=\frac{4\pi}{3}-\pi=\frac{\pi}{3}).

Step2: Use the cosine value of the reference angle and the sign in the third quadrant

We know that (\cos\frac{\pi}{3}=\frac{1}{2}). In the third quadrant, (\cos\theta<0). So (\cos\frac{4\pi}{3}=-\cos\frac{\pi}{3}).

Answer:

(-\frac{1}{2})