1. The problem asks to find the exact values of $\cos\left(\frac{4\pi}{3}\right)$ and $\sin\left(\frac{4\pi}{3}\right)$.\n\n2. Recall that $\frac{4\pi}{3}$ radians is in the third quadrant of the unit circle, where cosine is negative and sine is negative.\n\n3. The reference angle for $\frac{4\pi}{3}$ is $\pi - \frac{4\pi}{3} = \frac{\pi}{3}$.\n\n4. Using the unit circle values for $\frac{\pi}{3}$: $\cos\left(\frac{\pi}{3}\right) = \frac{1}{2}$ and $\sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2}$.\n\n5. Since $\frac{4\pi}{3}$ is in the third quadrant, both cosine and sine are negative, so:\n$$\cos\left(\frac{4\pi}{3}\right) = -\frac{1}{2}$$\n$$\sin\left(\frac{4\pi}{3}\right) = -\frac{\sqrt{3}}{2}$$\n\n6. Therefore, the exact values are:\n$\cos\left(\frac{4\pi}{3}\right) = -\frac{1}{2}$\n$\sin\left(\frac{4\pi}{3}\right) = -\frac{\sqrt{3}}{2}$