The problem asks to find the exact value of $\cos(\frac{2\pi}{3})$ without using a calculator.

1. Problem Description

The problem asks to find the exact value of $\cos(\frac{2\pi}{3})$ without using a calculator.

2. Solution Steps

First, we need to find the reference angle for $\frac{2\pi}{3}$. Since $\frac{2\pi}{3}$ is in the second quadrant, the reference angle is $\pi - \frac{2\pi}{3} = \frac{3\pi}{3} - \frac{2\pi}{3} = \frac{\pi}{3}$.

Now, we know that $\cos(\frac{\pi}{3}) = \frac{1}{2}$.

Since $\frac{2\pi}{3}$ is in the second quadrant, where cosine is negative, we have

$\cos(\frac{2\pi}{3}) = -\cos(\frac{\pi}{3}) = -\frac{1}{2}$.

3. Final Answer

The final answer is $-\frac{1}{2}$.