To evaluate the function \( f(x) = -\cos(x) \) at \( x = \frac{4\pi}{3} \), we first need to find the cosine of \( \frac{4\pi}{3} \).
The angle \( \frac{4\pi}{3} \) radians can be found in the unit circle. It is located in the third quadrant, where the cosine value is negative.
The reference angle for \( \frac{4\pi}{3} \) is calculated as follows: \[ \frac{4\pi}{3} - \pi = \frac{4\pi}{3} - \frac{3\pi}{3} = \frac{\pi}{3} \]
Now, we know: \[ \cos\left(\frac{\pi}{3}\right) = \frac{1}{2} \] Thus: \[ \cos\left(\frac{4\pi}{3}\right) = -\cos\left(\frac{\pi}{3}\right) = -\frac{1}{2} \]
Now we can substitute this value into the function: \[ f\left(\frac{4\pi}{3}\right) = -\cos\left(\frac{4\pi}{3}\right) = -\left(-\frac{1}{2}\right) = \frac{1}{2} \]
Therefore, the value of the function \( f(x) \) at \( x = \frac{4\pi}{3} \) is: \[ \boxed{\frac{1}{2}} \]