To solve for [tex]\(x\)[/tex] in the equation [tex]\(\cos{x} = -\frac{\sqrt{3}}{2}\)[/tex] on the interval [tex]\([ \pi, 2\pi ]\)[/tex], we need to find the values of [tex]\(x\)[/tex] in this interval where the cosine function takes the value [tex]\(-\frac{\sqrt{3}}{2}\)[/tex].
First, remember that [tex]\(\cos{x} = -\frac{\sqrt{3}}{2}\)[/tex] at specific known angles in the unit circle. These angles are usually determined based on the known values of the cosine function. Specifically:
[tex]\[ \cos{\left(\frac{5\pi}{6}\right)} = -\frac{\sqrt{3}}{2} \quad \text{and} \quad \cos{\left(\frac{7\pi}{6}\right)} = -\frac{\sqrt{3}}{2} \][/tex]
We need to check if these angles fall within our given interval [tex]\([\pi, 2\pi]\)[/tex].
The interval [tex]\([\pi, 2\pi]\)[/tex] corresponds to the angles from [tex]\(180^\circ\)[/tex] to [tex]\(360^\circ\)[/tex]:
- [tex]\(\frac{5\pi}{6} = 150^\circ\)[/tex] which is not within [tex]\([\pi, 2\pi]\)[/tex]
- [tex]\(\frac{7\pi}{6} = 210^\circ\)[/tex] which is within [tex]\([\pi, 2\pi]\)[/tex]
Since [tex]\(\frac{7\pi}{6}\)[/tex] (210 degrees) is within the given interval and satisfies the equation [tex]\(\cos{x} = -\frac{\sqrt{3}}{2}\)[/tex], this is the angle we are looking for.
Thus, the exact solution for the interval [tex]\(\pi \leq x \leq 2\pi\)[/tex] is:
[tex]\[ \boxed{\frac{7\pi}{6}} \][/tex]
