To solve the equation [tex]\(\cos x = -\frac{1}{2}\)[/tex] within the range [tex]\(0^\circ \leq x \leq 360^\circ\)[/tex], we need to identify the angles [tex]\(x\)[/tex] where the cosine of [tex]\(x\)[/tex] equals [tex]\(-\frac{1}{2}\)[/tex].
1. Identify the General Solutions:
The cosine function [tex]\(\cos x\)[/tex] yields a value of [tex]\(-\frac{1}{2}\)[/tex] at two specific angles within one full cycle of the unit circle (0 to 360 degrees). We can start by recognizing that the cosine function has symmetry and periodicity properties. Specifically, [tex]\(\cos x = -\frac{1}{2}\)[/tex] corresponds to certain standard angles within the second and third quadrants.
2. Determine the Standard Angles:
From trigonometric tables or the unit circle, we know that [tex]\(\cos 120^\circ = -\frac{1}{2}\)[/tex] and [tex]\(\cos 240^\circ = -\frac{1}{2}\)[/tex].
3. Confirm Angles in the Given Range:
It falls in the specified range of [tex]\(0^\circ \leq x \leq 360^\circ\)[/tex]. As such, these angles match the criteria given in the problem.
4. Verify With the Options Given:
Looking at the options provided:
- [tex]\(135^\circ, 225^\circ\)[/tex]
- [tex]\(210^\circ, 330^\circ\)[/tex]
- [tex]\(150^\circ, 210^\circ\)[/tex]
- [tex]\(120^\circ, 240^\circ\)[/tex]
The correct pairs [tex]\(\cos x = -\frac{1}{2}\)[/tex] within the specified range are found to be [tex]\(120^\circ, 240^\circ\)[/tex].
Thus, the solution to the equation [tex]\(\cos x = -\frac{1}{2}\)[/tex] within the range [tex]\(0^\circ \leq x \leq 360^\circ\)[/tex] is:
[tex]\[
x = 120^\circ \text{ and } x = 240^\circ
\][/tex]
Hence, the correct option is:
[tex]\[
\boxed{120^\circ, 240^\circ}
\][/tex]
