To solve the equation [tex]\(\cos(\theta) = -\frac{1}{2}\)[/tex] for [tex]\(\theta\)[/tex] in the interval [tex]\(0^\circ \leq \theta < 360^\circ\)[/tex], we need to find all angles that produce a cosine of [tex]\(-\frac{1}{2}\)[/tex]. Here is a detailed, step-by-step solution:
1. Understanding the Range of Cosine Values:
- Cosine values range from -1 to 1.
- The function [tex]\(\cos\)[/tex] is negative in the second and third quadrants of the unit circle.
2. Identifying Reference Angle:
- The cosine of [tex]\(-\frac{1}{2}\)[/tex] corresponds to an angle whose reference angle with [tex]\(\cos\)[/tex] is [tex]\(\frac{1}{2}\)[/tex].
- The reference angle for [tex]\(\cos^{-1}(\frac{1}{2})\)[/tex] is [tex]\(60^\circ\)[/tex], because [tex]\(\cos(60^\circ) = \frac{1}{2}\)[/tex].
3. Determining Relevant Angles:
- Since cosine is [tex]\(-\frac{1}{2}\)[/tex] in the second and third quadrants:
- In the second quadrant, the angle is [tex]\(180^\circ - 60^\circ = 120^\circ\)[/tex].
- In the third quadrant, the angle is [tex]\(180^\circ + 60^\circ = 240^\circ\)[/tex].
4. Validation:
- To confirm, we can check these angles:
- For [tex]\(\theta = 120^\circ\)[/tex]:
[tex]\[
\cos(120^\circ) = \cos(180^\circ - 60^\circ) = -\cos(60^\circ) = -\frac{1}{2}
\][/tex]
- For [tex]\(\theta = 240^\circ\)[/tex]:
[tex]\[
\cos(240^\circ) = \cos(180^\circ + 60^\circ) = -\cos(60^\circ) = -\frac{1}{2}
\][/tex]
- Both angles satisfy the equation [tex]\(\cos(\theta) = -\frac{1}{2}\)[/tex].
Thus, the angles [tex]\( \theta \)[/tex] that satisfy the given equation in the interval [tex]\(0^\circ \leq \theta < 360^\circ\)[/tex] are:
[tex]\[
\boxed{120.0^\circ \text{ and } 240.0^\circ}
\][/tex]
