To find all angles [tex]\(0^{\circ} \leq \theta < 360^{\circ}\)[/tex] that satisfy the equation [tex]\(\cos (\theta) = \frac{1}{2}\)[/tex], we will go through the following steps:
1. Identify the principal angles: The cosine function is positive in the first and fourth quadrants. We first determine the reference angle for [tex]\(\cos \theta = \frac{1}{2}\)[/tex].
2. Determine the reference angle: The reference angle is the angle formed with the x-axis, and for [tex]\(\cos^{-1}\left(\frac{1}{2}\right)\)[/tex], this is [tex]\(60^\circ\)[/tex]. Thus, one solution in the first quadrant is:
[tex]\[
\theta = 60^\circ
\][/tex]
3. Find all angles in the specified range: Since cosine is also positive in the fourth quadrant, we need another angle in that quadrant where cosine will still be [tex]\(\frac{1}{2}\)[/tex]. The angle in the fourth quadrant corresponding to [tex]\(60^\circ\)[/tex] can be found by subtracting [tex]\(60^\circ\)[/tex] from [tex]\(360^\circ\)[/tex]:
[tex]\[
\theta = 360^\circ - 60^\circ = 300^\circ
\][/tex]
4. List the unique solutions: We now have two angles within the range [tex]\(0^\circ \leq \theta < 360^\circ\)[/tex] that satisfy the given equation:
[tex]\[
\theta = 60^\circ \quad \text{and} \quad \theta = 300^\circ
\][/tex]
5. Round the solutions to the nearest tenth of a degree: Given in this problem, the answers are already in degrees that are whole numbers, thus no further rounding is required.
Therefore, the angles that satisfy the equation [tex]\(\cos (\theta) = \frac{1}{2}\)[/tex] within the specified range are:
[tex]\[
60.0^\circ \quad \text{and} \quad 300.0^\circ
\][/tex]
