To solve for the angles [tex]\(\theta\)[/tex] between [tex]\(0\)[/tex] and [tex]\(2\pi\)[/tex] where [tex]\(\cos(\theta) = \frac{1}{2}\)[/tex], we can use the unit circle.
1. Understanding the Problem:
The cosine function returns the x-coordinate of the corresponding point on the unit circle for an angle [tex]\(\theta\)[/tex]. We seek angles where this x-coordinate is [tex]\(\frac{1}{2}\)[/tex].
2. Identifying Key Angles:
- The cosine of [tex]\(\theta\)[/tex] is [tex]\(\frac{1}{2}\)[/tex] at [tex]\(\theta = \frac{\pi}{3}\)[/tex] in the first quadrant.
- Cosine is also positive in the fourth quadrant. To find the corresponding angle in the fourth quadrant, we take the full circle [tex]\(2\pi\)[/tex] and subtract the reference angle [tex]\(\frac{\pi}{3}\)[/tex].
3. Calculating the Angles:
- For the first quadrant:
[tex]\[
\theta_1 = \frac{\pi}{3}
\][/tex]
- For the fourth quadrant:
[tex]\[
\theta_2 = 2\pi - \frac{\pi}{3} = \frac{6\pi}{3} - \frac{\pi}{3} = \frac{5\pi}{3}
\][/tex]
4. Final Answer:
Combining these, we have the angles:
[tex]\[
\theta = \frac{\pi}{3}, \frac{5\pi}{3}
\][/tex]
Thus, the values of [tex]\(\theta\)[/tex] between [tex]\(0\)[/tex] and [tex]\(2\pi\)[/tex] for which [tex]\(\cos(\theta) = \frac{1}{2}\)[/tex] are:
[tex]\[
\theta = \frac{\pi}{3}, \frac{5\pi}{3}
\][/tex]
