To solve the equation [tex]\((2 \cos \theta - \sqrt{3})(2 \sin \theta - 1) = 0\)[/tex], we need to consider each factor separately and solve for [tex]\(\theta\)[/tex] within the interval [tex]\([0, 2\pi)\)[/tex].
### Factor 1: [tex]\(2 \cos \theta - \sqrt{3} = 0\)[/tex]
1. Set [tex]\(2 \cos \theta - \sqrt{3} = 0\)[/tex] and solve for [tex]\(\cos \theta\)[/tex].
[tex]\[
2 \cos \theta = \sqrt{3}
\][/tex]
[tex]\[
\cos \theta = \frac{\sqrt{3}}{2}
\][/tex]
2. Determine the values of [tex]\(\theta\)[/tex] within the interval [tex]\([0, 2\pi)\)[/tex] where [tex]\(\cos \theta = \frac{\sqrt{3}}{2}\)[/tex].
[tex]\[
\cos \theta = \frac{\sqrt{3}}{2} \text{ at } \theta = \frac{\pi}{6}, \frac{11\pi}{6}
\][/tex]
### Factor 2: [tex]\(2 \sin \theta - 1 = 0\)[/tex]
1. Set [tex]\(2 \sin \theta - 1 = 0\)[/tex] and solve for [tex]\(\sin \theta\)[/tex].
[tex]\[
2 \sin \theta = 1
\][/tex]
[tex]\[
\sin \theta = \frac{1}{2}
\][/tex]
2. Determine the values of [tex]\(\theta\)[/tex] within the interval [tex]\([0, 2\pi)\)[/tex] where [tex]\(\sin \theta = \frac{1}{2}\)[/tex].
[tex]\[
\sin \theta = \frac{1}{2} \text{ at } \theta = \frac{\pi}{6}, \frac{5\pi}{6}
\][/tex]
### Combine Solutions
We combine the solutions from each factor and ensure they are within the interval [tex]\([0, 2\pi)\)[/tex].
#### Solutions from [tex]\(2 \cos \theta - \sqrt{3} = 0\)[/tex]:
[tex]\[
\theta = \frac{\pi}{6}, \frac{11\pi}{6}
\][/tex]
#### Solutions from [tex]\(2 \sin \theta - 1 = 0\)[/tex]:
[tex]\[
\theta = \frac{\pi}{6}, \frac{5\pi}{6}
\][/tex]
Combine these solutions into a single set and sort them:
[tex]\[
\theta = \frac{\pi}{6}, \frac{5\pi}{6}, \frac{11\pi}{6}
\][/tex]
Thus, the final solutions to the equation [tex]\((2 \cos \theta - \sqrt{3})(2 \sin \theta - 1) = 0\)[/tex] in the interval [tex]\([0, 2\pi)\)[/tex] are:
[tex]\[
\theta = \frac{\pi}{6}, \frac{5\pi}{6}, \frac{11\pi}{6}
\][/tex]
