Consider the equation:
[tex]\[ 2 \sin^2 \theta = \sin \theta \][/tex]
We are tasked with finding all exact solutions [tex]\(\theta\)[/tex] on the interval [tex]\([0, 2\pi)\)[/tex].
1. Rewrite the Equation: Start by moving all terms to one side of the equation to set it to zero.
[tex]\[ 2 \sin^2 \theta - \sin \theta = 0 \][/tex]
2. Factor the Equation: Factor out [tex]\(\sin \theta\)[/tex] from both terms.
[tex]\[ \sin \theta (2 \sin \theta - 1) = 0 \][/tex]
This means that:
[tex]\[ \sin \theta = 0 \quad \text{or} \quad 2 \sin \theta - 1 = 0 \][/tex]
3. Solve for [tex]\(\sin \theta = 0\)[/tex]:
[tex]\[ \sin \theta = 0 \][/tex]
The solutions for this within the interval [tex]\([0, 2\pi)\)[/tex] are:
[tex]\[ \theta = 0, \pi \][/tex]
4. Solve for [tex]\(2 \sin \theta - 1 = 0\)[/tex]:
[tex]\[ 2 \sin \theta = 1 \][/tex]
[tex]\[ \sin \theta = \frac{1}{2} \][/tex]
The solutions for this within the interval [tex]\([0, 2\pi)\)[/tex] are:
[tex]\[ \theta = \frac{\pi}{6}, \frac{5\pi}{6} \][/tex]
5. Combine All Solutions: Collect all the solutions we have found.
[tex]\[ \theta = 0, \frac{\pi}{6}, \frac{5\pi}{6}, \pi \][/tex]
Thus, the exact solutions [tex]\(\theta\)[/tex] on the interval [tex]\([0, 2\pi)\)[/tex] are:
[tex]\[ \boxed{0, \frac{\pi}{6}, \frac{5\pi}{6}, \pi} \][/tex]
Hence, the correct answer is:
[tex]\[ \theta = 0, \frac{\pi}{6}, \frac{5 \pi}{6}, \pi \][/tex]
