To solve the equation:
[tex]\[ -1 + \sin(\theta) = -\sin(\theta) \][/tex]
over the interval [tex]\([0, 2\pi)\)[/tex], we can follow these steps:
1. Combine like terms:
[tex]\[ -1 + \sin(\theta) + \sin(\theta) = 0 \][/tex]
[tex]\[ -1 + 2\sin(\theta) = 0 \][/tex]
2. Isolate [tex]\(\sin(\theta)\)[/tex]:
[tex]\[ 2\sin(\theta) = 1 \][/tex]
3. Solve for [tex]\(\sin(\theta)\)[/tex]:
[tex]\[ \sin(\theta) = \frac{1}{2} \][/tex]
4. Determine the values of [tex]\(\theta\)[/tex] that satisfy [tex]\(\sin(\theta) = \frac{1}{2}\)[/tex] within the interval [tex]\([0, 2\pi)\)[/tex]:
The sine function equals [tex]\(\frac{1}{2}\)[/tex] at [tex]\(\theta = \frac{\pi}{6}\)[/tex] in the first quadrant and [tex]\(\theta = \frac{5\pi}{6}\)[/tex] in the second quadrant.
Hence, the solutions to the equation [tex]\(-1+\sin (\theta)=-\sin (\theta)\)[/tex] in the interval [tex]\([0, 2\pi)\)[/tex] are:
[tex]\[
\theta = \frac{\pi}{6}, \frac{5\pi}{6}
\][/tex]
