To find all solutions of the equation [tex]\(\csc \theta - 2 = 0\)[/tex] in the interval [tex]\([0, 2\pi)\)[/tex]:
1. Isolate [tex]\(\csc \theta\)[/tex]:
[tex]\[
\csc \theta - 2 = 0
\][/tex]
Adding 2 to both sides, we get:
[tex]\[
\csc \theta = 2
\][/tex]
2. Convert [tex]\(\csc \theta\)[/tex] into [tex]\(\sin \theta\)[/tex]:
Since [tex]\(\csc \theta = \frac{1}{\sin \theta}\)[/tex], we have:
[tex]\[
\frac{1}{\sin \theta} = 2
\][/tex]
3. Solve for [tex]\(\sin \theta\)[/tex]:
Taking the reciprocal of both sides, we get:
[tex]\[
\sin \theta = \frac{1}{2}
\][/tex]
4. Determine the values of [tex]\(\theta\)[/tex] where [tex]\(\sin \theta = \frac{1}{2}\)[/tex]:
The sine function equals [tex]\(\frac{1}{2}\)[/tex] at the following angles in the interval [tex]\([0, 2\pi)\)[/tex]:
[tex]\(\theta_1 = \frac{\pi}{6}\)[/tex] (or 30 degrees) and [tex]\(\theta_2 = \frac{5\pi}{6}\)[/tex] (or 150 degrees).
5. Express the solutions in terms of [tex]\(\pi\)[/tex]:
[tex]\[
\theta = \frac{\pi}{6}, \frac{5\pi}{6}
\][/tex]
Therefore, the solutions to [tex]\(\csc \theta - 2 = 0\)[/tex] in the interval [tex]\([0, 2\pi)\)[/tex] are:
[tex]\[
\theta = \frac{\pi}{6}, \frac{5\pi}{6}
\][/tex]
