To solve the given equation [tex]\( 4 \csc x + 6 = -2 \)[/tex] on the interval [tex]\([0, 2\pi)\)[/tex], follow these steps:
1. Isolate [tex]\(\csc x\)[/tex]:
[tex]\[
4 \csc x + 6 = -2
\][/tex]
Subtract 6 from both sides:
[tex]\[
4 \csc x = -2 - 6
\][/tex]
Simplify the right-hand side:
[tex]\[
4 \csc x = -8
\][/tex]
Divide both sides by 4:
[tex]\[
\csc x = -2
\][/tex]
2. Express [tex]\(\csc x\)[/tex] in terms of [tex]\(\sin x\)[/tex]:
Recall that [tex]\(\csc x\)[/tex] is the reciprocal of [tex]\(\sin x\)[/tex]:
[tex]\[
\csc x = \frac{1}{\sin x}
\][/tex]
Therefore:
[tex]\[
\frac{1}{\sin x} = -2
\][/tex]
Take the reciprocal of both sides to solve for [tex]\(\sin x\)[/tex]:
[tex]\[
\sin x = -\frac{1}{2}
\][/tex]
3. Determine values of [tex]\(x\)[/tex] where [tex]\(\sin x = -\frac{1}{2}\)[/tex]:
[tex]\(\sin x\)[/tex] is negative in the third and fourth quadrants. The reference angle for which [tex]\(\sin\)[/tex] equals [tex]\(\frac{1}{2}\)[/tex] is [tex]\(\frac{\pi}{6}\)[/tex].
- In the third quadrant, the angle is:
[tex]\[
\pi + \frac{\pi}{6} = \frac{6\pi}{6} + \frac{\pi}{6} = \frac{7\pi}{6}
\][/tex]
- In the fourth quadrant, the angle is:
[tex]\[
2\pi - \frac{\pi}{6} = \frac{12\pi}{6} - \frac{\pi}{6} = \frac{11\pi}{6}
\][/tex]
Thus, the solutions in the interval [tex]\([0, 2\pi)\)[/tex] are:
[tex]\[
x = \frac{7\pi}{6}, \frac{11\pi}{6}
\][/tex]
4. Compare solutions with given options:
- Option A: [tex]\(\frac{\pi}{3}, \frac{5\pi}{3}\)[/tex]
- Option B: [tex]\(\frac{2\pi}{3}, \frac{4\pi}{3}\)[/tex]
- Option C: [tex]\(\frac{\pi}{6}, \frac{5\pi}{6}\)[/tex]
- Option D: [tex]\(\frac{7\pi}{6}, \frac{11\pi}{6}\)[/tex]
The correct answer matches Option D:
[tex]\[
\boxed{\text{D}}
\][/tex]
