Let's solve the equation [tex]\( 7 \csc(x) - 3 = 11 \)[/tex] on the interval [tex]\([0, 2\pi)\)[/tex].
1. Given equation:
[tex]\[
7 \csc(x) - 3 = 11
\][/tex]
2. Isolate the cosecant function:
[tex]\[
7 \csc(x) - 3 = 11
\][/tex]
Add 3 to both sides:
[tex]\[
7 \csc(x) = 14
\][/tex]
Divide both sides by 7:
[tex]\[
\csc(x) = 2
\][/tex]
3. Express cosecant in terms of sine:
[tex]\[
\csc(x) = \frac{1}{\sin(x)}
\][/tex]
Substitute [tex]\(\csc(x)\)[/tex] with [tex]\(\frac{1}{\sin(x)}\)[/tex]:
[tex]\[
\frac{1}{\sin(x)} = 2
\][/tex]
Find the reciprocal to solve for [tex]\(\sin(x)\)[/tex]:
[tex]\[
\sin(x) = \frac{1}{2}
\][/tex]
4. Find solutions for [tex]\(\sin(x) = \frac{1}{2}\)[/tex] within the interval [tex]\([0, 2\pi)\)[/tex]:
The sine function equals [tex]\(\frac{1}{2}\)[/tex] at [tex]\( x = \frac{\pi}{6} \)[/tex] and [tex]\( x = \frac{5\pi}{6} \)[/tex].
So, the solutions to the equation on the interval [tex]\([0, 2\pi)\)[/tex] are:
[tex]\[
x = \frac{\pi}{6}, \frac{5\pi}{6}
\][/tex]
Therefore, [tex]\(x\)[/tex] is:
[tex]\[
x = \frac{\pi}{6}, \frac{5\pi}{6}
\][/tex]
