First, let's understand the problem we are dealing with: solving the equation [tex]\(\sin(4x) = \frac{1}{2}\)[/tex] within the interval [tex]\((0, 2\pi)\)[/tex].
1. Identifying Basic Solutions:
The equation [tex]\(\sin(\theta) = \frac{1}{2}\)[/tex] has specific angles that satisfy it. We know that:
[tex]\[
\theta = \frac{\pi}{6} + 2k\pi \quad \text{and} \quad \theta = \frac{5\pi}{6} + 2k\pi
\][/tex]
where [tex]\(k\)[/tex] is any integer.
2. Relating [tex]\(\theta\)[/tex] to [tex]\(4x\)[/tex]:
We rewrite the given equation in terms of [tex]\(\theta\)[/tex]:
[tex]\[
4x = \frac{\pi}{6} + 2k\pi \quad \text{and} \quad 4x = \frac{5\pi}{6} + 2k\pi
\][/tex]
3. Solving for [tex]\(x\)[/tex]:
To find [tex]\(x\)[/tex], we divide both sides by 4:
[tex]\[
x = \frac{\pi}{24} + \frac{k\pi}{2} \quad \text{and} \quad x = \frac{5\pi}{24} + \frac{k\pi}{2}
\][/tex]
4. Determining Valid [tex]\(x\)[/tex] Values Within the Interval [tex]\((0, 2\pi)\)[/tex]:
We need to find [tex]\(x\)[/tex] values that lie within the interval [tex]\((0, 2\pi)\)[/tex]. We test various integer values of [tex]\(k\)[/tex].
- For [tex]\(\frac{\pi}{24} + \frac{k\pi}{2}\)[/tex]:
[tex]\[
\begin{aligned}
&k = 0:\quad x = \frac{\pi}{24} \\
&k = 1:\quad x = \frac{\pi}{24} + \frac{\pi}{2} = \frac{13\pi}{24} \\
&k = 2:\quad x = \frac{\pi}{24} + \pi = \frac{25\pi}{24} \\
&k = 3:\quad x = \frac{\pi}{24} + \frac{3\pi}{2} = \frac{37\pi}{24}
\end{aligned}
\][/tex]
All these values are within [tex]\((0, 2\pi)\)[/tex].
- For [tex]\(\frac{5\pi}{24} + \frac{k\pi}{2}\)[/tex]:
[tex]\[
\begin{aligned}
&k = 0:\quad x = \frac{5\pi}{24} \\
&k = 1:\quad x = \frac{5\pi}{24} + \frac{\pi}{2} = \frac{17\pi}{24} \\
&k = 2:\quad x = \frac{5\pi}{24} + \pi = \frac{29\pi}{24} \\
&k = 3:\quad x = \frac{5\pi}{24} + \frac{3\pi}{2} = \frac{41\pi}{24}
\end{aligned}
\][/tex]
All these values are within [tex]\((0, 2\pi)\)[/tex].
5. Counting the Solutions:
Combining all the valid [tex]\(x\)[/tex] values, we have 8 solutions in total for the given equation in the specified interval.
Thus, the number of solutions to the equation [tex]\(\sin(4x) = \frac{1}{2}\)[/tex] within the interval [tex]\((0, 2\pi)\)[/tex] is:
[tex]\[
\boxed{8}
\][/tex]
