Sure, let's solve the equation [tex]\(\sin(2x) = 2\cos(x)\)[/tex] for [tex]\(x\)[/tex] in the interval [tex]\([0, 2\pi)\)[/tex].
### Step-by-Step Solution:
1. Rewrite the Equation:
The equation we are dealing with is:
[tex]\[
\sin(2x) = 2\cos(x)
\][/tex]
Using the double-angle identity for sine, we can rewrite [tex]\(\sin(2x)\)[/tex] as [tex]\(2\sin(x)\cos(x)\)[/tex]:
[tex]\[
2\sin(x)\cos(x) = 2\cos(x)
\][/tex]
2. Isolate Terms:
Divide both sides of the equation by 2:
[tex]\[
\sin(x)\cos(x) = \cos(x)
\][/tex]
3. Solve for [tex]\(\cos(x)\)[/tex]:
We can rearrange this equation to isolate [tex]\(\cos(x)\)[/tex]:
[tex]\[
\sin(x)\cos(x) - \cos(x) = 0
\][/tex]
Factor out [tex]\(\cos(x)\)[/tex]:
[tex]\[
\cos(x)(\sin(x) - 1) = 0
\][/tex]
4. Find the Zeros:
Now set each factor to zero and solve for [tex]\(x\)[/tex]:
[tex]\[
\cos(x) = 0 \quad \text{or} \quad \sin(x) - 1 = 0
\][/tex]
- For [tex]\(\cos(x) = 0\)[/tex]:
[tex]\[
x = \frac{\pi}{2}, \frac{3\pi}{2} \quad \text{within the interval} \; [0, 2\pi)
\][/tex]
- For [tex]\(\sin(x) = 1\)[/tex]:
[tex]\[
x = \frac{\pi}{2} \quad \text{within the interval} \; [0, 2\pi)
\][/tex]
5. Combine Solutions:
From these calculations we find:
[tex]\[
x = \frac{\pi}{2}, \frac{3\pi}{2}
\][/tex]
However, we need to verify that both of these values satisfy the original equation [tex]\(\sin(2x) = 2\cos(x)\)[/tex]. For [tex]\(x = \frac{\pi}{2}\)[/tex]:
[tex]\(\sin\left(2 \cdot \frac{\pi}{2}\right) = \sin(\pi) = 0\)[/tex]
[tex]\(2\cos\left(\frac{\pi}{2}\right) = 2 \cdot 0 = 0\)[/tex]
Therefore, [tex]\(x = \frac{\pi}{2}\)[/tex] is a solution.
For [tex]\(x = \frac{3\pi}{2}\)[/tex]:
[tex]\(\sin\left(2 \cdot \frac{3\pi}{2}\right) = \sin(3\pi) = 0\)[/tex]
[tex]\(2\cos\left(\frac{3\pi}{2}\right) = 2 \cdot 0 = 0\)[/tex]
Therefore, [tex]\(x = \frac{3\pi}{2}\)[/tex] is also a solution.
### Conclusion
The solutions to the equation [tex]\(\sin 2x = 2\cos x\)[/tex] in the interval [tex]\([0, 2\pi)\)[/tex] are:
[tex]\[
\left\{ \frac{\pi}{2}, \frac{3\pi}{2} \right\}
\][/tex]
So, the correct answer is:
[tex]\[
\boxed{\left\{ \frac{\pi}{2}, \frac{3\pi}{2} \right\}}
\][/tex]
