To solve the equation [tex]\(3 \tan(x) = 2 \sin(2x)\)[/tex] in the interval [tex]\([0, 2\pi)\)[/tex], we need to follow these steps:
1. Use the double-angle identity:
Recall the double-angle identity for sine: [tex]\(\sin(2x) = 2 \sin(x) \cos(x)\)[/tex].
Substitute this identity into the equation:
[tex]\[
3 \tan(x) = 2 \cdot 2 \sin(x) \cos(x)
\][/tex]
This simplifies to:
[tex]\[
3 \tan(x) = 4 \sin(x) \cos(x)
\][/tex]
2. Express [tex]\(\tan(x)\)[/tex] in terms of [tex]\(\sin(x)\)[/tex] and [tex]\(\cos(x)\)[/tex]:
Recall that [tex]\(\tan(x) = \frac{\sin(x)}{\cos(x)}\)[/tex]. Substitute this into the equation:
[tex]\[
3 \frac{\sin(x)}{\cos(x)} = 4 \sin(x) \cos(x)
\][/tex]
3. Simplify the equation:
Multiply both sides by [tex]\(\cos(x)\)[/tex] to eliminate the fraction:
[tex]\[
3 \sin(x) = 4 \sin(x) \cos^2(x)
\][/tex]
4. Isolate [tex]\(\sin(x)\)[/tex]:
Factor out [tex]\(\sin(x)\)[/tex]:
[tex]\[
\sin(x) \left(3 - 4 \cos^2(x)\right) = 0
\][/tex]
This gives us two cases to consider:
1. [tex]\(\sin(x) = 0\)[/tex]
2. [tex]\(3 - 4 \cos^2(x) = 0\)[/tex]
5. Solve [tex]\(\sin(x) = 0\)[/tex]:
[tex]\(\sin(x) = 0\)[/tex] at:
[tex]\[
x = 0, \pi, 2\pi
\][/tex]
However, since we are only considering the interval [tex]\([0, 2\pi)\)[/tex], we exclude [tex]\(2\pi\)[/tex]. Thus:
[tex]\[
x = 0, \pi
\][/tex]
6. Solve [tex]\(3 - 4 \cos^2(x) = 0\)[/tex]:
Rearrange the equation:
[tex]\[
4 \cos^2(x) = 3
\][/tex]
Divide by 4:
[tex]\[
\cos^2(x) = \frac{3}{4}
\][/tex]
Take the square root of both sides:
[tex]\[
\cos(x) = \pm \sqrt{\frac{3}{4}} = \pm \frac{\sqrt{3}}{2}
\][/tex]
Therefore, [tex]\(x\)[/tex] must satisfy:
[tex]\[
\cos(x) = \frac{\sqrt{3}}{2} \quad \text{or} \quad \cos(x) = -\frac{\sqrt{3}}{2}
\][/tex]
7. Determine the corresponding [tex]\(x\)[/tex] values:
[tex]\[
\cos(x) = \frac{\sqrt{3}}{2} \implies x = \frac{\pi}{6}, \frac{11\pi}{6}
\][/tex]
[tex]\[
\cos(x) = -\frac{\sqrt{3}}{2} \implies x = \frac{5\pi}{6}, \frac{7\pi}{6}
\][/tex]
8. Combine all solutions:
Collect all valid solutions in the interval [tex]\([0, 2\pi)\)[/tex]:
[tex]\[
x = 0, \frac{\pi}{6}, \frac{5\pi}{6}, \pi, \frac{7\pi}{6}, \frac{11\pi}{6}
\][/tex]
Thus, the solutions to the equation [tex]\(3 \tan(x) = 2 \sin(2x)\)[/tex] in the interval [tex]\([0, 2\pi)\)[/tex] are:
[tex]\[
\boxed{0, \frac{\pi}{6}, \frac{5\pi}{6}, \pi, \frac{7\pi}{6}, \frac{11\pi}{6}}
\][/tex]
