To solve the equation \(\sqrt{3} \tan(3\theta) + 1 = 0\) in the interval \([0, 2\pi)\), let's follow these steps:

### 1. Simplify the Equation
First, isolate \(\tan(3\theta)\):
\[
\sqrt{3} \tan(3\theta) = -1
\]
\[
\tan(3\theta) = -\frac{1}{\sqrt{3}}
\]

### 2. Identify the General Solution
The value \(\tan(3\theta) = -\frac{1}{\sqrt{3}}\) corresponds to angles where the tangent is \(-\frac{1}{\sqrt{3}}\). We know that:
\[
\tan(\theta) = \tan\left(\frac{\pi}{6}\right)
\]
Thus:
\[
3\theta = \pi - \frac{\pi}{6} + n\pi \quad \text{(for any integer \(n\))}
\]
\[
3\theta = \frac{5\pi}{6} + n\pi
\]
\[
\theta = \frac{5\pi}{18} + \frac{n\pi}{3}
\]

### 3. Consider the Interval \([0, 2\pi)\)
To find all solutions for \(\theta\) within the interval \([0, 2\pi)\), substitute different integer values of \(n\):

- For \(n = 0\):
\[
\theta = \frac{5\pi}{18}
\]
- For \(n = 1\):
\[
\theta = \frac{5\pi}{18} + \frac{\pi}{3} = \frac{5\pi}{18} + \frac{6\pi}{18} = \frac{11\pi}{18}
\]
- For \(n = 2\):
\[
\theta = \frac{5\pi}{18} + \frac{2\pi}{3} = \frac{5\pi}{18} + \frac{12\pi}{18} = \frac{17\pi}{18}
\]
- For \(n = 3\):
\[
\theta = \frac{5\pi}{18} + \pi = \frac{5\pi}{18} + \frac{18\pi}{18} = \frac{23\pi}{18}
\]
- For \(n = 4\):
\[
\theta = \frac{5\pi}{18} + \frac{4\pi}{3} = \frac{5\pi}{18} + \frac{24\pi}{18} = \frac{29\pi}{18}
\]
- For \(n = 5\):
\[
\theta = \frac{5\pi}{18} + \frac{5\pi}{3} = \frac{5\pi}{18} + \frac{30\pi}{18} = \frac{35\pi}{18}
\]
- For \(n = 6\):
\[
\theta = \frac{5\pi}{18} + 2\pi = \frac{5\pi}{18} + \frac{36\pi}{18} = \frac{41\pi}{18}
\]

### 4. Filter Valid Solutions
We need \(\theta\) to be within the interval \([0, 2\pi)\), which is \([0, \frac{36\pi}{18})\). Therefore, the valid solutions are:
\[
\theta = \frac{5\pi}{18}, \frac{11\pi}{18}, \frac{17\pi}{18}, \frac{23\pi}{18}, \frac{29\pi}{18}, \frac{35\pi}{18}
\]

### Final Answer
The solutions for \(\theta\) in the interval \([0, 2\pi)\) are:
\[
\boxed{\frac{5\pi}{18}, \frac{11\pi}{18}, \frac{17\pi}{18}, \frac{23\pi}{18}, \frac{29\pi}{18}, \frac{35\pi}{18}}
\]

Would you like further details or have any questions?

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**Next Steps**:
1. How does the period of a trigonometric function affect its solutions?
2. What is the general form for solutions of a tangent equation?
3. How does the tangent function behave in different quadrants?
4. How do phase shifts influence trigonometric equations?
5. Why does \(\tan(x)\) have a period of \(\pi\) rather than \(2\pi\)?
6. How do we handle equations involving multiple angles like \(3\theta\)?
7. What role does symmetry play in solving trigonometric equations?
8. How can you verify solutions of trigonometric equations?

**Tip**: Always check the interval of the solutions when solving trigonometric equations, especially for multiple angles.

Learn how to solve the equation \( \sqrt{3} \tan(3\theta) + 1 = 0 \) within the interval [0, 2\pi). Detailed step-by-step solution with multiple angle identities used in trigonometry. Suitable for advanced high school students.

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