To solve the equation $3\cot x - 1 = 0$ on the interval $x \in [0, 2\pi]$:

Step 1: Isolate $\cot x$

$3\cot x = 1 \implies \cot x = \frac{1}{3}.$

Step 2: Express in terms of $\tan x$

Since $\cot x = \frac{1}{\tan x}$, we have: $\tan x = 3.$

Step 3: Solve for $x$

The general solution for $\tan x = 3$ is: $x = \arctan(3) + n\pi, \quad n \in \mathbb{Z}.$

Using a calculator: $\arctan(3) \approx 1.25 \, \text{radians}.$

Step 4: Find solutions in $[0, 2\pi]$

The tangent function is periodic with a period of $\pi$. Therefore, the two solutions in the interval $[0, 2\pi]$ are:

1. $x_1 = 1.25$,
2. $x_2 = 1.25 + \pi \approx 1.25 + 3.14 = 4.39$.

Final Answer:

The solutions to $3\cot x - 1 = 0$ on $[0, 2\pi]$ are: $x \approx 1.25 \, \text{radians}, \quad x \approx 4.39 \, \text{radians}.$

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Would you like a detailed explanation or have additional questions? Here are some related inquiries:

1. How is $\cot x$ related to $\tan x$?
2. Why does the tangent function have a period of $\pi$?
3. How to derive $\arctan(3)$ without a calculator?
4. What are other trigonometric equations with similar methods of solution?
5. How does the graph of $\tan x$ help in solving such problems?

Tip: When solving trigonometric equations, always verify your solutions within the specified interval.