We are given the trigonometric equation:

$3 \cot(x) - 1 = 0$

Step 1: Isolate $\cot(x)$

Start by solving for $\cot(x)$ in the equation:

$3 \cot(x) = 1$

Divide both sides by 3:

$\cot(x) = \frac{1}{3}$

Step 2: Use the cotangent identity

Recall that $\cot(x)$ is the reciprocal of $\tan(x)$, so:

$\cot(x) = \frac{1}{\tan(x)}$

Thus, the equation becomes:

$\frac{1}{\tan(x)} = \frac{1}{3}$

Step 3: Solve for $\tan(x)$

Taking the reciprocal of both sides:

$\tan(x) = 3$

Step 4: Find the general solution for $\tan(x) = 3$

The general solution for $\tan(x) = a$ is given by:

$x = \tan^{-1}(a) + n\pi \quad \text{where} \quad n \in \mathbb{Z}$

For $\tan(x) = 3$, we first find the principal value of $x$:

$x = \tan^{-1}(3)$

Using a calculator or reference table:

$x \approx 1.249 \text{ radians}$

Step 5: Find the specific solutions within the interval $0 \leq x < 2\pi$

Since the tangent function has a period of $\pi$, the solutions are:

$x = 1.249 + n\pi$

Now, consider the values of $n$ that give solutions within the range $0 \leq x < 2\pi$.

• For $n = 0$, $x = 1.249$
• For $n = 1$, $x = 1.249 + \pi \approx 1.249 + 3.1416 = 4.3916$

Thus, the solutions within the interval $0 \leq x < 2\pi$ are:

$x \approx 1.249 \, \text{radians} \quad \text{and} \quad x \approx 4.392 \, \text{radians}$

Final Answer:

The solutions to the equation $3 \cot(x) - 1 = 0$ in the interval $0 \leq x < 2\pi$ are:

$x \approx 1.249 \, \text{radians} \quad \text{and} \quad x \approx 4.392 \, \text{radians}$

---

Would you like further clarification on any part of the solution? Or do you have any other questions?

Related Questions:

1. How do you find solutions for other trigonometric functions like sine and cosine?
2. What is the relationship between tangent and cotangent functions?
3. How can you use the unit circle to find trigonometric solutions?
4. What is the effect of changing the interval for solving trigonometric equations?
5. How do you solve a trigonometric equation when multiple functions are involved?

Tip: When solving trigonometric equations, always remember the periodicity of the functions (e.g., sine, cosine, tangent, and cotangent) to find all possible solutions within a given interval.