We are given the equation:

\[
\sin(3x) - \cos(3x) = 0
\]

### Step-by-step solution:

1. **Move \(\cos(3x)\) to the other side:**

\[
\sin(3x) = \cos(3x)
\]

2. **Divide both sides by \(\cos(3x)\), provided that \(\cos(3x) \neq 0\):**

\[
\frac{\sin(3x)}{\cos(3x)} = 1
\]

This gives:

\[
\tan(3x) = 1
\]

3. **Solve for \(3x\):**

The general solution for \(\tan(\theta) = 1\) is:

\[
3x = \frac{\pi}{4} + n\pi, \quad \text{where } n \text{ is any integer.}
\]

4. **Solve for \(x\):**

\[
x = \frac{\pi}{12} + \frac{n\pi}{3}, \quad \text{where } n \text{ is any integer.}
\]

Thus, the general solution is:

\[
x = \frac{\pi}{12} + \frac{n\pi}{3}, \quad n \in \mathbb{Z}
\]

Would you like more details or a specific range of solutions?

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Here are 5 related questions to further explore this:

1. How do you solve a similar equation for \(\tan(3x) = -1\)?
2. What is the solution to \(\sin(3x) + \cos(3x) = 0\)?
3. Can you explain the general solution for \(\tan(\theta) = 1\)?
4. What are the specific solutions for \(x\) in the interval \([0, 2\pi]\)?
5. How would the solution change if the equation was \(\sin(2x) - \cos(2x) = 0\)?

**Tip:** When solving trigonometric equations, always check for periodicity and use general solutions when necessary.

Learn how to solve the trigonometric equation sin 3x - cos 3x = 0 with a detailed step-by-step explanation. This solution explores key concepts like transforming trigonometric functions and using the general solution for tan(θ) = 1. Suitable for students in Grades 10-12.

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