To solve the equation \( 3\sin{x} = 2\cos{x} \), follow these steps:

### Step 1: Divide both sides by \( \cos{x} \)
\[
\frac{3\sin{x}}{\cos{x}} = \frac{2\cos{x}}{\cos{x}}
\]
This simplifies to:
\[
3\tan{x} = 2
\]
(where \( \tan{x} = \frac{\sin{x}}{\cos{x}} \)).

### Step 2: Solve for \( \tan{x} \)
\[
\tan{x} = \frac{2}{3}
\]

### Step 3: Find the general solution for \( x \)
The general solution for \( \tan{x} = \frac{2}{3} \) is:
\[
x = \tan^{-1}\left(\frac{2}{3}\right) + n\pi
\]
where \( n \) is any integer.

### Step 4: Approximate the principal value
Using a calculator:
\[
x \approx \tan^{-1}\left(\frac{2}{3}\right) \approx 0.588 \, \text{radians}
\]

Thus, the general solution is:
\[
x \approx 0.588 + n\pi
\]

Would you like more details or have any questions?

Here are five related questions to expand on this concept:
1. How would the solution change if it were \( 3\sin{x} = 2\sin{x} \)?
2. Can you solve \( 2\tan{x} + 3 = 0 \)?
3. How does the general solution of \( \sin{x} = \cos{x} \) differ from this?
4. What happens when solving \( 3\sin{x} = -2\cos{x} \)?
5. Can you solve \( \tan{2x} = 1 \)?

**Tip:** When solving trigonometric equations, look for opportunities to use identities like \( \tan{x} = \frac{\sin{x}}{\cos{x}} \) to simplify.

Learn how to solve the trigonometric equation 3sin x = 2cos x step by step using trigonometric identities and the general solution formula. This problem demonstrates solving for x using the tangent identity and covers related concepts in trigonometry. Suitable for students in Grades 10-12.

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