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.