The Laplace Transform of a function \( f(t) \), denoted by \( \mathcal{L}[f(t)] \), is given by:

\[
\mathcal{L}[f(t)] = \int_0^\infty e^{-st} f(t) \, dt
\]

For the given function \( f(t) = \sin(3t) \), we can use the standard Laplace Transform formula for \( \sin(at) \), which is:

\[
\mathcal{L}[\sin(at)] = \frac{a}{s^2 + a^2}
\]

In this case, \( a = 3 \). Therefore, the Laplace Transform of \( \sin(3t) \) is:

\[
\mathcal{L}[\sin(3t)] = \frac{3}{s^2 + 9}
\]

### Final Answer:
\[
\mathcal{L}[\sin(3t)] = \frac{3}{s^2 + 9}
\]

Would you like more details or have any questions?

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Here are some related questions to deepen your understanding:

1. What is the Laplace transform of \( \cos(3t) \)?
2. How do you apply the Laplace transform to \( \sin(at) + \cos(at) \)?
3. What is the inverse Laplace transform of \( \frac{3}{s^2 + 9} \)?
4. How does the Laplace transform handle functions with different frequencies?
5. What is the Laplace transform of \( t\sin(3t) \)?

**Tip:** The Laplace transform is a powerful tool for solving differential equations, especially in systems involving initial conditions.

Discover how to find the Laplace Transform of sin(3t) using the formula \( \mathcal{L}[\sin(at)] = \frac{a}{s^2 + a^2} \). This example demonstrates key concepts in Laplace Transforms and is suitable for students studying signal processing or differential equations.

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