To find the Laplace transform of \( \sin(3at) \), we can use the standard Laplace transform formula for sine functions.

### Formula:

The Laplace transform of \( \sin(bt) \) is given by:

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

### Applying the Formula:

In this case, \( b = 3a \), so:

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

### Simplifying:

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

So the Laplace transform of \( \sin(3at) \) is:

\[
\boxed{\frac{3a}{s^2 + 9a^2}}
\]

Do you have any further questions or need more details on this?

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Here are 5 related questions:

1. What is the Laplace transform of \( \cos(2bt) \)?
2. How would you find \( \mathcal{L}\{e^{2t} \sin(5t)\} \)?
3. Can you calculate the inverse Laplace transform of \( \frac{7}{s^2 + 49} \)?
4. What is the Laplace transform of \( \sin(bt) + \cos(bt) \)?
5. How does the Laplace transform handle complex exponentials like \( e^{at} \sin(bt) \)?

**Tip:** The Laplace transform can simplify the process of solving differential equations by converting them into algebraic equations, which are often easier to solve.

Learn how to compute the Laplace transform of sin(3at) using the standard Laplace transform formula for sine functions. This problem involves trigonometric functions and their transformations into the s-domain, relevant for advanced high school or college-level students.

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