Find the Laplace transform of the function: [ f(t) = \sin(3t)

The Laplace transform of a function \( f(t) \) is defined as:

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

For the function \( f(t) = \sin(3t) \), we use the known Laplace transform formula for \( \sin(at) \):

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

In this case, \( a = 3 \), so the Laplace transform of \( \sin(3t) \) is:

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

Thus, the Laplace transform of \( \sin(3t) \) is:

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

Would you like further details or have any questions?

Here are 5 related questions you might find useful:
1. What is the Laplace transform of \( \cos(3t) \)?
2. How do you find the inverse Laplace transform of \( \frac{3}{s^2 + 9} \)?
3. What is the Laplace transform of \( \sin(kt) \) for any constant \( k \)?
4. Can you solve the Laplace transform of \( e^{2t} \sin(3t) \)?
5. What is the Laplace transform of a shifted sine function \( \sin(3(t - 2)) \)?

**Tip**: Remember that the Laplace transform is linear, meaning you can break down sums of functions into individual transforms.

Find the Laplace transform of the function: f(t) = sin(3t)

This page provides a detailed solution for finding the Laplace transform of the trigonometric function sin(3t). The problem applies the known Laplace transform formula for sin(at) and simplifies the expression to find the result. Suitable for undergraduate students studying calculus or differential equations.

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