To find the Laplace transform of $\sin(\pi t)$, we can use the standard formula for the Laplace transform of $\sin(at)$:

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

Step-by-Step Solution:

Given function: $f(t) = \sin(\pi t)$

1. Identify the value of $a$: $a = \pi$

2. Apply the Laplace transform formula: $\mathcal{L}\{\sin(\pi t)\} = \frac{\pi}{s^2 + \pi^2}$

Thus, the Laplace transform of $\sin(\pi t)$ is: $\mathcal{L}\{\sin(\pi t)\} = \frac{\pi}{s^2 + \pi^2}$

Would you like more details on any part of this process, or do you have any questions?

Here are some related questions you might find helpful:

1. What is the Laplace transform of $\cos(at)$?
2. How do you find the Laplace transform of hyperbolic functions like $\sinh(at)$?
3. Can you explain the significance of the Laplace transform in solving harmonic oscillation problems?
4. What are the differences in Laplace transforms between sine and cosine functions?
5. How do initial conditions affect the Laplace transform of a function?

Tip: When dealing with trigonometric functions in Laplace transforms, always identify the frequency parameter $a$ correctly, as it directly influences the transform's denominator.