To find the Laplace transform of \( 3 \sin(200t + \frac{\pi}{3}) \), we can use the following property of Laplace transforms for sinusoids:

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

Where:
- \( a \) is the angular frequency, in this case \( a = 200 \),
- \( \phi \) is the phase shift, in this case \( \phi = \frac{\pi}{3} \),
- \( s \) is the Laplace variable.

So for \( 3 \sin(200t + \frac{\pi}{3}) \), the Laplace transform is:

\[
\mathcal{L}\{ 3 \sin(200t + \frac{\pi}{3}) \} = 3 \cdot \frac{200 \sin(\frac{\pi}{3}) + s \cos(\frac{\pi}{3})}{s^2 + 200^2}
\]

Let's break this down:

- \( \sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2} \),
- \( \cos\left(\frac{\pi}{3}\right) = \frac{1}{2} \).

Substitute these values:

\[
\mathcal{L}\{ 3 \sin(200t + \frac{\pi}{3}) \} = 3 \cdot \frac{200 \cdot \frac{\sqrt{3}}{2} + s \cdot \frac{1}{2}}{s^2 + 200^2}
\]

Simplifying:

\[
\mathcal{L}\{ 3 \sin(200t + \frac{\pi}{3}) \} = \frac{3 \left(100 \sqrt{3} + \frac{s}{2}\right)}{s^2 + 40000}
\]

This is the Laplace transform of the given function.

Would you like further details on how I arrived at this result or have any questions?

Here are some related questions for further exploration:
1. What is the Laplace transform of \( \cos(at + \phi) \)?
2. How does the Laplace transform handle time shifts?
3. What is the inverse Laplace transform of the result?
4. How does the Laplace transform simplify the process of solving differential equations?
5. What are the conditions under which the Laplace transform exists?

**Tip:** Memorizing the basic Laplace transform formulas for sine and cosine functions can help solve a variety of time-domain problems easily.

Find the Laplace transform of 3sin(200t + pi/3) using the Laplace transform properties for sinusoidal functions. This problem explains key mathematical concepts such as trigonometry, Laplace transforms, and provides a formula-based approach suitable for undergraduate students.

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