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To find the Fourier sine series for the given piecewise function \( f(x) \) on the interval \( [0, \pi] \), we need to follow these steps:

### 1. Fourier Sine Series General Formula

For a function \( f(x) \) defined on \( [0, \pi] \), the Fourier sine series is given by:

\[
f(x) \sim \sum_{n=1}^{\infty} b_n \sin\left(\frac{n\pi x}{\pi}\right)
\]
where
\[
b_n = \frac{2}{\pi} \int_0^{\pi} f(x) \sin\left(\frac{n\pi x}{\pi}\right) dx
\]

In this case, the function is defined as:

\[
f(x) =
\begin{cases} 
0 & 0 \leq x < \frac{\pi}{2} \\
\pi - x & \frac{\pi}{2} \leq x \leq \pi
\end{cases}
\]

### 2. Calculate \( b_n \)
We will break the integral into two parts based on the piecewise function.

\[
b_n = \frac{2}{\pi} \left( \int_0^{\frac{\pi}{2}} 0 \cdot \sin\left(\frac{n\pi x}{\pi}\right) dx + \int_{\frac{\pi}{2}}^{\pi} (\pi - x) \sin\left(\frac{n\pi x}{\pi}\right) dx \right)
\]

The first integral evaluates to zero because the function is zero over the interval \( [0, \frac{\pi}{2}] \).

Thus, we are left with:
\[
b_n = \frac{2}{\pi} \int_{\frac{\pi}{2}}^{\pi} (\pi - x) \sin\left(n x\right) dx
\]

### 3. Solve the Integral

We now need to compute this integral. Let me calculate this step for you.

The integral evaluates to the following result for \( b_n \):

\[
b_n = \frac{2}{\pi} \left( \frac{\pi \cos\left(\frac{n\pi}{2}\right)}{2n} + \frac{\sin\left(\frac{n\pi}{2}\right)}{n^2} - \frac{\sin(n\pi)}{n^2} \right)
\]

Since \( \sin(n\pi) = 0 \) for all integers \( n \), this term vanishes. The final expression for \( b_n \) simplifies to:

\[
b_n = \frac{2}{\pi} \left( \frac{\pi \cos\left(\frac{n\pi}{2}\right)}{2n} + \frac{\sin\left(\frac{n\pi}{2}\right)}{n^2} \right)
\]

This gives the coefficients for the Fourier sine series of the given piecewise function.

Would you like further details on this solution or clarification on any steps? Here are some related questions:

1. How do Fourier series coefficients change for different boundary conditions?
2. Can we apply this method to other piecewise functions?
3. What is the physical interpretation of the Fourier sine series?
4. How does the series converge to the original function?
5. What are the orthogonality conditions for sine functions in Fourier series?

**Tip**: When solving Fourier series problems, symmetry in the function can often simplify calculations, so always check for even or odd symmetries first!

Find the Fourier sine series on [0, π] of the function f(x) = {0, 0 ≤ x < π/2; π - x, π/2 ≤ x ≤ π}

Learn how to compute the Fourier sine series for a piecewise function defined on [0, π]. This solution involves breaking the integral into two parts and calculating the Fourier coefficients for the sine series representation. Perfect for university-level students studying advanced calculus and Fourier analysis.

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