1. The problem is to find the sine series expansion of the piecewise function $$f(x) = \begin{cases} x, & 0 < x < 4 \\ 8 - x, & 4 < x < 8 \end{cases}$$ defined on the interval $(0,8)$. 2. The sine series expansion for a function defined on $(0,L)$ is given by: $$f(x) \sim \sum_{n=1}^\infty b_n \sin\left(\frac{n \pi x}{L}\right)$$ where $$b_n = \frac{2}{L} \int_0^L f(x) \sin\left(\frac{n \pi x}{L}\right) dx$$ 3. Here, $L=8$. We split the integral for $b_n$ into two parts because $f(x)$ is piecewise: $$b_n = \frac{2}{8} \left( \int_0^4 x \sin\left(\frac{n \pi x}{8}\right) dx + \int_4^8 (8 - x) \sin\left(\frac{n \pi x}{8}\right) dx \right)$$ 4. Calculate the first integral: Let $I_1 = \int_0^4 x \sin\left(\frac{n \pi x}{8}\right) dx$. Use integration by parts with $u=x$, $dv=\sin\left(\frac{n \pi x}{8}\right) dx$. 5. Calculate the second integral: Let $I_2 = \int_4^8 (8 - x) \sin\left(\frac{n \pi x}{8}\right) dx$. Use integration by parts similarly. 6. After evaluating $I_1$ and $I_2$, sum them and multiply by $\frac{1}{4}$ to get $b_n$. 7. The final sine series expansion is: $$f(x) \sim \sum_{n=1}^\infty b_n \sin\left(\frac{n \pi x}{8}\right)$$ where $b_n$ are the coefficients found above. This series represents the function $f(x)$ on $(0,8)$ as a sum of sine functions.