To find the Fourier series for the given function $f(t)$, we will need to calculate its Fourier coefficients, which include both the $a_n$ (cosine) and $b_n$ (sine) terms, along with the constant $a_0$.

1. Restating the Function:

The function is periodic with period $4\pi$ since it satisfies $f(t) = f(t + 4\pi)$. The piecewise definition of the function over one period is:

-t + 2\pi & \text{for} \quad -2\pi < t < 0 \\ -t - 2\pi & \text{for} \quad 0 < t < 2\pi \end{cases}$$ The function is odd, as can be seen from the symmetry: $$f(-t) = -f(t).$$ Because the function is odd, there will be no cosine terms in the Fourier series ($$a_n = 0$$ for all $$n$$). ### 2. **Fourier Series Formula:** For an odd function with period $$4\pi$$, the Fourier series will take the form: $$f(t) = \sum_{n=1}^{\infty} b_n \sin\left( \frac{n\pi t}{2\pi} \right)$$ where the sine coefficients $$b_n$$ are given by: $$b_n = \frac{2}{4\pi} \int_{-\pi}^{\pi} f(t) \sin\left(\frac{n\pi t}{\pi} \right) dt$$