(a) Sketch a graph of three cycles of the function with period \(2 \pi\) given by $$ f(t)=1-\frac{|t|}{\pi}, \quad-\pi \leq t<\pi $$ (b) Find its Fourier series representation.

(a) Sketch the function: Now sketching the first cycle, we have a straight line that starts at 1 when t = -π and decreases linearly to 0 at t = 0, then increases linearly back to 1 at t = π. For the second cycle, we have the same graph shifted to the right by 2π; this means that the graph in this cycle starts at 1 when t = π and decreases linearly to 0 at t = 2π, then increases linearly back to 1 at 3π. And for the third cycle, the graph is the same as the first cycle shifted to the left by 2π; this means that the graph starts at 1 when t = -3π and decreases linearly to 0 at t = -2π, then increases linearly back to 1 at t = -π. Combining these three cycles, we get a periodic linear function consisting of the three cycles as explained. (b) Final Fourier series representation: After calculating a0 and an, let's say: $$ a_0 = A \\ a_n = B_n $$ Substituting these values, the Fourier series representation for the given function is as follows: $$f(t) = \frac{A}{2}+\sum_{n=1}^{\infty} B_n \cos(nt)$$ where A and Bn are the calculated values of the coefficients a0 and an, respectively.