Let's break down the problem step by step.

The Problem:

You are asked to calculate the Fourier series representation of the given function $f(t)$ from the graph, up to at least the 5th harmonic of its fundamental frequency. You are required to integrate over the whole cycle to determine the Fourier coefficients.

Understanding the Graph:

From the graph:

• The function appears to be periodic with a period $T = 2\pi$.
• $f(t)$ has specific values at critical points:
  • $f(0) = 0$
  • $f(\pi) = 1$
  • $f(2\pi) = 0$
  • $f(3\pi) = -1$

Fourier Series:

The Fourier series for a periodic function $f(t)$ with period $T = 2\pi$ is given by:

$f(t) = a_0 + \sum_{n=1}^{\infty} \left( a_n \cos\left( n \omega_0 t \right) + b_n \sin\left( n \omega_0 t \right) \right)$ where $\omega_0$ is the fundamental frequency, $\omega_0 = \frac{2\pi}{T}$.

In this case, $T = 2\pi$, so $\omega_0 = 1$.

The Fourier coefficients $a_0$, $a_n$, and $b_n$ are given