Explanation:

The trigonometric Fourier series represents a periodic function as a sum of sine and cosine functions. It is given by the formula:

\[f(x) = \dfrac{a_0}{2} + \sum_{n=1}^{\infty} \left( a_n \cos(nx) + b_n \sin(nx) \right)\]

where \(a_0\), \(a_n\), and \(b_n\) are the Fourier coefficients.

An even function is a function that satisfies the condition \(f(x) = f(-x)\) for all \(x\) in its domain. This means that the function is symmetrical about the y-axis. Examples of even functions include \(f(x) = \cos(x)\) and \(f(x) = x^2\).

Even function and its Fourier series:

When a function is even, its Fourier series simplifies because the odd harmonic terms vanish. This is because the sine function is an odd function, satisfying the condition \(\sin(-x) = -\sin(x)\). Therefore, for an even function, the sine terms in the Fourier series will always be zero.

Reason for absence of sine terms:

The absence of sine terms in the trigonometric Fourier series of an even function can be understood as follows:

1. For an even function \(f(x)\), we have \(f(x) = f(-x)\) for all \(x\) in the domain.
2. When we substitute \(-x\) into the Fourier series formula, we get:

\[f(-x) = \dfrac{a_0}{2} + \sum_{n=1}^{\infty} \left( a_n \cos(-nx) + b_n \sin(-nx) \right)\]

3. Since the cosine function is an even function, it satisfies the condition \(\cos(-x) = \cos(x)\). Therefore, the cosine terms in the Fourier series remain the same when substituting \(-x\).
4. However, the sine function is an odd function, satisfying the condition \(\sin(-x) = -\sin(x)\). Therefore, the sine terms in the Fourier series change sign when substituting \(-x\).
5. Since \(f(-x) = f(x)\), the sine terms in the Fourier series must be zero to satisfy this condition. Otherwise, the function would not be symmetrical about the y-axis.

Conclusion:

In conclusion, the trigonometric Fourier series of an even function does not have sine terms. This is because even functions are symmetrical about the y-axis, and the sine terms change sign when substituting \(-x\), leading to their cancellation in the Fourier series.