Introduction:
The Trigonometric Fourier series is a mathematical representation of a periodic time function. It decomposes a periodic function into a sum of sine and cosine functions of different frequencies and amplitudes. The coefficients of these sine and cosine terms determine the shape of the periodic function.

Explanation:
The Trigonometric Fourier series of a periodic time function can have only dc and cosine terms. This means that the series will not contain any sine terms.

DC Term:
The dc term represents the average value of the periodic function over one period. It is a constant term and does not vary with time. The dc term is represented by the coefficient a0 in the Fourier series.

Cosine Terms:
The cosine terms in the Fourier series represent the harmonics of the periodic function. These terms are oscillatory functions with different frequencies and amplitudes. The cosine terms are represented by the coefficients an in the Fourier series.

Sine Terms:
The sine terms in the Fourier series represent the phase shifts of the periodic function. These terms are also oscillatory functions with different frequencies and amplitudes. The sine terms are represented by the coefficients bn in the Fourier series.

Why only DC and Cosine terms:
The reason behind having only dc and cosine terms in the Trigonometric Fourier series is due to the nature of the periodic time function. If the function is an even function, meaning it is symmetric about the y-axis, then the Fourier series will only have cosine terms and a dc term. This is because the sine function is an odd function and will be zero for an even function.

If the function is an odd function, meaning it is anti-symmetric about the y-axis, then the Fourier series will only have sine terms and a dc term. This is because the cosine function is an even function and will be zero for an odd function.

However, if the periodic function is neither even nor odd, then the Fourier series will have both cosine and sine terms, along with the dc term.

Conclusion:
In conclusion, the Trigonometric Fourier series of a periodic time function can have only dc and cosine terms if the function is even, only sine terms if the function is odd, and both cosine and sine terms if the function is neither even nor odd.