Consider the rings $S$ and $S^{\prime }$, of real and complex trigonometric polynomials over the field ${Q}$ and its algebraic extension ${Q}(i)$ respectively. Then $S$ is an FFD, whereas $S^{\prime}$ is a Euclidean domain. We discuss irreducible elements of $S$ and $S^{\prime}$, and prove a few results on the trigonometric polynomial rings $T$ and $T^{\prime}$ introduced by G. Picavet and M. Picavet in [Trigonometric polynomial rings, Commutative ring theory, Lecture notes on Pure Appl. Math., Marcel Dekker, Vol. 231 (2003), 419-433]. We consider several examples and discuss the trigonometric polynomials in terms of irreducibles (atoms), to study the construction of these polynomials from irreducibles, which gives a geometric view of this study.