<p>Variation table of when <i>b</i> < 0, , and <i>k</i> < 0.</p>

Context: Many models based on S-shaped functions demonstrate their advantages in non-homogeneous Poisson process software reliability modeling. However, three well-known types have been used without deep mathematical evaluation. Furthermore, some other promising S-shaped functions should be aimed at.
Objectives: (1) Transform the sine function to change the domain and co-domain without losing the S-shaped pattern, and (2) Evaluate four types of S-shaped functions mathematically, including three well-known types and this newly considered type of transformation form of the trigonometric function.
Methods: The transformation is taken by a four-step process, including horizontal and vertical shift, horizontal and vertical scale, to maintain the S-shaped form. The mathematical evaluation is performed using numerical analysis techniques in four aspects of function: special cases, domain, range, and limitations.
Results: The first contribution is a deep mathematical appreciation of three well-known S-shaped functions. The second is the mathematical transformation of the trigonometric function to meet the real assumption. The last are the advantages and the applicability of this transformation in software reliability modeling.
Conclusion: Besides three well-used S-shaped functions, the transformation of the trigonometric function has a new advantage. Most importantly, there is a possibility of using the trigonometric function to introduce a new software reliability model.