Which mathematical theory focuses on the relationships between angles and sides particularly in triangles?

The chosen answer, which is trigonometry, centers on the study of the relationships between the angles and sides of triangles. The foundation of trigonometry lies in its ability to provide a systematic way to link the angles of a triangle with the lengths of its sides, particularly in right-angled triangles. This is achieved through fundamental functions such as sine, cosine, and tangent, which are defined based on these relationships.

Trigonometry is not only pivotal in geometry but also plays a critical role in various fields such as physics, engineering, and computer science, where understanding the properties and functions of triangles is essential.

In contrast, probability theory is concerned with the analysis of random phenomena, and real analysis deals with the properties of real numbers and functions of real variables, neither of which addresses the geometric relationships in triangles. Complex analysis, similarly, focuses on functions of complex variables and their applications but does not pertain specifically to triangles. Thus, trigonometry is the distinct theory that effectively encompasses the relationships between angles and sides in triangles.