Trigonometry is concerned with measurements of angles about a central point (or of arcs of circles centered at that point) and quantities, geometrical and otherwise, that depend on the sizes of such angles (or the lengths of the corresponding arcs). It is one of those subjects that has become a standard part of the toolbox of every scientist and applied mathematician. Why is it so valuable?
There is a key geometrical feature of the measurement of angles, or arcs that are traced out, about a point in the plane: as we might expect, the sizes of such angles (or the lengths of such arcs) grow as one end of the arc moves counterclockwise around its circle (while the other remains fixed); but when the moving point returns to the place of the fixed point after making a full turn around the circle, it continues to retrace the same path through another turn, and another and another as the angle/arc grows ever-larger. In other words, periodic behavior is at the core of the relationships between angles and arcs and the measurements we associate with them.
As a consequence, wherever the mathematical description of cyclical phenomena is needed, trigonometric functions make appearances in pure mathematics and in applications of mathematics to the sciences. An introduction to trigonometry is a staple of the mathematics curriculum in high schools and colleges, many of whose students later study calculus and other forms of mathematical analysis in which periodic phenomena are explored.
Daniel E. Otero (Xavier University), "Teaching and Learning the Trigonometric Functions through Their Origins," Convergence (March 2020)