Figure 1: Defining the (circular) trigonometric functions via a circle.
What is the fundamental idea in trigonometry? Although often introduced as the study of triangles, trigonometry is really the study of circles.
One way to construct the trigonometric functions is as follows:
Notice the key role that arclength plays in this construction. To measure an angle, and hence to define the trigonometric functions, one must know how to measure arclength. 1)
See Circle Geometry for further details.