A curve is fixed an tangent to the curve at an adjustable point is observed. One observes the constancy of the ratio of the attached triangle as well as the angle. Thus one may “well-definedly” associate the Ratio to the Angle. Concretely, one defines: tan(Angle) = Ratio

Vary (1) the point on the curve, P, (2) the size of the triangle, and observe/compute (by hand or with a calculator, etc.)

the Angle, theta.

the Ratio, Opp/Adj.

Notice how under changes in the size of the triangle, these two do not change. Compute tan(theta). What does one observe?