This app illustrates the geometric connection between the unit hyperbola and the hyperbolic sine in cosine functions. These relationships are analogous to those between the unit circle and the trigonometric sine and cosine functions.

Drag the point P along the hyperbola. The x and y coordinates of this point are equal to cosh(2A) and sinh(2A), respectively, where A is the area bounded by the x-axis and the graphs of the line through the origin and the point P and the graph of the hyperbola, as you can verify directly by performing the calculations above the graph. The same relationship exists between the unit circle and the cosine and sine functions.