| A standard fact from trigonometry is that the sine and
cosine functions define a parametric description of the unit circle x2 + y2 = 1. In other words, any point on the unit circle can be represented in the form (cos
t, sin t).
Because of the Pythagorean Identity, i.e., sin2
t + cos2
t = 1, this point is easily seen to be on the unit circle.
In a similar fashion, we may define the unit hyperbola by x2 - y2 = 1. Like all hyperbolas, the unit hyperbola has two branches. In this case, there is one branch in the left-hand plane and one branch in the right-hand plane. The right branch has a parametric description as (cos t, sin t), where cosh and sinh are the hyperbolic cosine and hyperbolic sine functions, respectively. We may define these hyperbolic functions in terms of the exponential function. In particular, cosh(x) = (ex + e-x)/2 and sinh(x) = (ex - e-x)/2. ?Using these descriptions, it is easy to see that cosh2 t - sinh2 t = 1. In the applet at left, you may examine the parametric descriptions of both the unit circle and unit hyperbola. By pressing the appropriate button, you may switch back and forth between the descriptions. Pressing these buttons will also reset the applet. By moving the slide below the graph, you may move the point along the curve. |