The hyperbolic functions cosh(t) and sinh(t) parameterise (one branch of) the standard hyperbola: as [i]t[/i] varies, the point (cosh(t), sinh(t)) traces out the right-hand branch of the hyperbola [math]x^2-y^2=1[/math]. Similarly to how cos(t) and sin(t) parameterise the unit circle. But what does the parameter [i]t[/i] represent? In the case of the unit circle, [i]t[/i] has a clear interpretation: the angle from the x-axis to the point (cos(t), sin(t)). But what about in the hyperbolic case? This applet explores a geometric interpretation of [i]t[/i], in the parameterisation of the standard hyperbola using cosh and sinh, and the parameterisation of the unit circle using cos and sin.[br][br]Initially the applet shows the standard hyperbola[math]x^2-y^2=1[/math], a point (cosh(t), sinh(t)) on the hyperbola, and the area enclosed between the x axis, the hyperbola and the line from the origin to (cosh(t), sinh(t)). Drag the point to move it, or type in a new value of t in the textbox. What do you notice about the area and the value of [i]t[/i]?[br][br]Click 'Show circle' to show the unit circle [math]x^2+y^2=1[/math] and a point (cos(t), sin(t)) on the circle. What do you notice about the area and the value of [i]t[/i]?[br][br]Challenge: prove what you've observed.