# Exploring areas with hyperbolic and trigonometric functions

This applet explores a geometric interpretation of the parameter t in the parameterisation of the standard hyperbola using cosh and sinh.

The hyperbolic functions $\cosh(t)$ and $\sinh(t)$ parameterise (one branch of) the standard hyperbola: as $t$ varies, the point $(\cosh(t), \sinh(t))$ traces out the right-hand branch of the hyperbola $x^2-y^2 =1$. Similarly to how $\cos(t)$ and $\sin(t)$ parameterise the unit circle. But what does the parameter $t$ represent? In the case of the unit circle, $t$ 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 $t$, in the parameterisation of the standard hyperbola using cosh and sinh, and the parameterisation of the unit circle using cos and sin.

Initially the applet shows the standard hyperbola $x^2-y^2=1$, 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 $t$?

Click ‘Show circle’ to show the unit circle $x^2+y^2=1$ and a point $(\cos(t), \sin(t))$ on the circle. What do you notice about the area and the value of $t$?

Challenge: prove what you’ve observed.

Other resources:

Geogebratube page for this applet