Exploring areas with hyperbolic and trigonometric functions
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 . 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, 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 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.