What t and θ represent in the different parametric equations for a hyperbola

In FP3 you learn parametric equations for the hyperbola

x2a2y2b2 = 1

x = a cosh t
y = b sinh t

or

x = a sec θ
y = b tan θ

Here is what θ represents, as an “angle-stamp” for each point on the hyperbola (illustrated for the case a = b; the angle of the asymptotes will be different for a ≠ b, but the animation always “flips” at θ = π/2 or 90 degrees and θ = 3π/2 or 270 degrees).

t is a sort of “area-stamp”

analogous to the “area-stamp” you can use for points on a circle

The relation between the two is:

sin θ = tanh t

⁡θ is the slope of the segment from the origin to (cosht,sinht)

The cosh t/ sinh t equations represent only the right-hand side of the hyperbola, but the sec θ/ tan θ equations represent both sides.