= Hyperbolic angle
{title2=$t=2A$}
A <hyperbolic angle> $t$ is twice the signed <area> $A$ enclosed by the two rays from the origin to $(1,0)$ and $(\cosh t,\sinh t)$ and the intervening arc of the unit <hyperbola>. Counterclockwise traversal is positive: $t>0$ gives an ordinary <area> $t/2$ above the horizontal axis, and $t<0$ gives signed <area> $t/2$ below it. This parallels the <unit circle>, whose sector has <area> $\theta/2$ for an <angle> $\theta$ measured in <radians>.
To check the <area> interpretation, write $x(u)=\cosh u$ and $y(u)=\sinh u$. <Green's theorem> expresses the signed <area> as a <line integral>; the straight radial edges contribute zero, leaving
$$
A(t)=\frac12\int_0^t\bigl(x(u)y'(u)-y(u)x'(u)\bigr)\,du=\frac12\int_0^t\bigl(\cosh^2u-\sinh^2u\bigr)\,du=\frac t2.
$$
Thus $t$ is an <area> parameter, rather than the ordinary <angle> of the ray. For $t>0$, that ordinary <angle> is $\arctan(\tanh t)$; it stays below $\pi/4$ even as the <hyperbolic angle> grows without bound.
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