A hyperbolic angle in standard position is the angle at (0,0) between the ray to (1,1) and the ray to (x,1/x) where x > 1.
The magnitude of the hyperbolic angle is the area of the corresponding hyperbolic sector which is log x.
Note that unlike circular angle, hyperbolic angle is unbounded, as is the function logx, a fact related to the unbounded nature of the harmonic series.
The hyperbolic equivalent of the fundamental trigonometric identity is cosh
Hyperbolic functions are also useful because they occur in the solutions of some simple linear differential equations, notably that defining the shape of a hanging cable, the catenary.
The parameter t is not a circular angle, but rather a hyperbolicangle which represents twice the area between the x-axis, the hyperbola and the straight line which links the origin with the point (cosh t, sinh t) on the hyperbola.
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