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Unit hyperbola
The unit hyperbola is described by the equation
This is very similar to the unit circle, and there are also hyperbolic functions defined for it, that again are very similar to the trigonometric functions.
Explanation
The coordinates (x, y) of a point on the standard hyperbola are connected to the size of the blue-colored surface by
and
For the inverse functions you get
and
and because this concerns areas they have got the names area hyperbolic cosine and area hyperbolic sine, what you write as
and