Hyperbolic sector

A hyperbolic sector is a region of the Cartesian plane {(x,y)} bounded by rays from the origin to two points (a, 1/a) and (b, 1/b) and by the rectangular hyperbola xy = 1 (or the corresponding region when this hyperbola is rescaled and its orientation is altered by a rotation leaving the center at the origin, as with the unit hyperbola).

A hyperbolic sector in standard position has a = 1 and b > 1 .

Hyperbolic sectors are the basis for the hyperbolic functions.

The area of a hyperbolic sector in standard position is ln b .

Proof: Integrate under 1/x from 1 to b, add triangle {(0, 0), (1, 0), (1, 1)}, and subtract triangle {(0, 0), (b, 0), (b, 1/b)}.

When in standard position, a hyperbolic sector corresponds to a positive hyperbolic angle at the origin, with the measure of the latter being defined as the area of the former.

When in standard position, a hyperbolic sector determines a hyperbolic triangle, the right triangle with one vertex at the origin, base on the diagonal ray y = x, and third vertex on the hyperbola

with the hypotenuse being the segment from the origin to the point (x, y) on the hyperbola. The length of the base of this triangle is

and the altitude is

where u is the appropriate hyperbolic angle.

The analogy between circular and hyperbolic functions was described by Augustus De Morgan in his Trigonometry and Double Algebra (1849).William Burnside used such triangles, projecting from a point on the hyperbola xy = 1 onto the main diagonal, in his article "Note on the addition theorem for hyperbolic functions".

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