The hyperbolic cosine and hyperbolic sine can be understood geometrically in terms of areas of hyperbolic sectors. The unit hyperbola is defined by the equation x^2 - y^2 = 1, and the right branch (x ≥ 1) of the unit hyperbola is parameterized by x = cosh(t) and y = sinh(t), where t ranges through the set of all real numbers.
If P = (cosh(p), sinh(p)) and Q = (cosh(q), sinh(q)) are two points on this curve, then a hyperbolic sector OPQ is formed by the arc PQ and two segments OP and OQ, joining the endpoints of the arc to the origin. The area of the hyperbolic sector OPQ is |p-q|/2.
In the interactive figure shown below, the area of the blue sector is |t|/2, and the area of the yellow sector is also |t|/2. The sectors can be altered by dragging the points A and B. This figure provides a visual interpretation of the addition laws for the hyperbolic cosine and hyperbolic sine.
This figure is a web supplement for our article, A geometric approach to the natural exponential function, which is in preparation.