In Excel, the hyperbolic functions COSH, SINH, and TANH all take a number representing a hyperbolic angle as input. A hyperbolic angle is defined by the area of the sector on the right branch of the unit hyperbola x² - y² = 1, formed by the origin, the point (1,0), and a point on the unit hyperbola. For example, a hyperbolic angle of 1 corresponds to the sector formed on the unit hyperbola with an area of one-half.
In general, you can think about a hyperbolic angle forming a point on the unit hyperbola, where COSH and SINH give the coordinates of the point:
A negative hyperbolic angle corresponds to a point with a negative y-coordinate.
The area of the sector is half the angle's value to align the hyperbolic functions with their circular counterparts: cosine and sine. This is because the area formed by a circular angle on the unit circle is one-half the angle's value.
Dividing by two makes the area of a hyperbolic angle equal to that of a circular angle. For example, the geometry of the hyperbolic and circular functions for the angle a=1 is shown below.
Unlike a circular angle, whose point rotates periodically around the circle as the angle grows, a hyperbolic angle diverges toward infinity as it increases positively and toward negative infinity as it increases negatively.
Images courtesy of wumbo.net.
