Hyperbolic Functions
Hyperbolic functions are defined in terms of certain combinations of ex and e−x. These functions arise naturally in various engineering and physics applications, including the study of water waves and vibrations of elastic membranes.
Another common use for a hyperbolic function is the representation of a hanging chain or cable, also known as a catenary. If we introduce a coordinate system so that the low point of the chain lies along the y-axis, we can describe the height of the chain in terms of a hyperbolic function.

Using the definition of cosh(x) and principles of physics, it can be shown that the height of a hanging chain can be described by the function h(x)=acosh(x/a)+c for certain constants a and c.
hyperbolic functions
Hyperbolic cosine
Hyperbolic sine
Hyperbolic tangent
Hyperbolic cosecant
Hyperbolic secant
Hyperbolic cotangent
The name cosh rhymes with “gosh,” whereas the name sinh is pronounced “cinch.” Tanh, sech, csch, and coth are pronounced “tanch,” “seech,” “coseech,” and “cotanch,” respectively.
But why are these functions called hyperbolic functions?
To answer this question, consider the quantity cosh2t−sinh2t. Using the definition of cosh and sinh, we see that
This identity is the analog of the trigonometric identity cos2t+sin2t=1. Here, given a value t, the point (x,y)=(cosht,sinht) lies on the unit hyperbola x2−y2=1 (Figure 7).

If you think hyperbolic functions look a lot like trigonometric ones, you’re not wrong! They share similar properties because they’re both connected to the concept of the exponential function ex. Remember, while trigonometric functions relate to the unit circle, hyperbolic functions are associated with the unit hyperbola.
Graphs of Hyperbolic Functions
The graphs of coshx and sinhx, can be derived by observing how they relate to exponential functions.
As x approaches towards infinity, both functions approach 12ex because the term e−x becomes negligible.
In contrast, as x moves towards negative infinity, coshx mirrors 12e−x, while sinhx mirrors −12e−x.
Therefore, the graphs 12ex,12e−x, and −12e−x provide a roadmap for sketching the graphs.
When graphing tanhx, we note that its value starts at 0 when x is 0 and then ascends towards 1 or descends towards −1 as x goes to positive or negative infinity, respectively.
The graphs of the other three hyperbolic functions can be sketched using the graphs of coshx,sinhx, and tanhx (Figure 8).
