By now you should have begun to feel fairly familiar with the hyperbolic functions. You've seen their connection with exponential functions, logarithms and the trig functions. You've differentiated them and integrated some of them and looked at their inverse functions. But I haven't said why they are called hyperbolic functions.
As you might guess, it's because they have something to do with the "hyperbola". This curve is one of the so-called "conic sections" (which is a particular group of curves that you see if you cut up a cone..). It looks like this:
Another member of the group of conic sections is the circle. You might remember that the equation for a circle in Cartesian coordinates is: x2+y2=a2, where a is a constant and is the length of the circle's radius.
Each of the conic sections has an equation. The equation for the hyperbola is this:
Here a and b can take any values, and varying a and b varies the shape of the hyperbola. In particular, consider the rectangular hyperbola, for which a and b have the same value, and let's say the value is 1. Then the equation above becomes:
x2-y2=1.
Now recall the identity relating sinh and cosh, we'll use t as the variable to avoid confusion, since we've got an "x" already in the last equation
cosh2(t)-sinh2(t)=1.
Notice the similarity between those last two equations. It suggests that in some way we can say that the x is like the cosh(t) and the y is like the sinh(t). (In the sense that the role played by the "x" in the first equation is the same as the role played by the cosh(t) in the second equation, and similarly for the y and the sinh(t)).
We can look at that similarity in a different and more useful way as follows.
We can think of x and y as being functions of the variable t, defined by x(t)=cosh(t) and y(t)=sinh(t). That's building on the feeling in the previous paragraph that "x is like cosh(t)".
Then if we allow t to take different values, and see what values we get for x and y, we will find that because we have constrained x and y to be cosh(t) and sinh(t), they must always satisfy x2-y2=1 (because cosh2(t)-sinh2(t)=1).
So, whatever value we choose for t, the point (x,y) will lie on the hyperbola. And if we let t go through all possible values, the (x,y) points will trace out the hyperbola.
Actually only the right-hand half of the curve is produced, since cosh(t), and hence x, is always positive.
This idea of making the coordinates (x,y) be functions of another variable, t, is called parameterising the curve. The variable t is called the parameter.
The equations: "x=cosh(t), y=sinh(t)" are called the parametric equations for the right-hand branch of the hyperbola.
So to sum up, the functions sinh and cosh are called "hyperbolic" because they define the parametric form for the hyperbola.
For exactly the same reason, the trigonometric functions sine and cosine, which satisfy sin2(t)+cos2(t)=1, are sometimes called the "circular functions", as they provide the parametric form for the circle: x=sin(t),y=cos(t). As you vary t, the (x,y) points trace out a circle.
Another parametric representation for the hyperbola (which gives both branches) is the following:
x=a sec(t), y=b tan(t).
This works because sec2(t)-tan2(t)=1.