The hyperbolic functions have similar names to the trigonometric functions, but they are defined
in terms of the exponential function. In this unit I define the three main hyperbolic functions,
and sketch their graphs. I also discuss some identities relating these functions, and mention
their inverse functions and reciprocal functions.
The hyperbolic functions cosh x and sinh x are defined using the exponential function ex. We
shall start with cosh x. This is defined by the formula
cosh x = (ex + e−x)/2.
We can use our knowledge of the graphs of ex and e−x to sketch the graph of cosh x. First, let
us calculate the value of cosh 0. When x = 0, ex = 1 and e−x = 1.
cosh 0 =(ex + e−x)/2 = (1 + 1)/2 = 1 .
Next, let us see what happens as x gets large. We shall rewrite cosh x as
cosh x =e x/2 + e−x/2.
To see how this behaves as x gets large, recall the graphs of the two exponential functions.
y = ex/2
As x gets larger, ex increases quickly, but −x decreases quickly. So the second part of the sum
e , ex/2 + e−x /2 gets very small as x gets large. Therefore, as x gets larger, cosh x gets closer and
closer to ex/2. We write this as
cosh x ≈
for large x.
But the graph of cosh x will always stay above the graph of ex/2. This is because, even though
(e−x)/2 (the second part of the sum) gets very small, it is always greater than zero. As x gets
larger and larger the difference between the two graphs gets smaller and smaller.
As x becomes more negative, ex increases quickly, but ex decreases
quickly, so the first part of the sum ex/2 + e−x/2 gets very small. As x gets more and more
negative, cosh x gets closer and closer to e−x/2. We write this as
cosh x ≈
for large negative x.
Again, the graph of cosh x will always stay above the graph of e−x/2 when x is negative. This is
because, even though ex/2 (the first part of the sum) gets very small, it is always greater than
zero. But as x gets more and more negative the difference between the two graphs gets smaller
We can now sketch the graph of cosh x. Notice the graph is symmetric about the y-axis, because
cosh x = cosh(−x).
The hyperbolic function f(x) = cosh x is defined by the formula
cosh x = (ex)/2 + e−x/2
The function satisfies the conditions cosh 0 = 1 and cosh x = cosh(−x). The graph of cosh x is always above the graphs of ex/2 and e−x/2.