Integration of Exponential, Logarithmic, and Hyperbolic Functions: Background You’ll Need 3

  • Identify hyperbolic functions their graphs, and understand their fundamental identities

Hyperbolic Functions

Hyperbolic functions are defined in terms of certain combinations of [latex]e^x[/latex] and [latex]e^{−x}[/latex]. 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 [latex]y[/latex]-axis, we can describe the height of the chain in terms of a hyperbolic function. 

A photograph of a spider web collecting dew drops.
Figure 6. The shape of a strand of silk in a spider’s web can be described in terms of a hyperbolic function. The same shape applies to a chain or cable hanging from two supports with only its own weight. (credit: “Mtpaley”, Wikimedia Commons)

Using the definition of [latex]\cosh(x)[/latex] and principles of physics, it can be shown that the height of a hanging chain can be described by the function [latex]h(x)=a \cosh(x/a)+c[/latex] for certain constants [latex]a[/latex] and [latex]c[/latex].

hyperbolic functions

Hyperbolic cosine

[latex]\cosh x=\large \frac{e^x+e^{−x}}{2}[/latex]

 

Hyperbolic sine

[latex]\sinh x=\large \frac{e^x-e^{−x}}{2}[/latex]

 

Hyperbolic tangent

[latex]\tanh x=\large \frac{\sinh x}{\cosh x} \normalsize = \large \frac{e^x-e^{−x}}{e^x+e^{−x}}[/latex]

 

Hyperbolic cosecant

[latex]\text{csch} \, x=\large \frac{1}{\sinh x} \normalsize = \large \frac{2}{e^x-e^{−x}}[/latex]

 

Hyperbolic secant

[latex]\text{sech} \, x=\large \frac{1}{\cosh x} \normalsize = \large \frac{2}{e^x+e^{−x}}[/latex]

 

Hyperbolic cotangent

[latex]\coth x=\large \frac{\cosh x}{\sinh x} \normalsize = \large \frac{e^x+e^{−x}}{e^x-e^{−x}}[/latex]

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 [latex]\cosh^2 t-\sinh^2 t[/latex]. Using the definition of [latex]\cosh[/latex] and [latex]\sinh[/latex], we see that

[latex]\cosh^2 t-\sinh^2 t=\large \frac{e^{2t}+2+e^{-2t}}{4}-\frac{e^{2t}-2+e^{-2t}}{4} \normalsize =1[/latex]

 

This identity is the analog of the trigonometric identity [latex]\cos^2 t+\sin^2 t=1[/latex]. Here, given a value [latex]t[/latex], the point [latex](x,y)=(\cosh t,\sinh t)[/latex] lies on the unit hyperbola [latex]x^2-y^2=1[/latex] (Figure 7).

An image of a graph. The x axis runs from -1 to 3 and the y axis runs from -3 to 3. The graph is of the relation “(x squared) - (y squared) -1”. The left most point of the relation is at the x intercept, which is at the point (1, 0). From this point the relation both increases and decreases in curves as x increases. This relation is known as a hyperbola and it resembles a sideways “U” shape. There is a point plotted on the graph of the relation labeled “(cosh(1), sinh(1))”, which is at the approximate point (1.5, 1.2).
Figure 7. The unit hyperbola [latex]\cosh^2 t-\sinh^2 t=1[/latex].

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 [latex]e^x[/latex]. Remember, while trigonometric functions relate to the unit circle, hyperbolic functions are associated with the unit hyperbola.

Graphs of Hyperbolic Functions

The graphs of [latex]\cosh x[/latex] and [latex]\sinh x[/latex], can be derived by observing how they relate to exponential functions.

As [latex]x[/latex] approaches towards infinity, both functions approach [latex]\frac{1}{2}e^x[/latex] because the term [latex]e^{−x}[/latex] becomes negligible.

In contrast, as [latex]x[/latex] moves towards negative infinity, [latex]\cosh x[/latex] mirrors [latex]\frac{1}{2}e^{−x}[/latex], while [latex]\sinh x[/latex] mirrors [latex]-\frac{1}{2}e^{−x}[/latex].

Therefore, the graphs [latex]\frac{1}{2}e^x, \, \frac{1}{2}e^{−x}[/latex], and [latex]−\frac{1}{2}e^{−x}[/latex] provide a roadmap for sketching the graphs.

When graphing [latex]\tanh x[/latex], we note that its value starts at [latex]0[/latex] when [latex]x[/latex] is [latex]0[/latex] and then ascends towards [latex]1[/latex] or descends towards  [latex]-1[/latex] as [latex]x[/latex] goes to positive or negative infinity, respectively.

The graphs of the other three hyperbolic functions can be sketched using the graphs of [latex]\cosh x, \, \sinh x[/latex], and [latex]\tanh x[/latex] (Figure 8).

An image of six graphs. Each graph has an x axis that runs from -3 to 3 and a y axis that runs from -4 to 4. The first graph is of the function “y = cosh(x)”, which is a hyperbola. The function decreases until it hits the point (0, 1), where it begins to increase. There are also two functions that serve as a boundary for this function. The first of these functions is “y = (1/2)(e to power of -x)”, a decreasing curved function and the second of these functions is “y = (1/2)(e to power of x)”, an increasing curved function. The function “y = cosh(x)” is always above these two functions without ever touching them. The second graph is of the function “y = sinh(x)”, which is an increasing curved function. There are also two functions that serve as a boundary for this function. The first of these functions is “y = (1/2)(e to power of x)”, an increasing curved function and the second of these functions is “y = -(1/2)(e to power of -x)”, an increasing curved function that approaches the x axis without touching it. The function “y = sinh(x)” is always between these two functions without ever touching them. The third graph is of the function “y = sech(x)”, which increases until the point (0, 1), where it begins to decrease. The graph of the function has a hump. The fourth graph is of the function “y = csch(x)”. On the left side of the y axis, the function starts slightly below the x axis and decreases until it approaches the y axis, which it never touches. On the right side of the y axis, the function starts slightly to the right of the y axis and decreases until it approaches the x axis, which it never touches. The fifth graph is of the function “y = tanh(x)”, an increasing curved function. There are also two functions that serve as a boundary for this function. The first of these functions is “y = 1”, a horizontal line function and the second of these functions is “y = -1”, another horizontal line function. The function “y = tanh(x)” is always between these two functions without ever touching them. The sixth graph is of the function “y = coth(x)”. On the left side of the y axis, the function starts slightly below the boundary line “y = 1” and decreases until it approaches the y axis, which it never touches. On the right side of the y axis, the function starts slightly to the right of the y axis and decreases until it approaches the boundary line “y = -1”, which it never touches.
Figure 8. The hyperbolic functions involve combinations of [latex]e^x[/latex] and [latex]e^{−x}[/latex].

Identities Involving Hyperbolic Functions

Just as trigonometric functions have identities that allow for the simplification and transformation of expressions, hyperbolic functions also possess their own set of identities.

hyperbolic function identities

Hyperbolic Reflection Identities:

  • [latex]\cosh(−x)=\cosh x[/latex]
  • [latex]\sinh(−x)=−\sinh x[/latex]

Hyperbolic Pythagorean Identities:

  • [latex]\cosh^2 x-\sinh^2 x=1[/latex]

Hyperbolic Squared Identities:

  • [latex]1-\tanh^2 x=\text{sech}^2 x[/latex]
  • [latex]\coth^2 x-1=\text{csch}^2 x[/latex]

Hyperbolic Addition Formulas:

  • [latex]\sinh(x \pm y)=\sinh x \cosh y \pm \cosh x \sinh y[/latex]
  • [latex]\cosh (x \pm y)=\cosh x \cosh y \pm \sinh x \sinh y[/latex]

Exponential Definitions of Hyperbolic Functions

  • [latex]\cosh x+\sinh x=e^x[/latex]
  • [latex]\cosh x-\sinh x=e^{−x}[/latex]
  1. Simplify [latex]\sinh(5 \ln x)[/latex].
  2. If [latex]\sinh x=\frac{3}{4}[/latex], find the values of the remaining five hyperbolic functions.