**Introduction to the Six Hyperbolic Functions**

There are six *hyperbolic functions* that are similar in some ways to the circular functions. They are known as the *hyperbolic sine, hyperbolic cosine, hyperbolic tangent, hyperbolic cosecant, hyperbolic secant,* and*hyperbolic cotangent*. In formulas and equations, they are abbreviated sinh, cosh, tanh, csch, sech, and coth respectively.

The hyperbolic functions are based on certain characteristics of the *unit hyperbola*, which has the equation *x*^{2} - *y*^{2} = 1 in rectangular coordinates. Hyperbolic functions are used in certain engineering applications.

The circular functions operate on angles. In theory, the hyperbolic functions do too. Units are generally not mentioned for the quantities on which the hyperbolic functions operate, but they are understood to be in radians. Greek symbols are not always used to denote these variables. Plain lowercase English italicized *x* and *y* are common. Some mathematicians, scientists, and engineers prefer to use *u* and *v* . Once in a while you’ll come across a paper where the author uses the lowercase italicized Greek alpha ( *α* ) and beta ( *β* ) to represent the angles in hyperbolic functions.

**Powers Of ***e*

Once we define the hyperbolic sine and the hyperbolic cosine of a quantity, the other four hyperbolic functions can be defined, just as the circular tangent, cosecant, secant, and cotangent follow from the circular sine and cosine.

In order to clearly define what is meant by the hyperbolic sine and the hyperbolic cosine, we use *base-e exponential functions* . These revolve around a number that is denoted e. This number has some special properties. It is an *irrational number* —a number that can’t be precisely expressed as a ratio of two whole numbers. The best we can do is approximate it. (The term “irrational,” in mathematics, means “not expressible as a ratio of whole numbers.” It does not mean “unreasonable” or “crazy.”)

If you have a calculator with a function key marked “ *e *^{x} ” you can determine the value of e to several decimal places by entering the number 1 and then hitting the “ *e *^{x} ” key. If your calculator does not have an “ *e *^{x} ” key, it should have a key marked “ln” which stands for *natural logarithm* , and a key marked “inv” which stands for *inverse* . To get e from these keys, enter the number 1, and then hit “inv” and “ln” in succession. You should get a number whose first few digits are 2.71828.

If you want to determine the value of e ^{x} for some quantity *x* other than 1, you should enter the value *x* and then hit either the “ *e *^{x} ” key or else hit the “inv” and “ln” keys in succession, depending on the type of calculator you have. In order to find e ^{– x} , find e ^{x} first, and then find the reciprocal of this by hitting the “1/x” key.

If your calculator lacks exponential or natural logarithm functions, it is time for you to go out and buy one. Most personal computers have calculator programs that can be placed in “scientific mode,” where these functions are available.

**The Six**

**Two To Start**

Let *x* be a real number. The hyperbolic sine and the hyperbolic cosine can be defined in terms of powers of *e* , like this:

sinh *x* = ( *e* ^{x} – *e* ^{– x} )/2

cosh *x* = ( *e* ^{x} + e ^{– x} )/2

If these look intimidating, just remember that using them involves nothing more than entering numbers into a calculator and hitting certain keys in the correct sequence.

In a theoretical course, you will find other ways of expressing the hyperbolic sine and cosine functions, but for our purposes, the above two formulas are sufficient.

**The Other Four**

The remaining four hyperbolic functions follow from the hyperbolic sine and the hyperbolic cosine, like this:

tanh *x* = sinh *x* /cosh *x*

csch *x* = 1/sinh *x*

sech *x* = 1/cosh *x*

coth *x* = cosh *x* /sinh *x*

In terms of exponential functions, they are expressed this way:

tanh *x* = ( *e* ^{x} – *e* ^{– x} )/( *e* ^{x} + *e* ^{– x} )

csch *x* = 2/( *e* ^{x} – *e* ^{– x} )

sech *x* = 2/( *e* ^{x} + *e* ^{– x} )

coth *x* = ( *e* ^{x} + *e* ^{– x} )/( *e* ^{x} – *e* ^{– x} )

Now let’s look at the graphs of the six hyperbolic functions. As is the case with the inverses of the circular functions, the domain and/or range of the inverse of a hyperbolic function may have to be restricted to ensure that there is never more than one ordinate ( *y* value) for a given abscissa ( *x* value).

**Hyperbolic Sine**

Figure 4-1 is a graph of the function *y* = sinh *x* . Its domain and range both extend over the entire set of real numbers.

**Fig. 4-1.** Graph of the hyperbolic sine function.

**Hyperbolic Cosine**

Figure 4-2 is a graph of the function *y* = cosh *x* . Its domain extends over the whole set of real numbers, and its range is the set of real numbers *y* greater than or equal to 1.

**Fig. 4-2** . Graph of the hyperbolic cosine function.

**Hyperbolic Tangent**

Figure 4-3 is a graph of the function *y* = tanh *x* . Its domain encompasses the entire set of real numbers. The range of the hyperbolic tangent function is limited to the set of real numbers *y* between, but not including, –1 and 1; that is, –1 < *y* < 1.

**Fig. 4-3.** Graph of the hyperbolic tangent function.

**Hyperbolic Cosecant**

Figure 4-4 is a graph of the function *y* = csch *x* . Its domain encompasses the set of real numbers *x* such that *x* ≠ 0. The range of the hyperbolic cotangent function encompasses the set of real numbers *y* such that *y* ≠ 0.

**Fig. 4-4.** Graph of the hyperbolic cosecant function.

**Hyperbolic Secant**

Figure 4-5 is a graph of the function *y* = sech *x.* Its domain encompasses the entire set of real numbers. Its range is limited to the set of real numbers *y* greater than 0 but less than or equal to 1; that is, 0 < *y* ≤ 1.

**Fig. 4-5.** Graph of the hyperbolic secant function.

**Hyperbolic Cotangent**

Figure 4-6 is an approximate graph of the function *y* = coth *x* . Its domain encompasses the entire set of real numbers *x* such that *x* ≠ 0. The range of the hyperbolic cotangent function encompasses the set of real numbers *y* less than –1 or greater than 1; that is, *y* < –1 or *y* > 1.

**Fig. 4-6.** Graph of the hyperbolic cotangent function.

**The Six Hyperbolic Functions Practice Problems**

**Practice 1**

Why does the graph of *y* = csch *x* “blow up” when *x* = 0? Why is csch *x* not defined when *x* = 0?

**Solution 1**

Remember that the hyperbolic cosecant (csch) is the reciprocal of the hyperbolic sine (sinh). If *x* = 0, then sinh *x* = 0, as you can see from Fig. 4-1.

**Fig. 4-1.** Graph of the hyperbolic sine function.

As *x* approaches zero (written *x* → 0) from either side, the value of the hyperbolic sine also approaches zero (sinh *x* → 0). Thus, csch *x* , which is equal to 1/(sinh *x* ) and is graphed in Fig. 4-4, grows without limit as *x* → 0 from either direction.

**Fig. 4-4.** Graph of the hyperbolic cosecant function.

The value of *y* “blows up” positively as *x* → 0 from the positive, or right, side (written *x* → 0 ^{+} ) and negatively as *x* → 0 from the negative, or left, side ( *x* → 0 ^{–} ). When *x* = 0, the reciprocal of the hyperbolic sine is not defined, because it is a quotient with 0 in the denominator.

**Practice 2**

What is the hyperbolic cotangent of 0? Express it in two ways.

**Solution 2**

This quantity is not defined. The easiest way to demonstrate this fact is to look at the graph of the hyperbolic cotangent function (Fig. 4-6).

**Fig. 4-6.** Graph of the hyperbolic cotangent function.

The graph of the function *y* = coth *x* “blows up” at *x* = 0. It doesn’t have a *y* value there.

We can also express coth 0 by first finding the values of sinh 0 and cosh 0 using the exponential definitions. Remember the formulas:

sinh *x* = ( *e* ^{x} = *e* ^{– x} )/2

cosh *x* = ( *e* ^{x} = *e* ^{– x} )/2

If *x* = 0, then *e *^{x} = 1 and *e *^{–x} = 1. Therefore:

sinh 0 = (1 – 1)/2 = 0/2 = 0

cosh 0 = (1 + 1)/2 = 2/2 = 1

The hyperbolic cotangent is the hyperbolic cosine divided by the hyperbolic sine:

coth 0 = cosh 0/sinh 0 = 1/0

This expression is undefined, because it is a quotient with 0 in the denominator.

Practice problems for these concepts can be found at: Hyperbolic Functions Practice Test