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# The Six Hyperbolic Functions Help (page 2)

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By — McGraw-Hill Professional
Updated on Oct 3, 2011

## 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 xe 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 xe x )/( e x + e x )

csch x = 2/( e xe x )

sech x = 2/( e x + e x )

coth x = ( e x + e x )/( e xe 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.

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