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The Six Hyperbolic Functions Help (page 2)

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.

Hyperbolic Functions The Hyper Six Hyperbolic Sine

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.

Hyperbolic Functions The Hyper Six Hyperbolic Cosine

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.

Hyperbolic Functions The Hyper Six Hyperbolic Tangent

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.

Hyperbolic Functions The Hyper Six Hyperbolic Cotangent

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.

Hyperbolic Functions The Hyper Six Hyperbolic Cotangent

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.

Hyperbolic Functions The Hyper Six Hyperbolic Cotangent

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

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