**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.

**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 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 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 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 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.

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