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The Six Hyperbolic Functions Help (page 3)

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

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.

Hyperbolic Functions The Hyper Six Hyperbolic Sine

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.

Hyperbolic Functions The Hyper Six Hyperbolic Cotangent

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).

Hyperbolic Functions The Hyper Six Hyperbolic Cotangent

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

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