Education.com
Try
Brainzy
Try
Plus

Hyperbolic Inverses Help (page 2)

(not rated)
By McGraw-Hill Professional
Updated on Aug 30, 2011

Hyperbolic Arctangent

Figure 4-9 is a graph of the function y = arctanh x (or y = tanh –1 x ). The domain is limited to real numbers x such that –1 < x < 1. The range of the hyperbolic arctangent function spans the entire set of real numbers.

Fig. 4-9. Graph of the hyperbolic arctangent function.

Arccosecant, Arcsecant, and Arccotangent

Hyperbolic Arccosecant

Figure 4-10 is a graph of the function y = arccsch x (or y = csch –1 x ). Both the domain and the range of the hyperbolic arccosecant function include all real numbers except zero.

Fig. 4-10. Graph of the hyperbolic arccosecant function.

Hyperbolic Arcsecant

Figure 4-11 is a graph of the function y = arcsech x (or y = sech –1 x ). The domain of this function is limited to real numbers x such that 0 < x ≤ 1. The range of the hyperbolic arcsecant function is limited to the non-negative reals, that is, to real numbers y such that y ≥ 0.

Fig. 4-11. Graph of the hyperbolic arcsecant function.

Hyperbolic Arccotangent

Figure 4-12 is a graph of the function y = arccoth x (or y = coth –1 x ). The domain of this function includes all real numbers x such that x < –1 or x > 1. The range of the hyperbolic arccotangent function includes all real numbers except zero.

Fig. 4-12. Graph of the hyperbolic arccotangent function.

Hyperbolic Inverses Practice Problems

Practice 1

What is the value of arcsinh 0? Use a calculator if you need it.

Solution 1

From the graph in Fig. 4-7, it appears that it ought to be 0. We can verify this by using the formula above along with a calculator if needed:

If you’ve had any experience with logarithms, you don’t need a calculator to do the above calculation, because you already know that the natural logarithm of 1 is equal to 0.

Practice 2

What is the value of arccsch 1? Use a calculator if you need it. Use the logarithm-based formulas to determine the answer, and express it to three decimal places.

Solution 2

From the graph in Fig. 4-10, we can guess that arccsch 1 ought to be a little less than 1.

Fig. 4-10. Graph of the hyperbolic arccosecant function.

Let’s use the formula above and find out:

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

150 Characters allowed

Related Questions

Q:

See More Questions
Top Worksheet Slideshows