Hyperbolic Inverses Help (page 2)
Introduction Hyperbolic Inverses—Logarithms
Each of the six hyperbolic functions has an inverse relation. These are known as the hyperbolic arcsine, hyperbolic arccosine, hyperbolic arctangent, hyperbolic arccosecant, hyperbolic arcsecant, and hyperbolic arccotangent. In formulas and equations, they are abbreviated arcsinh or sinh –1 , arccosh or cosh –1 , arctanh or tanh –1 , arccsch or csch –1 , arcsech or sech –1 , and arccoth or coth –1 respectively. These relations become functions when their domains are restricted as shown in the graphs of Figs. 4-7 through 4-12.
The Natural Logarithm
Now it is time to learn a little about logarithms. It is common to write “the natural logarithm of x ” as “ln x .” This function is the inverse of the base- e exponential function. The natural logarithm function and the base- e exponential function “undo” each other. Suppose x and v are real numbers, and y and u are positive real numbers. If e x = y , then x = ln y , and if ln u = v , then u = e v .
The natural logarithm function is useful in expressing the inverse hyperbolic functions, just as the exponential function can be used to express the hyperbolic functions.
You can find the natural logarithm of a specific number using a calculator. Enter the number for which you want to find the natural logarithm, and then hit the “ln” key. Beware: the logarithm of 0 or any negative real number is not defined in the set of real numbers.
Hyperbolic Inverses As Logarithms
You can find hyperbolic inverses of specific quantities using a calculator that has the “ln” function. Here are the expressions for the hyperbolic inverses, in terms of natural logarithms. (The ½ power represents the square root.)
arcsinh x = ln [ x + ( x 2 + 1) 1/2 ]
arccosh x = ln [x + ( x 2 – 1) 1/2 ]
arctanh x = 0.5 ln [(1 + x )/(1 – x )]
arccsch x = ln [ x –1 + ( x –2 + 1) 1/2 ]
arcsech x = ln [ x –1 + ( x –2 – 1) 1/2 ]
arctanh x = 0.5 ln [( x + 1)/( x – 1 )]
In these expressions, the values 0.5 represent exactly ½. The formulas are a little bit messy, but if you plug in the numbers and take your time doing the calculations, you shouldn’t have trouble. Be careful about the order in which you perform the operations. Perform the operations in the innermost sets of parentheses or brackets first, and then work outward.
Let’s see what the graphs of the inverse hyperbolic functions look like.
Arcsine, Arccosine, and Arctangent
Figure 4-7 is a graph of the function y = arcsinh x (or y = sinh –1 x ). Its domain and range both encompass the entire set of real numbers.
Figure 4-8 is a graph of the function y = arccosh x (or y = cosh –1 x ). The domain includes real numbers x such that x ≥ 1. The range of the hyperbolic arccosine function is limited to the non-negative reals, that is, to real numbers y such that y ≥ 0.
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.
Arccosecant, Arcsecant, and Arccotangent
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.
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.
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.
Hyperbolic Inverses Practice Problems
What is the value of arcsinh 0? Use a calculator if you need it.
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.
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.
From the graph in Fig. 4-10, we can guess that arccsch 1 ought to be a little less than 1.
Let’s use the formula above and find out:
Practice problems for these concepts can be found at: Hyperbolic Functions Practice Test
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