Question

Evaluate the function. If the value is not a rational number, round your answer to three decimal places. (a) \(\cosh 0\) (b) \(\operatorname{sech} 1\)

Short answer

\(\cosh 0 = 1\) and \(\operatorname{sech} 1 \approx 0.648\)

Step-by-step solution

Find \(\cosh 0\)

To find \(\cosh 0\), remember the definition of the hyperbolic cosine function \(\cosh x = \frac{e^x + e^{-x}}{2}\). Using this definition, when \(x = 0\), \(\cosh 0 = \frac{e^0 + e^-0}{2} = \frac{1 + 1}{2}= 1\).

Evaluate \(\operatorname{sech} 1\)

Next, to find \(\operatorname{sech} 1\), as mentioned earlier, remember the definition of the hyperbolic secant function \(\operatorname{sech} x = \frac{1}{\cosh x}\). Using this definition, when \(x = 1\), \(\operatorname{sech} 1 = \frac{1}{\cosh 1}\). The value of \(\cosh 1\) is approximately 1.54308, so \(\operatorname{sech} 1 = \frac{1}{1.54308} \approx 0.648\). This answer is rounded to three decimal places as instructed because it's not a rational number.

Key concepts

Hyperbolic Cosine

Understanding the hyperbolic cosine function, or \\(\cosh x\)\, is key to exploring the realm of hyperbolic functions, which are analogs to the well-known trigonometric functions but for a hyperbola rather than a circle.

The formula for hyperbolic cosine is \\(\cosh x = \frac{e^x + e^{-x}}{2}\), where \(e\) is the base of the natural logarithm, approximately equal to 2.71828. This function represents the average of \(e^x\) and \(e^{-x}\), which can be seen as growth and decay rates in certain natural processes.

When evaluating \(\cosh 0\), the function simplifies remarkably as both \(e^0\) and \(e^{-0}\) equate to 1. Thus, the hyperbolic cosine of zero is indeed \(\frac{1+1}{2} = 1\). This not only demonstrates the symmetry of the hyperbolic cosine function but also indicates how it mirrors the traditional cosine function at certain points, specifically that both \(\cos 0\) and \(\cosh 0\) equal to 1.

Hyperbolic Secant

The hyperbolic secant, denoted as \\(\operatorname{sech} x\), is another important member of the hyperbolic function family and is defined as the reciprocal of the hyperbolic cosine.

The direct formula to compute the hyperbolic secant is \\(\operatorname{sech} x = \frac{1}{\cosh x}\). This reciprocal nature means when \(\cosh x\) is large, \(\operatorname{sech} x\) will be small, and vice versa. This is akin to the relationship between the trigonometric secant and cosine.

For the specific exercise of evaluating \(\operatorname{sech} 1\), we first need the value of \(\cosh 1\). The resulting value of \(\cosh 1\) is not a rational number, so the calculated \(\operatorname{sech} 1\) will similarly be irrational and approximated to three decimal places for practicality, yielding approximately 0.648. The understanding and computation of \(\operatorname{sech} x\) are essential in various fields such as calculus, complex analysis, and in solutions of certain differential equations.

Rational Numbers

The concept of rational numbers is foundational in mathematics. A number is considered \rational\ if it can be expressed as the quotient of two integers, where the denominator is non-zero; that is, in the form of \(\frac{a}{b}\) where \(a\) and \(b\) are integers and \(b eq 0\).

Rational numbers include integers, fractions, and finite or repeating decimals. Importantly, every rational number can be precisely expressed without the need for rounding.

In contrast, an \(\em{irrational number}\), such as \(\sqrt{2}\) or \(\pi\), cannot be represented as an exact ratio of two integers. In the context of our exercise, this distinction is crucial when determining whether to present the final answer as an exact value or a rounded approximation. For the hyperbolic secant of 1, we deal with an irrational outcome and hence provide a rounded value. Understanding what makes a number rational or irrational helps students make informed decisions on how to express their answers and deepens their comprehension of number theory.