Question

A calculator has a built-in \(\sinh ^{-1} x\) function, but no \(\operatorname{csch}^{-1} x\) function. How do you evaluate csch \(^{-1} 5\) on such a calculator?

Short answer

Answer: \(\operatorname{csch}^{-1} 5 \approx 0.201\).

Step-by-step solution

Write the relationship between csch and sinh functions

Recall that the hyperbolic cosecant function is the reciprocal of the hyperbolic sine function, i.e., \(\operatorname{csch} x = \frac{1}{\sinh x}\). Therefore, we can rewrite \(\operatorname{csch}^{-1} 5\) as \(\operatorname{csch}^{-1} 5 = \sinh^{-1} \left(\frac{1}{5}\right)\).

Evaluate the expression on the calculator

Now, using the calculator, evaluate the expression \(\sinh^{-1} \left(\frac{1}{5}\right)\). The calculator should provide the result: \(\sinh^{-1} \left(\frac{1}{5}\right) \approx 0.201\), which means \(\operatorname{csch}^{-1} 5 \approx 0.201\).

Key concepts

Hyperbolic Cosecant Function

The hyperbolic cosecant function, represented as \textbf{csch(x)}, is one of the basic hyperbolic functions that mirrors the properties of the classical trigonometric functions but in the context of hyperbolas instead of circles. The hyperbolic cosecant is particularly the reciprocal of the hyperbolic sine function. So for any non-zero number 'x', the relationship is given by:

\[\begin{equation}\text{csch(x)} = \frac{1}{\sinh(x)}\end{equation}\]Just as the cosecant function is significant in trigonometry, the hyperbolic cosecant has its importance in hyperbolic function calculus. In various disciplines like physics, engineering, and mathematics, the function is used to describe curves, model certain types of wave phenomena, and solve differential equations reflecting real-world scenarios.

Understanding the hyperbolic cosecant function starts with familiarizing yourself with its basic properties, graphs, and the relationship with its hyperbolic 'sine' counterpart. Once you grasp the basic relationship, like how the hyperbolic cosecant is the multiplicative inverse of the hyperbolic sine, calculations, including those involving complex numbers, become more manageable.

Hyperbolic Sine Function

The hyperbolic sine function, denoted as \textbf{sinh(x)}, is one of the fundamental hyperbolic functions closely related to the hyperbolic cosine function. It is defined for any real number 'x' and mirrors the properties of a sine wave, except that the shape of this function is akin to a hyperbola, hence the name. Its definition using exponential functions is as follows:

\[\begin{equation}\sinh(x) = \frac{e^x - e^{-x}}{2}\end{equation}\]Where e represents the base of the natural logarithm.This function is central to the study of hyperbolic geometry and calculus, as well as in the solution of certain differential equations. The relationship between the hyperbolic sine and its reciprocal function, the hyperbolic cosecant, is essential when you encounter a situation where a calculator or a computer program lacks a direct function for computing csch(x). It's important to remember that these hyperbolic functions behave similarly to their trigonometric counterparts, but with notable differences in their graphs and mathematical behaviors due to their definitions through exponential functions.

Inverse Hyperbolic Functions

Inverse hyperbolic functions are the inverses of hyperbolic functions and are used to solve for angles (usually represented by a hyperbolic angle) when the hyperbolic function value is known. They are analogous to the arcfunctions in trigonometry - like arctangent or arcsin - but apply to hyperbolic functions such as sinh, cosh, and tanh.

For instance, the inverse hyperbolic sine function, written as \textbf{sinh^-1(x)} or \textbf{arsinh(x)}, allows you to find the hyperbolic angle for which the hyperbolic sine equals 'x'. This is vitally important when we need to switch between different types of hyperbolic functions during computation, such as when a calculator lacks a specific hyperbolic function, as seen in the original exercise.

When dealing with inverse hyperbolic functions, one key feature to keep in mind is that they can typically be expressed in terms of logarithms, making their calculation possible even without a dedicated calculator function. This is illustrated in the representation:

\[\begin{equation} sinh^{-1}(x) = \ln(x + \sqrt{x^2 + 1}) \end{equation}\]In handling these functions, familiarity with algebraic manipulations, log properties, and the exponential function's inverses is beneficial. Whether it's for simplifying expressions, integrating, or solving equations, understanding inverse hyperbolic functions extends the toolbox for tackling a vast range of mathematical problems.