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Chapter 16: Problem 6

Evaluate coth \(0.38\), correct to 3 decimal places.

Short Answer

Expert verified
Evaluate using the hyperbolic identity or a calculator for quick approximation.

Step by step solution

01

Recall the Definition of Hyperbolic Cotangent

The hyperbolic cotangent function, denoted as \( \text{coth}(x) \), is the reciprocal of the hyperbolic tangent. It is defined as:\[ \text{coth}(x) = \frac{\cosh(x)}{\sinh(x)} \]where \( \cosh(x) = \frac{e^x + e^{-x}}{2} \) and \( \sinh(x) = \frac{e^x - e^{-x}}{2} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Hyperbolic Functions
Hyperbolic functions are analogs of the trigonometric functions but are based on hyperbolas instead of circles. They are particularly useful in various areas of mathematics, including calculus and complex analysis, because of their similar properties to trigonometric functions.

The primary hyperbolic functions are:
  • Hyperbolic Sine: \( \sinh(x) = \frac{e^x - e^{-x}}{2} \)
  • Hyperbolic Cosine: \( \cosh(x) = \frac{e^x + e^{-x}}{2} \)
  • Hyperbolic Tangent: \( \tanh(x) = \frac{\sinh(x)}{\cosh(x)} \)

These functions describe the shape of a hyperbola in much the same way that sine and cosine describe the shape of a circle.

A vital thing to understand about hyperbolic functions is that they arise naturally in the solutions of many types of differential equations, especially those that involve exponential growth or decay.
Reciprocal Functions
Reciprocal functions play a significant role in mathematics by showing the inverse relationship between different values. For every non-zero number \( x \), the reciprocal is \( \frac{1}{x} \).

In the context of hyperbolic functions, reciprocal functions create the hyperbolic cosecant, secant, and cotangent functions. Let's focus on the hyperbolic cotangent function:
  • Hyperbolic Cotangent: \( \text{coth}(x) = \frac{1}{\tanh(x)} = \frac{\cosh(x)}{\sinh(x)} \)
The formula \( \text{coth}(x) = \frac{1}{\tanh(x)} \) demonstrates how the hyperbolic cotangent is the reciprocal of the hyperbolic tangent.

Reciprocal functions help in switching perspectives in mathematics, often simplifying calculations and revealing inherent properties of the function it inverses.
Mathematical Definitions
Understanding mathematical definitions is crucial for comprehending and applying concepts effectively. A mathematical definition provides a clear, unambiguous meaning to a mathematical term.

In this context, let's consider the definitions of hyperbolic cotangent:
  • \( \text{coth}(x) = \frac{\cosh(x)}{\sinh(x)} \) defines it as the ratio of the hyperbolic cosine to the hyperbolic sine.
  • Similarly, \( \cosh(x) \) and \( \sinh(x) \) are defined based on the exponential function: \ \( \cosh(x) = \frac{e^x + e^{-x}}{2} \) \ and \ \( \sinh(x) = \frac{e^x - e^{-x}}{2} \).

These foundational definitions facilitate solving mathematical problems and conducting further analysis. By using these precise definitions, we can understand the behavior of these functions over real and complex domains.

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Most popular questions from this chapter

Solve the equation \(\operatorname{cosech} y=-0.4458\), correct to 4 significant figures.

A chain hangs in the form given by \(y=40 \operatorname{ch} \frac{x}{40}\). Determine, correct to 4 significant figures, (a) the value of \(y\) when \(x\) is 25 , and (b) the value of \(x\) when \(y=54.30\)
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