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Chapter 2: Problem 81

In each of the following exercises a value of one of the six hyperbolic functions of \(x\) is given. Find the remaining five. (a) \(\cosh x=\frac{5}{4}\) (b) \(\sinh x=\frac{8}{15}\) (c) \(\tanh x=-\frac{7}{25}\) (d) \(\operatorname{sech} x=\frac{5}{13}\) (e) \(\operatorname{cosech} x=-\frac{3}{4}\) (f) coth \(x=\frac{13}{12}\)

Short Answer

Expert verified
Each part calculates the remaining hyperbolic functions using identities and given values.

Step by step solution

01

Understanding the problem

We are given values of one of the six hyperbolic functions for several exercises. Our task is to determine the values of the remaining five hyperbolic functions (sinh, cosh, tanh, sech, cosech, coth) based on the given information.
02

Solving part (a)

Given \( \cosh x = \frac{5}{4} \). Use the identity \( \cosh^2 x - \sinh^2 x = 1 \) to find \( \sinh x \):1. \( \sinh^2 x = \cosh^2 x - 1 = \left( \frac{5}{4} \right)^2 - 1 = \frac{25}{16} - 1 = \frac{9}{16} \)2. Hence, \( \sinh x = \frac{3}{4} \) or \( \sinh x = -\frac{3}{4} \).Finding tanh, sech, cosech, and coth using \( \cosh x = \frac{5}{4} \) and \( \sinh x = \frac{3}{4} \):- \( \tanh x = \frac{\sinh x}{\cosh x} = \frac{3/4}{5/4} = \frac{3}{5} \)- \( \operatorname{sech} x = \frac{1}{\cosh x} = \frac{4}{5} \)- \( \operatorname{cosech} x = \frac{1}{\sinh x} = \frac{4}{3} \)- \( \operatorname{coth} x = \frac{\cosh x}{\sinh x} = \frac{5}{3} \)
03

Solving part (b)

Given \( \sinh x = \frac{8}{15} \). Use the identity \( \cosh^2 x - \sinh^2 x = 1 \) to find \( \cosh x \):1. \( \cosh^2 x = 1 + \sinh^2 x = 1 + \left( \frac{8}{15} \right)^2 = 1 + \frac{64}{225} = \frac{289}{225} \)2. Hence, \( \cosh x = \frac{17}{15} \).Calculate other functions:- \( \tanh x = \frac{8}{17} \)- \( \operatorname{sech} x = \frac{15}{17} \)- \( \operatorname{cosech} x = \frac{15}{8} \)- \( \operatorname{coth} x = \frac{17}{8} \)
04

Solving part (c)

Given \( \tanh x = -\frac{7}{25} \). Use \( \tanh x = \frac{\sinh x}{\cosh x} \) and \( \operatorname{sech} x = \sqrt{1 - \tanh^2 x} \).1. \( \tanh^2 x = \left( -\frac{7}{25} \right)^2 = \frac{49}{625} \)2. \( \operatorname{sech} x = \sqrt{1 - \frac{49}{625}} = \sqrt{\frac{576}{625}} = \frac{24}{25} \)3. \( \cosh x = \frac{25}{24} \)4. \( \sinh x = \tanh x \times \cosh x = -\frac{7}{25} \times \frac{25}{24} = -\frac{7}{24} \)Other functions:- \( \operatorname{cosech} x = -\frac{24}{7} \)- \( \operatorname{coth} x = -\frac{25}{7} \)
05

Solving part (d)

Given \( \operatorname{sech} x = \frac{5}{13} \). Find \( \cosh x \) which is the reciprocal:1. \( \cosh x = \frac{13}{5} \).2. Use \( \operatorname{sech}^2 x = 1 - \tanh^2 x \) and rearrange \( \sinh x = \sqrt{\cosh^2 x - 1} \).Calculations:- \( \sinh^2 x = \left( \frac{13}{5} \right)^2 - 1 = \frac{169}{25} - 1 = \frac{144}{25} \)- \( \sinh x = \frac{12}{5} \)- \( \tanh x = \frac{12}{13} \)- \( \operatorname{cosech} x = \frac{5}{12} \)- \( \operatorname{coth} x = \frac{13}{12} \)
06

Solving part (e)

Given \( \operatorname{cosech} x = -\frac{3}{4} \). Find \( \sinh x = -\frac{4}{3} \) because \( \operatorname{cosech} x = \frac{1}{\sinh x} \).1. \( \cosh^2 x = 1 + \sinh^2 x = 1 + \left( -\frac{4}{3} \right)^2 = \frac{25}{9} \)2. \( \cosh x = \frac{5}{3} \)Continue with other calculations:- \( \tanh x = -\frac{4}{5} \)- \( \operatorname{sech} x = \frac{3}{5} \)- \( \operatorname{coth} x = -\frac{3}{4} \)
07

Solving part (f)

Given \( \operatorname{coth} x = \frac{13}{12} \). This means \( \coth x = \frac{\cosh x}{\sinh x} \).1. Use \((\operatorname{coth}x)^2 = \coth^2 x - 1 = \frac{\cosh^2 x - \sinh^2 x}{\sinh^2 x}\) to find \( \sinh x \) first: \( \sinh x = \frac{12}{\sqrt{25}} = \frac{12}{5} \)2. Then, \( \cosh x = \frac{13}{5} \)Calculate other functions:- \( \tanh x = \frac{12}{13} \)- \( \operatorname{sech} x = \frac{5}{13} \)- \( \operatorname{cosech} x = \frac{5}{12} \)

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Key Concepts

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

cosh
Hyperbolic cosine, denoted as \( \cosh(x) \), is one of the foundational hyperbolic functions, which gives us the average of the exponential function raised to a power and its reciprocal. It's defined as: \[\cosh(x) = \frac{e^x + e^{-x}}{2}\]This function is analogous to the cosine function in trigonometry but applies to hyperbolic angles. Just like cosine measures circular angles, hyperbolic cosine measures hyperbolic angles.
  • \( \cosh(x) \) is always greater than or equal to 1 for any real value of \( x \).
  • This function is symmetrical, meaning \( \cosh(-x) = \cosh(x) \).
In solving problems, \( \cosh(x) \) is often paired with \( \sinh(x) \) by the identity \( \cosh^2(x) - \sinh^2(x) = 1 \), which helps in computing values of other hyperbolic functions when one is known.
sinh
The hyperbolic sine function \( \sinh(x) \) represents the difference between the exponential of \( x \) and its reciprocal, divided by 2. It is defined as: \[\sinh(x) = \frac{e^x - e^{-x}}{2}\]This function, analogous to the sine function in trigonometry, implies a relationship with hyperbolic asymptotes rather than circular ones.
  • Unlike \( \cosh(x) \), \( \sinh(x) \) can be both positive and negative depending on the value of \( x \).
  • It is an odd function, so \( \sinh(-x) = -\sinh(x) \).
The relationship \( \cosh^2(x) - \sinh^2(x) = 1 \) is pivotal in finding \( \cosh(x) \) when \( \sinh(x) \) is known, or vice versa. This identity is frequently used to solve equations involving hyperbolic functions.
tanh
Tanh, or hyperbolic tangent, links \( \sinh(x) \) and \( \cosh(x) \) by their ratio: \[\tanh(x) = \frac{\sinh(x)}{\cosh(x)}\]This function behaves similarly to the tangent function in trigonometry. It approaches asymptotes as \( x \) increases or decreases towards infinity.
  • Values of \( \tanh(x) \) range from -1 to 1, making it a useful function for normalization in computation.
  • It is an odd function, satisfying \( \tanh(-x) = -\tanh(x) \).
In practical problems, knowing \( \tanh(x) \) allows us to backtrack and find \( \sinh(x) \) and \( \cosh(x) \) using the formula for \( \operatorname{sech}(x) = \sqrt{1 - \tanh^2(x)} \). This relationship highlights its importance in solving identities and equations based around hyperbolic functions.
sech
The hyperbolic secant, labeled \( \operatorname{sech}(x) \), is the reciprocal of \( \cosh(x) \): \[\operatorname{sech}(x) = \frac{1}{\cosh(x)}\]It bears resemblance to the secant function in circular trigonometry but applies to hyperbolic angles.
  • \( \operatorname{sech}(x) \) only has values between 0 and 1.
  • The function is even, which means \( \operatorname{sech}(-x) = \operatorname{sech}(x) \).
The function is particularly useful in situations where we need to simplify expressions involving \( \cosh(x) \). By simplifying \( \operatorname{sech}^2(x) = 1 - \tanh^2(x) \), it helps find connections with other hyperbolic functions.
cosech
Cosech, the hyperbolic counterpart of cosecant, and written as \( \operatorname{cosech}(x) \), is the reciprocal of \( \sinh(x) \): \[\operatorname{cosech}(x) = \frac{1}{\sinh(x)}\]It is similar to the trigonometric cosecant but set amidst hyperbolic functions.
  • The domain of \( \operatorname{cosech}(x) \) excludes zero because \( \sinh(0) = 0 \), making \( \operatorname{cosech}(x) \) undefined there.
  • \( \operatorname{cosech}(-x) = -\operatorname{cosech}(x) \), indicating it is an odd function.
Its role in calculations involves being paired with \( \sinh(x) \) to establish identity relationships, thereby allowing calculation of associated hyperbolic functions when given a corresponding value.
coth
Hyperbolic cotangent, or \( \operatorname{coth}(x) \), is defined as the ratio of \( \cosh(x) \) to \( \sinh(x) \): \[\operatorname{coth}(x) = \frac{\cosh(x)}{\sinh(x)}\]This is similar to the cotangent function of circular trigonometry but applied in a hyperbolic context.
  • \( \operatorname{coth}(x) \) exists for all \( x eq 0 \) and takes any real value rather than being restricted between \(-1\) and \(1\).
  • It is an odd function, thus \( \operatorname{coth}(-x) = -\operatorname{coth}(x) \).
Knowing \( \operatorname{coth}(x) \) aids in finding \( \tanh(x) \) through complementing identities such as \( \coth^2(x) - \operatorname{cosech}^2(x) = 1 \), serving as an essential tool to derive other hyperbolic functions when needed.

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Assuming that all the numbers given are correctly rounded, calculate the positive root together with its error bound of the quadratic equation $$ 1.4 x^{2}+5.7 x-2.3=0 $$ Give your answer also as a correctly rounded number.

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