Problem 51 - College Trigonometry

Question

In Exercises 47 - 52 , we explore the hyperbolic cosine function, denoted \(\cosh (t)\), and the hyperbolic sine function, denoted \(\sinh (t)\), defined below: $$ \cosh (t)=\frac{e^{t}+e^{-t}}{2} \text { and } \sinh (t)=\frac{e^{t}-e^{-t}}{2} $$ Four other hyperbolic functions are waiting to be defined: the hyperbolic secant \(\operatorname{sech}(t)\) the hyperbolic cosecant \(\operatorname{csch}(t)\), the hyperbolic tangent \(\tanh (t)\) and the hyperbolic cotangent \(\operatorname{coth}(t) .\) Define these functions in terms of \(\cosh (t)\) and \(\sinh (t)\), then convert them to formulas involving \(e^{t}\) and \(e^{-t}\). Consult a suitable reference (a Calculus book, or this entry on the hyperbolic functions) and spend some time reliving the thrills of trigonometry with these 'hyperbolic' functions.

Step-by-Step Solution

Answer

Hyperbolic functions: \(\operatorname{sech}(t)\), \(\operatorname{csch}(t)\), \(\tanh(t)\), \(\operatorname{coth}(t)\) are reciprocal or ratios of \(\cosh(t)\) and \(\sinh(t)\).

Step 1: Define Hyperbolic Secant Function

The hyperbolic secant function, \( \operatorname{sech}(t) \), is defined as the reciprocal of the hyperbolic cosine function. Therefore, \[ \operatorname{sech}(t) = \frac{1}{\cosh(t)}. \] Using the definition of \( \cosh(t) \), this becomes \[ \operatorname{sech}(t) = \frac{2}{e^t + e^{-t}}. \]

Step 2: Define Hyperbolic Cosecant Function

The hyperbolic cosecant function, \( \operatorname{csch}(t) \), is defined as the reciprocal of the hyperbolic sine function. Therefore, \[ \operatorname{csch}(t) = \frac{1}{\sinh(t)}. \] Using the definition of \( \sinh(t) \), this becomes \[ \operatorname{csch}(t) = \frac{2}{e^t - e^{-t}}. \]

Step 3: Define Hyperbolic Tangent Function

The hyperbolic tangent function, \( \tanh(t) \), is defined as the ratio of the hyperbolic sine to the hyperbolic cosine. Therefore, \[ \tanh(t) = \frac{\sinh(t)}{\cosh(t)}. \] Substituting the definitions of \( \sinh(t) \) and \( \cosh(t) \), we get \[ \tanh(t) = \frac{\frac{e^t - e^{-t}}{2}}{\frac{e^t + e^{-t}}{2}} = \frac{e^t - e^{-t}}{e^t + e^{-t}}. \]

Step 4: Define Hyperbolic Cotangent Function

The hyperbolic cotangent function, \( \operatorname{coth}(t) \), is defined as the reciprocal of the hyperbolic tangent. Therefore, \[ \operatorname{coth}(t) = \frac{\cosh(t)}{\sinh(t)}. \] Substituting the definitions of \( \cosh(t) \) and \( \sinh(t) \), we get \[ \operatorname{coth}(t) = \frac{\frac{e^t + e^{-t}}{2}}{\frac{e^t - e^{-t}}{2}} = \frac{e^t + e^{-t}}{e^t - e^{-t}}. \]

Key Concepts

Understanding Hyperbolic Cosine

The hyperbolic cosine function, denoted as \( \cosh(t) \), is one of the fundamental hyperbolic functions used in calculus and trigonometry. It is defined by the formula:

• \( \cosh (t) = \frac{e^{t} + e^{-t}}{2} \)

This function is similar to the cosine function in trigonometry but operates on hyperbolic angles.
The hyperbolic cosine is an even function, meaning \( \cosh(-t) = \cosh(t) \) for any real number \( t \).
This property makes it symmetric about the y-axis.
It's worth noting that unlike the circular cosine, which oscillates, the hyperbolic cosine produces a kind of 'U' shape when graphed, with a minimum value of 1 at \( t = 0 \).
The curve increases exponentially as \( t \) goes to infinity in either direction.

Understanding Hyperbolic Sine

The hyperbolic sine function, denoted as \( \sinh(t) \), complements the hyperbolic cosine in the family of hyperbolic functions. It is expressed as:

• \( \sinh (t) = \frac{e^{t} - e^{-t}}{2} \)

Unlike the hyperbolic cosine, \( \sinh(t) \) is an odd function. This means that \( \sinh(-t) = - \sinh(t) \).
This crucial property gives it symmetry about the origin, with the graph of \( \sinh(t) \) resembling a stretched-out version of the sine wave.
Hyperbolic sine takes values from negative to positive infinity as \( t \) ranges over all real numbers.
This growth pattern can be vital when analyzing certain real-life phenomena, such as the shape of hanging cables or the curvature of space-time in general relativity.

Hyperbolic Functions in Calculus

In calculus, hyperbolic functions play a significant role, much like their trigonometric counterparts.
They are employed when solving differential equations, particularly in scenarios involving exponential growth or decay.
Derivative formulas for hyperbolic functions resemble those of trigonometric ones:

• \( \frac{d}{dt}[\cosh(t)] = \sinh(t) \)
• \( \frac{d}{dt}[\sinh(t)] = \cosh(t) \)

These simple derivative relationships make hyperbolic functions very convenient in various calculus applications.
They often simplify the integration of products and quotients that include exponential terms.
Moreover, the hyperbolic identities, like \( \cosh^2(t) - \sinh^2(t) = 1 \), parallel the identity \( \cos^2(\theta) + \sin^2(\theta) = 1 \) in trigonometry.
Such similarities between trigonometric and hyperbolic functions make them invaluable for transition in complex variable calculus.

Hyperbolic Functions Meet Trigonometry

Trigonometry and hyperbolic functions may come from different areas of mathematics, but they share many similar features.
The primary difference lies in the argument type: trigonometric functions handle circular angles, while hyperbolic functions relate to hyperbolic angles.
However, as we've seen with identities, both systems share similar properties, making them easy to remember and apply.
The relationships between these functions form the basis for various transformations and approximations used across several branches of mathematics and physics.

• Like their trigonometric counterparts, hyperbolic functions can be expressed in terms of exponential functions.
• Additionally, the hyperbolic functions interact with each other through identities that mirror those seen in trigonometry.

For students, understanding these properties and relationships enriches their comprehension and enhances their problem-solving capabilities in advanced mathematics.