Problem 47 - 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: $$ \cosh (t)=\frac{e^{t}+e^{-t}}{2} \text { and } \sinh (t)=\frac{e^{t}-e^{-t}}{2} $$ Using a graphing utility as needed, verify that the domain of \(\cosh (t)\) is \((-\infty, \infty)\) and the range of \(\cosh (t)\) is \([1, \infty)\)

Step-by-Step Solution

Answer

The domain of \(\cosh(t)\) is \((-\infty, \infty)\) and the range is \([1, \infty)\).

Step 1: Understanding the Hyperbolic Function

The hyperbolic cosine function is defined as \( \cosh(t) = \frac{e^t + e^{-t}}{2} \). We need to verify its domain and range.

Step 2: Determine the Domain of \(\cosh(t)\)

Since \( \cosh(t) \) is composed of exponential functions \( e^t \) and \( e^{-t} \), which are defined for all real numbers \( t \), \( \cosh(t) \) is also defined for all real numbers. Therefore, the domain of \( \cosh(t) \) is \(( -\infty, \infty) \).

Step 3: Analyze the Expression of \(\cosh(t)\)

Breaking down the expression, \( \cosh(t) = \frac{e^t + e^{-t}}{2} \), observe that both \(e^t\) and \(e^{-t}\) are positive for all real numbers \(t\). Hence, \( \cosh(t) \) is the average of two positive quantities.

Step 4: Calculate the Minimum Value of \(\cosh(t)\)

To find the minimum value, we note that \( \cosh(t) \geq 1 \). To verify this mathematically, use the fact that \( e^{-t} \) is at its maximum at \( t = 0 \), making \( \cosh(0) = 1 \). Thus, the minimum value of \( \cosh(t) \) is 1.

Step 5: Graphing \(\cosh(t)\) for Verification

Use a graphing utility to plot \( \cosh(t) = \frac{e^t + e^{-t}}{2} \). Observe that it is an even function, symmetric about the \(y\)-axis, and that it only takes values greater than or equal to 1.

Step 6: Conclusion on the Range of \(\cosh(t)\)

From the graph and analysis, confirm that \( \cosh(t) \) takes all values from \([1, \infty)\) since \( \cosh(t) \) never goes below 1 and increases without bounds as \( t \to \pm \infty \).

Key Concepts

Hyperbolic Cosine Function

The hyperbolic cosine function, denoted as \( \cosh(t) \), is an essential concept in mathematics linked to the behavior of hyperbolic curves. It is defined through exponential functions as follows: \[ \cosh(t) = \frac{e^t + e^{-t}}{2} \] Here, \( e^t \) and \( e^{-t} \) represent exponential growth and decay, respectively.

• The function \( \cosh(t) \) averages these two symmetrical exponential functions, resulting in values that depict characteristics of hyperbolic motion.
• This function exhibits certain unique properties, such as its symmetry about the y-axis, making it an even function.
• Understanding \( \cosh(t) \) is fundamental for tackling problems involving hyperbolic geometries and physics applications.

This leads to practical use in calculating distances and modeling systems in hyperbolic contexts, showcasing its importance in both theoretical and applied sciences.

Hyperbolic Sine Function

The hyperbolic sine function, noted as \( \sinh(t) \), complements the hyperbolic cosine function by similarly dealing with hyperbolic calculations. It is defined by: \[ \sinh(t) = \frac{e^t - e^{-t}}{2} \]

• Unlike \( \cosh(t) \), this function is related to the net difference in exponential changes, capturing both the increasing and decreasing dynamics of \( e^t \) and \( e^{-t} \).
• \( \sinh(t) \) is an odd function, evident from its symmetry about the origin. This means that \( \sinh(-t) = -\sinh(t) \).
• The behavior of \( \sinh(t) \) demonstrates how outputs can vary negatively and positively depending on \( t \), making it crucial for modeling oscillations and wave forms in hyperbolic terms.

With distinct applications in hyperbolic structures, understanding \( \sinh(t) \) aids in establishing more comprehensive models for physical and mathematical phenomena.

Domain and Range Analysis

Both the hyperbolic cosine and sine functions possess unique domains and ranges, which determine their applicability across various mathematical and real-world scenarios.

• The domain of \( \cosh(t) \) and \( \sinh(t) \) is all real numbers, \(( -\infty, \infty) \). This is due to their definitions involving the exponential functions \( e^t \) and \( e^{-t} \), which are defined for all real \( t \).
• For \( \cosh(t) \), the range is \((1, \infty)\), as the function never dips below 1. Its minimum at \( t = 0 \) is 1, and it tends toward infinity as \( t \) grows positive or negative.
• On the other hand, \( \sinh(t) \) encompasses all real numbers as its range, indicating its ability to output any real value given a real input \( t \).

Understanding these aspects allows scientists and engineers to predict and exploit the behavior of these functions in technical designs and analyses.

Graphing Utility

Utilizing a graphing utility to visualize the hyperbolic sine and cosine functions enhances your understanding by providing a clear graphical representation.

• Plotting \( \cosh(t) = \frac{e^t + e^{-t}}{2} \) reveals its symmetric, bell-shaped curve that only includes values from 1 upwards.
• The symmetry about the y-axis is evident, confirming its status as an even function whose graph can be mirrored across the vertical axis.
• For \( \sinh(t) = \frac{e^t - e^{-t}}{2} \), the plot will illustrate symmetry about the origin, supporting its characteristic as an odd function.

Using a graphing utility bolsters conceptual understanding by affording a comprehensive view of how these functions behave over different intervals and investigating the influence of various parameters.