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Chapter 7: Problem 25

Verify the identify. \(\sinh (-t)=-\sinh t ;\) the hyperbolic sine function is odd.

Short Answer

Expert verified
To verify the given identity, we first recall the definition of the hyperbolic sine function: \(\sinh(t) = \frac{e^t - e^{-t}}{2}\). We then evaluate \(\sinh(-t) = \frac{e^{-t} - e^t}{2}\). Comparing the expressions, we see that \(\sinh(-t) = -\sinh(t)\), which means the hyperbolic sine function is odd and the given identity is verified.

Step by step solution

01

Recall the definition of the hyperbolic sine function

The hyperbolic sine function, \(\sinh(t)\), is defined as follows: \(\sinh(t) = \dfrac{e^t - e^{-t}}{2}\) Where \(t\) is a real number and \(e\) is the base of the natural logarithms, approximately equal to 2.71828.
02

Evaluate \(\sinh(-t)\)

Using the definition of the hyperbolic sine function, let's find the value of \(\sinh(-t)\): \(\sinh(-t) = \dfrac{e^{-t} - e^t}{2}\)
03

Show that \(\sinh(-t) = -\sinh(t)\)

Now, we want to show that \(\sinh(-t)\) is equal to \(-\sinh(t)\). To do this, we will first multiply the definition of \(\sinh(t)\) by -1: \(-\sinh(t) = -\dfrac{e^t - e^{-t}}{2}\) Now, let's rewrite the equation and try to make it equal to the expression for \(\sinh(-t)\), which we found in step 2: \(-\sinh(t) = \dfrac{-e^t + e^{-t}}{2}\) Comparing the expressions from step 2 and step 3, we see that they are the same: \(\sinh(-t) = -\sinh(t)\) This equality holds true for all values of t, which means the hyperbolic sine function is indeed an odd function. Therefore, the given identity has been verified.

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Key Concepts

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

Hyperbolic Sine Function
The hyperbolic sine function, denoted as \(\sinh(t)\), may sound intimidating, but it's quite straightforward once you break it down. This function is part of the hyperbolic functions family, which are analogues of the trigonometric functions for the imaginary unit of complex numbers. The formula for \(\sinh(t)\) is written as:\[\sinh(t) = \dfrac{e^t - e^{-t}}{2}\]where \(t\) is a real number, and \(e\) is the base of the natural logarithms, approximately 2.71828.
This function is essential in many areas of mathematics and physics, representing scenarios that involve hyperbolic geometry, such as the shapes of cables in suspension bridges.
Odd Function
An odd function, in mathematics, has a unique property: its graph is symmetric with respect to the origin. In simpler terms, for any input \(t\), an odd function satisfies the condition:
  • \(f(-t) = -f(t)\)
The hyperbolic sine function \(\sinh(t)\) belongs to this category. Why is that so? If you substitute \(-t\) into the function, you find:
  • \(\sinh(-t) = \dfrac{e^{-t} - e^t}{2}\)
  • This matches \(-\sinh(t)\) because flipping the signs of terms in the numerator effectively gives you \(-\dfrac{e^t - e^{-t}}{2}\).
This verification shows that \(\sinh(t)\) is indeed an odd function, maintaining symmetry across the origin.
Mathematical Proof
In mathematics, proving something requires a logical sequence that connects a truth or statement to a conclusion. Proofs are essential to verify properties and theorems, providing confidence that they hold universally under defined conditions.
The textbook solution establishes \(\sinh(t)\) as an odd function through mathematical proof. Starting with the known equation of the hyperbolic sine, it explores \(\sinh(-t)\) from scratch, constructing:
  • Compute \(\sinh(-t) = \dfrac{e^{-t} - e^t}{2}\)
  • Compare with \(-\sinh(t) = -\dfrac{e^t - e^{-t}}{2}\)
  • Recognize matching terms when rearranged
The proof demonstrates that the expression \(\sinh(-t) = -\sinh(t)\) aligns perfectly, confirming the characteristic of an odd function.
Definition of Hyperbolic Sine
The definition of the hyperbolic sine function \(\sinh(t)\) is integral to understanding its properties. This definition explains how it is constructed using exponential functions:
  • \(\sinh(t) = \dfrac{e^t - e^{-t}}{2}\)
Here, the function employs both \(e^t\) and \(e^{-t}\), showcasing its dependence on the exponential growth and decay captured by these expressions.
  • This broadens the application of \(\sinh(t)\) to various fields such as hyperbolic geometry and theoretical physics.
  • The fascinating interplay of exponential terms gives rise to the unique symmetry and properties found in hyperbolic functions.
Understanding this definition permits deeper insights into how hyperbolic functions behave and their role in both complex analysis and real-world problems.

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Most popular questions from this chapter

Use a graphing utility to draw a figure that displays the graphs of \(f\) and \(g\). The figure should suggest that \(f\) and \(g\) are inverses. Show that this is true by verifying that \(f(g(x))=x\) for each \(x\) in the domain of \(g\). \(f(x)=e^{2 x}, \quad g(x)=\ln \sqrt{x} ; \quad x>0\).

Use a graphing utility to draw the graph of \(f\) Show that \(f\) is one-to-one by consideration of \(f^{\prime}\). Draw a figure that displays both the graph of \(f\) and the graph of \(f^{-1}\). $$f(x)=2-\cos 3 x , \quad 0 \leq x \leq \pi / 3$$

(i) Find the domain of \(f,(\) ii ) find the intervals on which the function increases and the intervals on which it decreases, (iii) find the extreme values, (iv) determine the concavity of the graph and find the points of inflection, and, finally, (v) sketch the graph, indicating asymptotes. $$f(x)=\ln \left[\frac{x^{3}}{x-1}\right]$$

Evaluate. $$\int_{1}^{2} 2^{-x} d x$$

Calculate. $$\int \frac{\sec ^{2} x}{9+\tan ^{2} x} d x$$.
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