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Chapter 0: Problem 75

Refer to the hyperbolic functions. Find all solutions of \(\sinh \left(x^{2}-1\right)=0\)

Short Answer

Expert verified
The solutions to the equation \(\sinh \left(x^{2}-1\right)=0\) are \(x = 1\) and \(x = -1\).

Step by step solution

01

Identifying and setting up the equation

We are given the equation \(\sinh \left(x^{2}-1\right)=0\) and we need to find all potential values of \(x\) that make this equation true.
02

Solving for y

Since \(\sinh y = 0\), we can conclude that \(y = 0\). This means \(x^{2}-1 = 0\).
03

Solving for x

Solving the equation \(x^{2}-1 = 0\) gives \(x^{2} = 1\), and therefore \(x = \pm\sqrt{1}\), which simplifies to \(x = 1\) and \(x = -1\).

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Key Concepts

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

Solving Equations
Solving equations involves finding the values of variables that satisfy given mathematical sentences. The trickiest part can often be recognizing which type of equation you are dealing with. In our exercise, we're dealing with a hyperbolic sine function, specifically \[\sinh(x^2 - 1) = 0\]To solve these types of problems:
  • First, identify the equation and isolate the hyperbolic function on one side.
  • Recognize that hyperbolic equations can often be simplified into algebraic ones.
Once the problem is simplified into something familiar, like a quadratic, you can use standard solution techniques for solving. This transformation process often renders confusing problems into something more manageable.
Sinh Function
The sinh function, short for hyperbolic sine, is a crucial part of hyperbolic functions. Like its trigonometric counterpart, the sinh function has unique properties that make it useful in various branches of mathematics and physics. For any real number, the hyperbolic sine is defined as:\[\sinh(y) = \frac{e^y - e^{-y}}{2}\]This equation indicates that \(\sinh(y)\) changes sign similarly to the regular sine function. In our specific exercise, we find the solution by setting \(\sinh(y) = 0\), which naturally leads to \(y = 0\). Knowing this property helps in swiftly navigating through the problem requirements.
Quadratic Equations
Quadratic equations are polynomial equations of degree 2, typically taking the form:\[ax^2 + bx + c = 0\]In this exercise, we reached a simpler quadratic equation after dealing with the hyperbolic sine: \[x^2 - 1 = 0\]To solve it:
  • Recognize it as a standard-form quadratic equation.
  • Factorize or rearrange the equation to easily find solutions.
  • Here, rewriting gives \((x - 1)(x + 1) = 0\), making solutions obvious.
From this factored form, the solutions to \(x\) are evident: \(x = 1\) and \(x = -1\). This approach highlights the simplicity of converting more complex equations into basic quadratic forms.

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Most popular questions from this chapter

Find a function of the form \(f(x)=a e^{b x}\) with the given function values. $$f(0)=3, f(3)=4$$

Use a graphing calculator or computer to determine the number of solutions of each equation, and numerically estimate the solutions \((x\) is in radians). $$\sin x=x^{2}$$

Adjust the graphing window to identify all vertical asympotes. $$f(x)=\frac{3}{x-1}$$

A fast-food restaurant gives every customer a game ticket. With each ticket, the customer has a 1 -in- 10 chance of winning a free meal. If you go 10 times, estimate your chances of winning at least one free meal. The exact probability is \(1-\left(\frac{9}{10}\right)^{10} .\) Compute this number and compare it to your guess.

Sketch a graph of the function showing all extreme, intercepts and asymptotes. $$f(x)=\frac{3}{x^{2}+4}$$
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