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Chapter 4: Problem 58

Show that \(1-x^{2} / 2<\cos x\) for all \(x \in(0, \infty).\)

Short Answer

Expert verified
The inequality \(1 - x^2 / 2 < cos(x)\) holds for all \(x \in (0, \infty)\) by showing that the difference between the two sides, through the Taylor's series for cosine, results in a sum of terms that are all positive.

Step by step solution

01

Initialize

To solve this, it's needed to use Taylor's Series for Cosine which is expressed by \(cos(x) = 1 - x^2/2! + x^4/4! - x^6/6! + ...\). By using that, the proof can be formulated.
02

Manipulate equation via Taylor's Series

Insert the Taylor's Series representation into the inequality to obtain \(1- x^2/2 < 1- x^2/2! + x^4/4! - x^6/6! + ...\). Simplify this to get \(0< x^4/4! - x^6/6! + ...\). The inequality holds as all terms on the right hand side are positive for \(x \in (0, \infty)\).
03

Proof

With the manipulation in the previous step, the proof was completed for all values of \(x \in (0, \infty)\).

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Key Concepts

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

Understanding the Cosine Function
The cosine function is a fundamental concept in trigonometry and a core component of many mathematical computations. It describes the relationship between the angle and the adjacent side of a right triangle. The cosine function is periodic, meaning it repeats its values in regular intervals.
  • Function Definition: The cosine of an angle is defined in the unit circle. It is the x-coordinate of a point where the terminal side of the angle intersects the circle.

  • Key Properties: Cosine is an even function, which means that \(\cos(-x) = \cos(x)\). Another important property is its range: the cosine function values oscillate between -1 and 1.

  • Taylor Series: The Taylor series for cosine, used in this exercise, expresses \(\cos(x) \approx 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \ldots\). This series gives us a polynomial that approximates the cosine function, and provides a powerful tool for calculus and analysis.
Exploring Inequalities
Inequalities are mathematical expressions that relate the relative size or order of two values. They are essential for comparisons and are represented using symbols like <, >, 鈮�, and 鈮�. Understanding how to manipulate and work with inequalities is crucial for solving many mathematical problems.
  • Types of Inequalities: An inequality can be linear, quadratic, or higher-order based on the degree of the expression. Each type has different methods for solving.

  • Simplifying Inequalities: To simplify and solve an inequality, you might need to add, subtract, multiply, or divide both sides by the same value. Keep in mind that multiplying or dividing by negative numbers reverses the inequality sign.

  • Inequalities in Context: In this exercise, the inequality \(1-x^{2}/2<\cos x\) is used to establish the validity through verification via Taylor series, demonstrating how inequalities play a role in defining boundaries in functions.
Introduction to Mathematical Proof
Mathematical proof is a method used to establish the truth of a mathematical statement beyond all doubt. Proofs employ logical reasoning from accepted principles, definitions, and previously proven statements.
  • Types of Proof: There are various proof techniques, such as direct proof, indirect proof (including proof by contradiction), and mathematical induction.

  • Structure of a Proof: A proof typically begins with known information or assumptions, follows a logically deduced path, and concludes with the statement being proven. It needs to be clear, precise, and justified with every step.

  • Application: In the given exercise, the inequality \(1-x^2/2 < \cos x\) for \(x \in (0, \infty)\) was proved using Taylor's series as a form of direct proof, by logically verifying each step, which underpins the entire argument.

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Most popular questions from this chapter

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