/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 49 In the circle in the accompanyin... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 49

In the circle in the accompanying figure, a central angle of measure \(\theta\) radians subtends a chord of length \(c(\theta)\) and a circular arc of length \(s(\theta) .\) Based on your intuition, what would you conjecture is the value of \(\lim _{\theta \rightarrow 0^{+}} c(\theta) / s(\theta) ?\) Verify your conjecture by computing the limit.

Short Answer

Expert verified
The limit is 1, as the chord length becomes nearly equal to the arc length for small \( \theta \).

Step by step solution

01

Understanding the problem

We are given a circle with a central angle \( \theta \) that subtends a chord of length \( c(\theta) \) and a circular arc of length \( s(\theta) \). We need to find the limit \( \lim_{\theta \to 0^+} \frac{c(\theta)}{s(\theta)} \), essentially how the length of the chord relates to the arc length as \( \theta \) approaches zero.
02

Using geometry relationships

As \( \theta \to 0^+ \), the chord \( c(\theta) \) becomes very close in length to the arc \( s(\theta) \) because the curve flattens. Since \( s(\theta) = r\theta \) where \( r \) is the radius, and \( c(\theta) \) is less than or equal to \( s(\theta) \), intuition suggests that \( c(\theta) \approx s(\theta) \). So, the ratio \( \frac{c(\theta)}{s(\theta)} \) approaches 1.
03

Mathematical form of a chord

The chord length \( c(\theta) \) in terms of radius \( r \) is given by the formula: \[ c(\theta) = 2r \sin\left(\frac{\theta}{2}\right) \] for an angle \( \theta > 0 \).
04

Evaluate the arc length

Recall, \( s(\theta) = r\theta \). This represents the linear distance along the arc set by the angle \( \theta \).
05

Substitute into the limit

The expression becomes: \[ \lim_{\theta \to 0^+} \frac{c(\theta)}{s(\theta)} = \lim_{\theta \to 0^+} \frac{2r \sin\left(\frac{\theta}{2}\right)}{r\theta} = \lim_{\theta \to 0^+} \frac{2 \sin\left(\frac{\theta}{2}\right)}{\theta} \].
06

Simplifying the limit expression

Using the substitution \( x = \frac{\theta}{2} \), then \( \theta = 2x \) and \[ \lim_{x \to 0^+} \frac{2 \sin(x)}{2x} = \lim_{x \to 0^+} \frac{\sin(x)}{x} \].
07

Apply the known limit of sin(x)/x

It is a well-known limit that \( \lim_{x \to 0} \frac{\sin(x)}{x} = 1 \). Applying this, we get: \[ \lim_{\theta \to 0^+} \frac{c(\theta)}{s(\theta)} = 1 \].

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Central Angles
The concept of a central angle is fundamental when working with circles in geometry. A central angle is an angle whose vertex is at the center of the circle and whose sides are radii extending to the circumference.
  • In our problem, the angle \( \theta \) is a central angle.
  • Central angles are important because they help us define other segments and arcs within the circle.
By changing the central angle \( \theta \), the lengths of the subtended chord and arc also change. This direct relationship is crucial when studying how the chord and arc behave when the circle's shape approaches a straight line as \( \theta \to 0^+ \). Understanding central angles is key to comprehending how the geometry of circle segments works, especially when calculating limits involving these segments.
Chord Length
The chord length in a circle represents the straight-line distance between two points on the circle's circumference. Unlike the arc, which follows the circle's curvature, the chord cuts straight across.
  • The length of the chord is expressed as \( c(\theta) = 2r \sin\left(\frac{\theta}{2}\right) \).
  • This formula makes use of the circle鈥檚 radius \( r \) and central angle \( \theta \).
A smaller central angle results in a shorter chord, which is why the chord approaches the arc's length when \( \theta \) becomes very small. As \( \theta \) approaches zero, the curve of the arc flattens, leading to the intriguing result that the chord length and arc length tend to converge.
Arc Length
Arc length is the measure of the distance along the portion of the circumference of a circle defined by a central angle. It represents the circle's path between two points.
  • The arc length is given by the formula \( s(\theta) = r\theta \).
  • It is directly proportional to the central angle \( \theta \), and the radius \( r \) remains as a constant.
When expressed in radians, the length of the arc becomes a linear relation, simplifying calculations. In our exercise, as \( \theta \to 0^+ \), the arc length \( s(\theta) = r\theta \) more closely approximates a straight line. This showcases how arc and chord lengths equate as the central angle diminishes, illustrating the elegant nature of trigonometric limits in calculus.
Trigonometric Limits
Trigonometric limits play a pivotal role in understanding calculus, particularly when calculating limits involving angles, like in circular measurements. One of the most important trigonometric limits is \[\lim_{x \to 0} \frac{\sin(x)}{x} = 1\]
  • This limit is crucial for solving problems involving small angles.
  • It simplifies expressions where the sine of an angle is divided by the angle itself as the angle approaches zero.
In the provided solution, this limit was used to evaluate the limit of \( \frac{c(\theta)}{s(\theta)} \) as \( \theta \to 0^+ \). By using the identity \( \theta = 2x \) where \( x = \frac{\theta}{2} \), we simplify the expression to a known limit problem. This demonstrates how understanding trigonometric functions and their limits aids in solving calculus problems efficiently.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Sketch the graphs of the curves \(y=1 / x, y=-1 / x\) and \(y=f(x),\) where \(f\) is a function that satisfies the inequalities $$ -\frac{1}{x} \leq f(x) \leq \frac{1}{x} $$ for all \(x\) in the interval \([1,+\infty) .\) What can you say about the limit of \(f(x)\) as \(x \rightarrow+\infty ?\) Explain your reasoning.

Find values of \(x,\) if any, at which \(f\) is not continuous. $$ f(x)=\frac{x+2}{x^{2}-4} $$

Find the limits. $$ \lim _{h \rightarrow 0} \frac{\sin h}{1-\cos h} $$

Suppose that \(M\) is a positive number and that for all real numbers \(x\), a function \(f\) satisfies $$ -M \leq f(x) \leq M $$ Then $$ \lim _{x \rightarrow 0} x f(x)=0 \quad \text { and } \quad \lim _{x \rightarrow+\infty} \frac{f(x)}{x}=0 $$

Evaluate the limit using an appropriate substitution. $$ \lim _{x \rightarrow+\infty}\left(1+\frac{2}{x}\right)^{x}[\text {Hint: } t=x / 2] $$
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Discrete Mathematics

Read Explanation

Calculus

Read Explanation

Statistics

Read Explanation

Mechanics Maths

Read Explanation

Probability and Statistics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.