/*! 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 59 Describe the graph given by the ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 8: Problem 59

Describe the graph given by the parametric equations \(x=n t\) and \(y=\sin t\) for positive integer \(n\).

Short Answer

Expert verified
The graph is a sinusoidal wave given by \(y = \sin\left(\frac{x}{n}\right)\), with period \(2\pi n\) and amplitude 1.

Step by step solution

01

Identify the Parametric Equations

We begin by identifying the given parametric equations. Here, we have \(x = n t\) and \(y = \sin t\). These equations indicate that \(x\) is directly proportional to \(t\) by a factor of \(n\), while \(y\) oscillates between -1 and 1 as \(t\) varies.
02

Understand the Relationship Between x and y

In order to describe the graph, we need to find a relationship between the variables \(x\) and \(y\). We know that \(t\) can be expressed as \(t = \frac{x}{n}\) from the equation for \(x\). Substituting this into the equation for \(y\), we get \(y = \sin\left(\frac{x}{n}\right)\).
03

Describe the Shape of the Graph

The equation \(y = \sin\left(\frac{x}{n}\right)\) forms a sinusoidal wave. The parameter \(n\) affects the frequency of the wave. Larger values of \(n\) will result in more oscillations over the same interval of \(x\). As \(n\) increases, the waves appear more compressed horizontally.
04

Note the Range of the Function

Regardless of the value of \(n\), the range of \(y = \sin\left(\frac{x}{n}\right)\) remains between -1 and 1, since the sine function only produces values within this range. The graph will oscillate vertically between these values as \(x\) runs over the real line. Thus, amplitude remains constant.
05

Identify the Periodicity

For \(y = \sin t\), the period is \(2\pi\). Therefore, for \(y = \sin\left(\frac{x}{n}\right)\), the period becomes \(2\pi n\). This means that the graph will complete one full cycle and start repeating every \(2\pi n\) units along the \(x\)-axis.

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.

Sinusoidal Wave
A sinusoidal wave is a smooth and repetitive oscillation that can be modeled using the sine function. In the context of the given parametric equations, the graph described by the relationship \(y = \sin\left(\frac{x}{n}\right)\) is sinusoidal. This wave-like shape is characterized by its peaks and troughs, repeating in a regular, predictable pattern.
Sinusoidal waves are common in nature, representing phenomena such as sound waves, light waves, and even tides. They are important in various scientific and engineering fields. Here, understanding the parametric graph involves recognizing the sine wave's nature, which oscillates between fixed bounds, providing a clear visual of oscillation over a linear path.
Frequency
The frequency of a wave refers to how often it oscillates or cycles over a given interval. For a sinusoidal wave represented by \(y = \sin\left(\frac{x}{n}\right)\), the frequency depends inversely on the parameter \(n\).
  • As \(n\) increases, the frequency of oscillation in relation to \(x\) becomes higher.
  • This means there will be more cycles over the same length when \(n\) is larger.
A higher frequency corresponds to more oscillations, which can be visualized as a more compressed wave along the horizontal \(x\)-axis. This concept of frequency is pivotal in understanding how the parameter \(n\) affects the wave's appearance when graphed.
Amplitude
The amplitude of a wave is the height of its peaks from its central axis. In the equation \(y = \sin\left(\frac{x}{n}\right)\), the sine function ensures that the amplitude remains constant at 1. This means that regardless of the value of \(n\) or the frequency of the wave, the wave will always oscillate between -1 and 1.
  • Amplitude is a measure of how far the wave moves from its central value, which in this case is 0.
  • Even as the frequency changes with different \(n\), the amplitude remains unchanged.
This constant amplitude is critical as it ensures that the wave's output values are consistent, allowing for predictable oscillation patterns as a key feature of sinusoidal waves.
Periodicity
Periodicity refers to the repeating nature of waves over specific intervals. For the sine function, the basic period is \(2\pi\). However, in the transformed function \(y = \sin\left(\frac{x}{n}\right)\), the periodicity changes.
  • The period of this sine wave becomes \(2\pi n\).
  • This period determines the distance over the \(x\)-axis that the wave takes to complete one full cycle.
As \(n\) increases, the period \(2\pi n\) becomes longer, indicating that it takes more units along the \(x\)-axis for the wave to repeat its pattern. Understanding periodicity is crucial in predicting and analyzing wave behavior, particularly in applications requiring precise timing, such as signal processing.

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

In Exercises 79 and 80, explain the mistake that is made. Convert the rectangular coordinate \((-\sqrt{3}, 1)\) to polar coordinates. Solution: Label \(x\) and \(y . \quad x=-\sqrt{3}, y=1\) Find \(r . \quad r=\sqrt{x^{2}+y^{2}}=\sqrt{3+1}=\sqrt{4}=2\) Find \(\theta\). $$ \begin{aligned} \tan \theta &=\frac{1}{-\sqrt{3}}=-\frac{1}{\sqrt{3}} \\ \theta &=\tan ^{-1}\left(-\frac{1}{\sqrt{3}}\right)=-\frac{\pi}{4} \end{aligned} $$ Write the point in polar \(\quad\left(2,-\frac{\pi}{4}\right)\) coordinates. This is incorrect. What mistake was made?

In Exercises 1-20, graph the curve defined by the following sets of parametric equations. Be sure to indicate the direction of movement along the curve. $$ x=3 \sin t, y=2 \cos t, t \text { in }[0,2 \pi] $$

In Exercises 1-20, graph the curve defined by the following sets of parametric equations. Be sure to indicate the direction of movement along the curve. $$ x=\sqrt{t}, y=t, t \text { in }[0,10] $$

For Exercises 77 and 78, refer to the following: The sword artistry of the samurai is legendary in Japanese folklore and myth. The elegance with which a samurai could wield a sword rivals the grace exhibited by modern figure skaters. In more modern times, such legends have been rendered digitally in many different video games (e.g., Ominusha). In order to make the characters realistically move across the screen-and in particular, wield various sword motions true to the legends-trigonometric functions are extensively used in constructing the underlying graphics module. One famous movement is a figure eight, swept out with two hands on the sword. The actual path of the tip of the blade as the movement progresses in this figure-eight motion depends essentially on the length \(L\) of the sword and the speed with which the figure is swept out. Such a path is modeled using a polar equation of the form \(r^{2}(\theta)=L \cos (A \theta)\) or \(r^{2}(\theta)=L \sin (A \theta), \theta_{1} \leq \theta \leq \theta_{2}\). Video Games. Write a polar equation that would describe the motion of a sword 12 units long that makes 8 complete motions in \([0,2 \pi]\).

For Exercises 77 and 78, refer to the following: The sword artistry of the samurai is legendary in Japanese folklore and myth. The elegance with which a samurai could wield a sword rivals the grace exhibited by modern figure skaters. In more modern times, such legends have been rendered digitally in many different video games (e.g., Ominusha). In order to make the characters realistically move across the screen-and in particular, wield various sword motions true to the legends-trigonometric functions are extensively used in constructing the underlying graphics module. One famous movement is a figure eight, swept out with two hands on the sword. The actual path of the tip of the blade as the movement progresses in this figure-eight motion depends essentially on the length \(L\) of the sword and the speed with which the figure is swept out. Such a path is modeled using a polar equation of the form \(r^{2}(\theta)=L \cos (A \theta)\) or \(r^{2}(\theta)=L \sin (A \theta), \theta_{1} \leq \theta \leq \theta_{2}\). Video Games. Graph the following equations: a. \(r^{2}(\theta)=5 \cos \theta, 0 \leq \theta \leq 2 \pi\) b. \(r^{2}(\theta)=5 \cos (2 \theta), 0 \leq \theta \leq \pi\) c. \(r^{2}(\theta)=5 \cos (4 \theta), 0 \leq \theta \leq \frac{\pi}{2}\) What do you notice about all of these graphs? Suppose that the movement of the tip of the sword in a game is governed by these graphs. Describe what happens if you change the domain in (b) and (c) to \(0 \leq \theta \leq 2 \pi\).
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Applied Mathematics

Read Explanation

Statistics

Read Explanation

Decision Maths

Read Explanation

Calculus

Read Explanation

Discrete Mathematics

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.