/*! 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 62 Graph the following equations. T... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 10: Problem 62

Graph the following equations. Then use arrows and labeled points to indicate how the curve is generated as \(\theta\) increases from 0 to \(2 \pi\). $$r=\frac{1}{1+2 \cos \theta}$$

Short Answer

Expert verified
Based on the solution provided, the short answer can be written as follows: The polar equation \(r = \frac{1}{1+2 \cos \theta}\) creates a curve formed by two loops as \(\theta\) increases from 0 to \(2 \pi\). The first smaller loop is formed when \(\theta\) ranges from 0 to \(\pi\), and the second larger loop, symmetric to the first one, is formed when \(\theta\) ranges from \(\pi\) to \(2 \pi\). The curve is generated in a counterclockwise direction, as indicated by arrows on the graph.

Step by step solution

01

Analyze the Polar Equation

We begin by analyzing the equation \(r = \frac{1}{1+2 \cos \theta}\). The main element here is the cosine function, which varies between -1 and 1 as \(\theta\) increases from 0 to \(2 \pi\). Let's analyze the equation as follows: - For \(\theta = 0\): \(\cos \theta = 1\), so \(r = \frac{1}{1+2 (1)} = \frac{1}{3}\) - For \(\theta = \frac{\pi}{2}\): \(\cos \theta = 0\), so \(r = \frac{1}{1+2 (0)} = 1\) - For \(\theta = \pi\): \(\cos \theta = -1\), so \(r = \frac{1}{1+2(-1)} = -1\) - For \(\theta = \frac{3 \pi}{2}\): \(\cos \theta = 0\), so \(r = \frac{1}{1+2 (0)} = 1\) Since the cosine function repeats itself in the range from 0 to \(2 \pi\), the curve will have a loop shape.
02

Plot the Polar Equation

To plot the polar equation \(r = \frac{1}{1+2 \cos \theta}\), we first create a polar grid. Next, we can choose values for \(\theta\) from 0 to \(2 \pi\) and calculate the corresponding \(r\) values using the polar equation. As a result, we obtain a series of points in polar coordinates. Connect the points one by one, following the order in which they appear as \(\theta\) increases. Observe that for \(\theta\) in the range from 0 to \(\pi\), the curve forms a smaller loop, while for \(\theta\) in the range from \(\pi\) to \(2 \pi\), the curve forms a larger, symmetric loop in relation to the first one.
03

Add Arrows and Labeled Points

Now that we have graphed the curve and identified how it evolves as \(\theta\) increases from 0 to \(2 \pi\), we can add arrows to indicate the direction in which the curve is generated. Following the sequence of points, place an arrowhead at the endpoint of each segment, pointing towards the next point. Label specific points on the curve corresponding to key angles, such as \(\theta = 0, \frac{\pi}{2}, \pi, \frac{3 \pi}{2}\), and \(2 \pi\). For each labeled point, indicate its polar coordinates, \((r, \theta)\), alongside the label. This completes the graphical representation of the polar equation, with arrows indicating how the curve is generated as \(\theta\) increases from 0 to \(2 \pi\).

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!

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

A general hypocycloid is described by the equations $$\begin{aligned}&x=(a-b) \cos t+b \cos \left(\frac{(a-b) t}{b}\right)\\\&y=(a-b) \sin t-b \sin \left(\frac{(a-b) t}{b}\right)\end{aligned}$$ Use a graphing utility to explore the dependence of the curve on the parameters \(a\) and \(b\)

Show that the equation \(r=a \cos \theta+b \sin \theta\) where \(a\) and \(b\) are real numbers, describes a circle. Find the center and radius of the circle.

Points at which the graphs of \(r=f(\theta)\) and \(r=g(\theta)\) intersect must be determined carefully. Solving \(f(\theta)=g(\theta)\) identifies some-but perhaps not all-intersection points. The reason is that the curves may pass through the same point for different values of \(\theta .\) Use analytical methods and a graphing utility to find all the intersection points of the following curves. \(r=1-\sin \theta\) and \(r=1+\cos \theta\)

Show that the polar equation $$r^{2}-2 r r_{0} \cos \left(\theta-\theta_{0}\right)=R^{2}-r_{0}^{2}$$ describes a circle of radius \(R\) whose center has polar coordinates \(\left(r_{0}, \theta_{0}\right)\).

Find an equation of the following curves, assuming the center is at the origin. Sketch a graph labeling the vertices, foci, asymptotes (if they exist), and directrices. Use a graphing utility to check your work. An ellipse with vertices (卤9,0) and eccentricity \(\frac{1}{3}\)
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Pure Maths

Read Explanation

Geometry

Read Explanation

Logic and Functions

Read Explanation

Decision Maths

Read Explanation

Calculus

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.