/*! 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 16 In Exercises 11-24, identify the... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 16

In Exercises 11-24, identify the conic and sketch its graph. $$ r=\frac{3}{3+\sin \theta} $$

Short Answer

Expert verified
The conic represented by the given polar equation is a circle. To sketch it, determine coordinates using various angles then plot these points on a polar grid and connect them to form a circle.

Step by step solution

01

Identify the Conic

Identify the form of the conic section from the equation. The equation \(r = \frac{3}{3 + \sin \theta}\) is in the form of a circle in polar coordinates. The denominator is of the form (1 + \sin \theta) or (1 - \sin \theta), or (1 + \cos \theta) or (1 - \cos \theta), which can be written in the form of \((1 卤 \cos \theta)\) or \((1 卤 \sin \theta)\) for circle conic sections in polar coordinates.
02

Sketch the Conic

On a polar coordinate grid, consider multiple angles (choosing various values for \theta) and calculate their corresponding radii as per the equation. Then mark those coordinates on the grid to sketch out the shape. \theta should be taken from 0 to 2\pi to cover all points. Trace the path of the points while increasing the angle, and you'll sketch the required circle.
03

Analyze the Sketch

Look at the graph just plotted, and notice that all the points should lie on the curve of a circle because we have identified the given polar equation as representing a circle. If the plotted points do not from a circle, consider checking calculations to correct any errors that may have occurred during determination of coordinates.

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.

Polar Coordinates
Polar coordinates provide a way of describing the location of a point that is different from the typical Cartesian coordinates. Instead of defining a location by its distance along the x-axis and y-axis, polar coordinates use a radius and an angle. The angle, denoted by \( \theta \), is measured from the positive x-axis, and the radius, denoted by \( r \), is the distance from the origin to the point.
  • The polar coordinate \((r, \theta)\) tells us the position of a point.
  • As \( \theta \) changes, the point sweeps around the origin at a fixed distance \( r \).
  • This is especially useful for circular and spiral patterns.
Understanding polar coordinates is key when dealing with conic sections because many conics can be more simply defined and visualized in this system. For example, the equation \(r = \frac{3}{3 + \sin \theta}\) describes a conic in polar form with varying radius as the angle changes.
Graphing Conics
Graphing conic sections involves plotting their form, whether they're circles, ellipses, parabolas, or hyperbolas. When dealing with polar equations, such as \(r = \frac{3}{3 + \sin \theta}\), you plot these graphs on a polar grid, which looks like a series of concentric circles.
  • Select different values of \( \theta \) and compute \( r \).
  • These \( (r, \theta) \) pairs become points on the polar grid.
  • Connect these points smoothly to visualize the shape.
When \( \theta \) varies from \(0\) to \(2\pi\), each position describes where the point will be. For a circle, this ensures a round, closed path. Polar grids aid in visualizing the full pattern, and you should aim for a smooth curve through the points for accurate sketching.
Identifying Conic Shapes
Identifying conic shapes involves recognizing their forms based on given equations. In polar coordinates, conics may look different from their Cartesian counterparts, but each still follows a recognizable pattern.
  • For circles, the polar equation often appears as \( r = \frac{k}{1 + a\sin \theta} \) or \( r = \frac{k}{1 + a\cos \theta} \).
  • This pattern tells you where the conic is centered and its size.
  • It's crucial to analyze the denominator to identify these traits.
In our example, \(r = \frac{3}{3 + \sin \theta}\), the denominator \(3 + \sin \theta\) and form resemble that of a circle, confirming the conic's circular nature. Knowing these patterns helps in quickly identifying conic sections, which is invaluable for solving and graphing equations 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

In Exercises 7-12, test for symmetry with respect to \(\theta=\pi / 2\), the polar axis, and the pole. \(r=5+4 \cos \theta\)

\(r^{2}=4 \sin \theta\)

In Exercises 25-28, use a graphing utility to graph the polar equation. Identify the graph. $$ r=\frac{5}{-4+2 \cos \theta} $$

\(r^{2}=16 \cos 2 \theta\)

In Exercises 79-82, find the exact value of the trigonometric function given that \(u\) and \(v\) are in Quadrant IV and \(\sin u=-\frac{3}{5}\) and \(\cos v=1 / \sqrt{2}\). $$ \sin (u+v) $$
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Calculus

Read Explanation

Statistics

Read Explanation

Mechanics Maths

Read Explanation

Pure Maths

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.