/*! 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 54 Give polar coordinates for their... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 10: Problem 54

Give polar coordinates for their centers and identify their radii. $$r=6 \sin \theta$$

Short Answer

Expert verified
Center: (0,3); Radius: 3.

Step by step solution

01

Identify the type of curve

The given equation is in polar form: \( r = 6 \sin \theta \). The form \( r = a \sin \theta \) represents a circle centered on the y-axis in polar coordinates.
02

Determine the circle's center

For the equation \( r = 6 \sin \theta \), the center is located at the point \( (0, a/2) \) in polar coordinates, which translates to \( (0, 3) \) since \( a = 6 \).
03

Calculate the circle's radius

The formula \( r = a \sin \theta \) also indicates that the radius of the circle is half of \( a \). Since \( a = 6 \), the radius is \( 6/2 = 3 \).

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.

Circle in Polar Coordinates
Polar coordinates offer a unique way to represent curves by using a fixed point and an angle from the point. In polar coordinates, equations like \( r = 6 \sin \theta \) describe circles.

This form \( r = a \sin \theta \) is specific for circles that are centered along the y-axis. Unlike Cartesian coordinates, where you have \( x^2 + y^2 = r^2 \), polar coordinates directly use the radius \( r \) and an angle \( \theta \).
  • The angle \( \theta \) is measured from the positive x-axis.
  • The radius \( r \) depends on \( \theta \), changing as the angle changes.
This method simplifies calculations and representations for circular paths and structures.
Radius Calculation
The radius in polar coordinates can sometimes be hidden in the equation, like in \( r = 6 \sin \theta \). To find the radius from this form, recognize that for \( r = a \sin \theta \), \( a \) determines size characteristics of the circle.

Here's how to calculate the radius:
  • Identify the parameter \( a \) from the given equation.
  • The radius of the circle is half the value of \( a \).
For the specific equation \( r = 6 \sin \theta \), the radius turns out to be \( \frac{6}{2} = 3 \). This gives you the exact size of the circle portrayed by the equation.
Equation of Circles
In polar coordinates, circle equations reveal characteristics at a glance. The equation \( r = 6 \sin \theta \) is expressed in the standard form \( r = a \sin \theta \), signifying special cases:

  • The circle is symmetric about the line \( \theta = \frac{\pi}{2} \).
  • The center is expressed as \( (0, \frac{a}{2}) \), polar form reflecting its vertical position.
This uses \( a/2 \) to determine the vertical offset of the circle from the origin.

From the equation, the steps to derive these characteristics become intuitive, reinforcing understanding of profile and placement in a circular definition. This form favors an analysis that aligns naturally with polar symmetry, offering a clear, succinct method to elaborate on circle dynamics.

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

Find the eccentricity of the hyperbola. Then find and graph the hyperbola's foci and directrices. $$8 y^{2}-2 x^{2}=16$$

Graph the lines and conic sections. $$r=3 \sec (\theta-\pi / 3)$$

Graph the lines and conic sections. $$r=1 /(1-\sin \theta)$$

Give the eccentricities of conic sections with one focus at the origin along with the directrix corresponding to that focus. Find a polar equation for each conic section. $$e=1, \quad y=2$$

Replace the polar equations with equivalent Cartesian equations. Then describe or identify the graph. $$r \sin \theta=\ln r+\ln \cos \theta$$
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Discrete Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Logic and Functions

Read Explanation

Probability and Statistics

Read Explanation

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.