/*! 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 70 Describe the graph of the polar ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 70

Describe the graph of the polar equation and find the corresponding rectangular equation. Sketch its graph. $$r=2 \csc \theta$$

Short Answer

Expert verified
The corresponding rectangular equation is \(y^4 + y^2(x^2 - 4) = 4(x^2)\). The graph of the equation is a straight line vertical to the pole at point (0,2) and (-0,2)

Step by step solution

01

Converting from Polar to Rectangular Coordinates

Firstly, we need to convert the polar equation \(r = 2 \csc \theta\) into its rectangular form. Using the trigonometric identity \(\sin \theta = \frac{1}{\csc \theta}\), the equation becomes \(r = 2 \sin \theta ^{-1}\). However, in rectangular coordinates, \(r = \sqrt{x^2 + y^2}\) and \(\sin \theta = \frac{y}{r}\). Substituting these equalities into the equation, we obtain: \( \sqrt{x^2 + y^2} = \frac{2r}{y} = \frac{2\sqrt{x^2 + y^2}}{y}\). After multiplying both sides of the equation by \(y\), the rectangular form of the equation is \(y\sqrt{x^2+y^2} = 2\sqrt{x^2+y^2}\).
02

Simplifying the Rectangular Equation

Simplify the equation obtained above to reduce it to a more familiar form. By clearing the radicals, we get \(y^2(x^2 + y^2) = 4(x^2 + y^2)\). This simplifies further to \(y^2x^2 + y^4 = 4x^2 + 4y^2\). Then, rearrange the equation to move all terms to one side, so it becomes \(y^4 + y^2(x^2 - 4) = 4(x^2)\). The resulting equation is not standard, but useful for the next sketching step.
03

Graphing the Polar Equation in Rectangular Coordinates

Now we sketch the polar equation in its rectangular form. Since the equation obtained is not standard, it would be more convenient if we sketch the polar equation directly and its graph would be a straight line vertical to the pole. So this graph will have all points satisfying the equation \(y = 2\), where \(y\) is the distance from the pole or origin and \(\theta\) is the angle.

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

\(r=1-2 \sin \theta\)

In Exercises 33-48, find a polar equation of the conic with its focus at the pole. $$ \begin{array}{lll} {\text { Conic }} & \text { Eccentricity } & \text { Directrix } \\ \text { Parabola } & e=1 & x=-1 \\ \end{array} $$

The comet Encke has an elliptical orbit with an eccentricity of \(e \approx 0.847\). The length of the major axis of the orbit is approximately \(4.42\) astronomical units. Find a polar equation for the orbit. How close does the comet come to the sun?

Consider the polar equation $$ r=\frac{4}{1-0.4 \cos \theta} $$ (a) Identify the conic without graphing the equation. (b) Without graphing the following polar equations, describe how each differs from the given polar equation. $$ r_{1}=\frac{4}{1+0.4 \cos \theta}, \quad r_{2}=\frac{4}{1-0.4 \sin \theta} $$ (c) Use a graphing utility to verify your results in part (b).

In Exercises 29-32, use a graphing utility to graph the rotated conic. $$ r=\frac{6}{2+\sin (\theta+\pi / 6)} $$
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Probability and Statistics

Read Explanation

Applied Mathematics

Read Explanation

Discrete Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Logic and Functions

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.