Logout Open In App
Open In App
Warning: foreach() argument must be of type array|object, bool given in /var/www/html/web/app/themes/studypress-core-theme/template-parts/header/mobile-offcanvas.php on line 20

Chapter 8: Problem 34

Convert the polar equation to rectangular form and sketch its graph. $$ r=2 \csc \theta $$

Short Answer

Expert verified
The rectangular form of the equation \(r = 2csc(\theta)\) is \(y = 2\). The graph is a horizontal line passing through (0, 2) and extending indefinitely in both the positive and negative x-directions.

Step by step solution

01

Convert the Polar to Rectangular Form

\(\) First, we need to convert the polar equation to rectangular form. Given \(r = 2csc(\theta)\) and we know that \(csc(\theta) = 1/sin(\theta)\), our equation becomes \(r = 2/sin(\theta)\).Now, multiply each side by \(sin(\theta)\) to clear the fraction:\(r* sin(\theta) = 2 \)We know that \(y=r*sin(\theta)\), thus the rectangular equation is \(y = 2 \).
02

Graph the Equation

Next, we graph the equation \(y = 2\), which is a horizontal line passing through the point (0, 2) on the y-axis. It extends indefinitely in the positive and negative x-directions.
03

Analyze the Graph

After graphing the equation, we see that it corresponds to an infinite line along y=2 in the rectangular coordinate system. The polar coordinates (r, \(\theta\)) of any point on this line will satisfy the equation \(r = 2csc(\theta)\), showing that our conversion is correct.

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.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

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 \(7-16,\) find the eccentricity and the distance from the pole to the directrix of the conic. Then sketch and identify the graph. Use a graphing utility to confirm your results. \(r=\frac{4}{1+2 \cos \theta}\)

Sketch the curve represented by the parametric equations (indicate the orientation of the curve), and write the corresponding rectangular equation by eliminating the parameter. $$ x=\sqrt[4]{t}, \quad y=3-t $$

Determine any differences between the curves of the parametric equations. Are the graphs the same? Are the orientations the same? Are the curves smooth? $$ \text { (a) } \begin{aligned} x &=2 \cos \theta \\ y &=2 \sin \theta \end{aligned} $$ $$ \begin{aligned} &\text { (b) } x=\sqrt{4 t^{2}-1} /|t|\\\ &y=1 / t \end{aligned} $$ $$ \text { (c) } \begin{aligned} x &=\sqrt{t} \\ y &=\sqrt{4-t} \end{aligned} $$ $$ \text { (d) } \begin{aligned} x &=-\sqrt{4-e^{2 t}} \\ y &=e^{t} \end{aligned} $$

Conjecture (a) Use a graphing utility to graph the curves represented by the two sets of parametric equations. \(x=4 \cos t \quad x=4 \cos (-t)\) \(y=3 \sin t \quad y=3 \sin (-t)\) (b) Describe the change in the graph when the sign of the parameter is changed. (c) Make a conjecture about the change in the graph of parametric equations when the sign of the parameter is changed. (d) Test your conjecture with another set of parametric equations.

In Exercises \(17-20,\) use a graphing utility to graph the polar equation. Identify the graph. \(r=\frac{-3}{2+4 \sin \theta}\)
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Mechanics Maths

Read Explanation

Probability and Statistics

Read Explanation

Decision Maths

Read Explanation

Statistics

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.

Sign-up for free