/*! 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 33 Test for symmetry and then graph... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 33

Test for symmetry and then graph each polar equation. $$r \cos \theta=-3$$

Short Answer

Expert verified
The polar equation \(r \cos \theta=-3\) is symmetric with respect to the pole, but not the polar axis. The graph of this equation represents a vertical line passing through -3 on the x-axis.

Step by step solution

01

Testing for symmetry

For \(r \cos \theta=-3\), go ahead and replace \(-r\) for \(r\) and \(-\theta\) for \(\theta\). The result is \(-r \cdot \cos(-\theta) = -3\). Due to the even symmetry of the cosine function, we get \(-r \cos \theta=-3\). This polar equation changes, so it's not symmetric about the polar axis.
02

Testing for pole symmetry

Replace \(\theta\) with \(-\theta\) in the original equation: \(r \cos(-\theta) = -3\), which simplifies to \(r \cos \theta = -3\). The equation doesn't change, so the graph is symmetric with respect to the pole.
03

Graphing the function

To graph \(r \cos \theta=-3\), recall that its rectangular form is \(x=-3\). This represents a vertical line cutting the x-axis at -3. In polar coordinates, it's an entire line going through pole (-3,0) and extending to the negative and positive direction of the y-axis. Graph the line on the coordinate system.

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

Solve and graph the solution set on a number line: $$ |2 x+3| \leq 13 $$

Explain how to find the quotient of two complex numbers in polar form.

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I'm working with a polar equation that failed the symmetry test with respect to \(\theta=\frac{\pi}{2},\) so my graph will not have this kind of symmetry.

Graph \(r_{1}\) and \(r_{2}\) in the same polar coordinate system. What is the relationship between the two graphs? $$r_{1}=4 \cos 2 \theta, r_{2}=4 \cos 2\left(\theta-\frac{\pi}{4}\right)$$

In Exercises \(69-76,\) find all the complex roots. Write roots in rectangular form. If necessary, round to the nearest tenth. The complex cube roots of 1
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Calculus

Read Explanation

Mechanics Maths

Read Explanation

Pure Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Probability and 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.