/*! 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 Convert each polar equation to a... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 70

Convert each polar equation to a rectangular equation. Then use a rectangular coordinate system to graph the rectangular equation. $$r=-4 \sin \theta$$

Short Answer

Expert verified
The polar equation \(r=-4 \sin \theta\) converts to \(x^2 + y^2 = 16 - x^2\) in the rectangular coordinate system. The plot of this equation in a rectangular coordinate system forms a circle.

Step by step solution

01

Conversion from Polar to Rectangular Coordinate System

The relationship between polar and rectangular coordinates is given as: \(x = r \cos \theta\) and \(y = r \sin \theta\). Now, remember from trigonometry that \(\sin^2 \theta + \cos^2 \theta = 1\). You can manipulate this equation to be: \(\sin^2 \theta = 1 - \cos^2 \theta\). Hence, substituting that into the equation, we get \(r=-4 \sqrt{1-\cos^2 \theta}\). Now, replace \(r\) and \(\theta\) with \(x\) and \(y\) respectively: \(x^2 + y^2 = 16(1 - x^2/16)\). Simplifying, this gives \(x^2 + y^2 = 16 - x^2\), hence the function in rectangular coordinates is \(x^2 + y^2 = 16 - x^2\).
02

Plotting the Graph

Next, we need to plot the function obtained in step 1 on a graph. Since it's a rectangle coordinate system, we will plot data points for various \(x\) and \(y\) values. The pattern should be a circle, as the equation represents a circle in rectangle 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!

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

Help you prepare for the material covered in the first section of the next chapter. Graph \(x+2 y=2\) and \(x-2 y=6\) in the same rectangular coordinate system. At what point do the graphs intersect?

Find the angle, in degrees, between \(\mathbf{v}\) and \(\mathbf{w}.\) $$\mathbf{v}=3 \cos \frac{5 \pi}{3} \mathbf{i}+3 \sin \frac{5 \pi}{3} \mathbf{j}, \quad \mathbf{w}=2 \cos \pi \mathbf{i}+2 \sin \pi \mathbf{j}$$

Use the dot product to determine whether v and w are orthogonal. $$\mathbf{v}=5 \mathbf{i}, \quad \mathbf{w}=-6 \mathbf{j}$$

Help you prepare for the material covered in the first section of the next chapter. a. Does (4,-1) satisfy \(x+2 y=2 ?\) b. Does (4,-1) satisfy \(x-2 y=6 ?\)

What are parallel vectors?
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Pure Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Discrete Mathematics

Read Explanation

Geometry

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.