/*! 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 75 Sketch a graph of the polar equa... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 8: Problem 75

Sketch a graph of the polar equation. $$ r^{2}=4 \cos 2 \theta $$

Short Answer

Expert verified
The polar equation \(r^{2}=4 \cos 2 \theta\) represents a lima莽on oriented on the x-axis. The graph is a circle centred at (2, 0) with radius 2.

Step by step solution

01

Understand the Standard Form of Polar Equation

A polar equation of the form \(r^{2}=a^{2} \cos(2 \theta)\) represents a lima莽on. The length \(a^{2}\) determines the size of the lima莽on, and the form \(\cos\) or \(\sin\) determines the orientation. In this case, the polar equation \(r^{2}=4 \cos 2 \theta\) is a lima莽on oriented on the x-axis.
02

Transform into Cartesian Coordinates

To create a graph more easily, convert the polar equation into Cartesian coordinates. In Cartesian coordinates, \(x = r \cos \theta\) and \(y = r \sin \theta\). Given the polar equation \(r^{2}=4 \cos 2 \theta\), it can be rewritten as \(x^{2}+ y^{2} = 4 x\).
03

Sketch the Graph

The graph of the equation \(x^{2}+ y^{2} = 4 x\) is a circle centred at (2, 0) with radius 2. This circle is the lima莽on described by the polar equation \(r^{2}=4 \cos 2 \theta\). Thus, the graph is a circle centred at (2, 0) with radius 2.

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

Graphical Reasoning In Exercises 1-4, use a graphing utility to graph the polar equation when (a) \(e=1,\) (b) \(e=0.5\) and \((\mathrm{c}) e=1.5 .\) Identify the conic. \(r=\frac{2 e}{1+e \sin \theta}\)

Show that the polar equation for \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\) is \(r^{2}=\frac{b^{2}}{1-e^{2} \cos ^{2} \theta} \cdot \quad\) Ellipse

Give the integral formulas for the area of the surface of revolution formed when the graph of \(r=f(\theta)\) is revolved about (a) the \(x\) -axis and (b) the \(y\) -axis.

$$ \text { State the definition of a smooth curve } $$

In Exercises \(27-38,\) find a polar equation for the conic with its focus at the pole. (For convenience, the equation for the directrix is given in rectangular form.) \(\frac{\text { Conic }}{\text { Parabola }} \quad \frac{\text { Eccentricity }}{e=1} \quad \frac{\text { Directrix }}{x=-1}\)
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Calculus

Read Explanation

Probability and Statistics

Read Explanation

Discrete Mathematics

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.