/*! 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 50 Explain why finding points of in... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 9: Problem 50

Explain why finding points of intersection of polar graphs may require further analysis beyond solving two equations simultaneously.

Short Answer

Expert verified
Finding points of intersection of polar graphs requires more than solving two equations simultaneously due to the presence of multiple representations of the same point.

Step by step solution

01

Understanding Polar Coordinates and Their Complexity

In polar coordinates, a point is represented by two parameters - radius (\( r \)) and angle (\( \theta \)). It's important to understand that, unlike Cartesian coordinates, a single point in polar coordinates is often represented by multiple pairs of (\( r, \theta \)).
02

Intersecting Points in Polar Coordinates

When two polar graphs intersect, we initially set the two equations equal to each other to find possible intersections. This gives us the initial set of candidates for points of intersection. However, the complexity of multi-representation in polar form means that these initially spotted intersection points may not be the only ones.
03

Further Investigation of Potential Intersection Points

We perform further analysis by checking all possible representations of these points as polar coordinates. This usually requires testing different values of \( \theta \), as each value of \( \theta \) typically corresponds to more than one value of \( r \). This process verifies if a point is actually a point of intersection or not.
04

Example

For instance, consider the polar graphs \( r = \cos(\theta) \) and \( r = \sin(2\theta) \). Upon solving these equations together, we get \( \cos(\theta) = \sin(2\theta) \) and can immediately spot \(\theta = 0\) as one of the solutions. This nevertheless isn't the only solution, as \( \theta = \frac{\pi}{4}, \frac{5\pi}{4} \) are also solutions. This shows the need to conduct further analysis beyond solving two equations simultaneously.

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

In Exercises 49 and \(50,\) find each scalar multiple of \(v\) and sketch its graph. \(\mathbf{v}=\langle 1,2,2\rangle\) (a) \(2 \mathbf{v}\) (b) \(-\mathbf{v}\) (c) \(\frac{3}{2} \mathbf{v}\) (d) \(0 \mathbf{v}\)

Find the vector \(z,\) given that \(\mathbf{u}=\langle 1,2,3\rangle\) \(\mathbf{v}=\langle 2,2,-1\rangle,\) and \(\mathbf{w}=\langle 4,0,-4\rangle\) \(\mathbf{z}=5 \mathbf{u}-3 \mathbf{v}-\frac{1}{2} \mathbf{w}\)

Determine whether \(\mathbf{u}\) and \(\mathbf{v}\) are orthogonal parallel, or neither. $$ \begin{array}{l} \mathbf{u}=-2 \mathbf{i}+3 \mathbf{j}-\mathbf{k} \\ \mathbf{v}=2 \mathbf{i}+\mathbf{j}-\mathbf{k} \end{array} $$

Find the magnitude of \(v\). Initial point of \(\mathbf{v}:(0,-1,0)\) Terminal point of \(\mathbf{v}:(1,2,-2)\)

Find the direction cosines of \(u\) and demonstrate that the sum of the squares of the direction cosines is 1. $$ \mathbf{u}=5 \mathbf{i}+3 \mathbf{j}-\mathbf{k} $$
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Geometry

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Pure Maths

Read Explanation

Logic and Functions

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.