/*! 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 22 Convert the Cartesian equation t... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 8: Problem 22

Convert the Cartesian equation to a Polar equation. $$ y=4 $$

Short Answer

Expert verified
Polar form: \(r \sin\theta = 4\).

Step by step solution

01

Understand the Cartesian Equation

The given Cartesian equation is a horizontal line, specifically the line where all points have a y-coordinate equal to 4. So, the equation is simply `y = 4`.
02

Recall Polar Coordinates and Relations

In polar coordinates, every point is represented by \(r, \theta\), where \(r\) is the distance from the origin, and \(\theta\) is the angle from the positive x-axis. The conversion from Cartesian to Polar involves the relations: \(x = r \cos\theta\) and \(y = r \sin\theta\).
03

Substitute Cartesian Conversion Formula

Replace \(y\) in the Cartesian equation with the polar coordinate equivalent \(r \sin\theta\). The equation becomes \(r \sin\theta = 4\).
04

Formulate the Polar Equation

The Polar equation corresponding to the Cartesian line \(y = 4\) is \(r \sin\theta = 4\). This represents all points at distance \(r\) such that their vertical component equals 4.

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!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Cartesian Equation
A Cartesian equation relies on the Cartesian coordinate system, which utilizes two perpendicular axes: the x-axis (horizontal) and y-axis (vertical). In this system, points in the plane are defined with pairs of values
  • (x, y), where 'x' represents the horizontal position,
  • and 'y' represents the vertical position.
The given Cartesian equation, \(y = 4\), signifies a horizontal line where the y-coordinate for every point on the line equals 4. This means, no matter the value of x, y will always remain 4, forming a perfectly straight line parallel to the x-axis.
Polar Equation
Moving into Polar coordinates, a Polar equation defines points based on their distance from the origin and the angle they make with the positive x-axis.In the polar system, every point on the plane is expressed as
  • \( r \), the distance from the origin,
  • \( \theta \), the angle.
The polar equation we derived from the Cartesian equation \(y = 4\) is \(r \sin\theta = 4\). This describes all points whose vertical distance, or height, from the x-axis, when expressed in terms of polar coordinates, equals 4 regardless of its distance \(r\) from the origin.The equation reflects the idea that each point maintains a specific height of 4 from x-axis at any angle \(\theta\), dictating a uniform distance from a horizontal plane.
Coordinate Conversion
Converting between Cartesian and Polar equations demands an understanding of both systems and the connection between them.Here are the foundational steps:
  • The main conversions are given by the formulas: \(x = r \cos\theta\) and \(y = r \sin\theta\).
  • To convert the equation \(y = 4\) in Cartesian, substitute \(y = r \sin\theta\).
Thus, \(y = r \sin\theta = 4\) leads us to a polar equation expressing the same line concept in Polar terms.These transformations highlight the relationship that any point expressed in one coordinate system can equally be depicted in the other, allowing for flexible representations and interpretations based on the context or requirement.

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

An airplane needs to head due north, but there is a wind blowing from the southwest at \(60 \mathrm{~km} / \mathrm{hr}\). The plane flies with an airspeed of \(550 \mathrm{~km} / \mathrm{hr}\). To end up flying due north, the pilot will need to fly the plane how many degrees west of north?

Compute each of the following, simplifying the result into \(a+b i\) form. $$ (4+4 i)^{6} $$

Rewrite each complex number into polar \(r e^{i \theta}\) form. $$ 4+7 i $$

Sketch a graph of the polar equation. $$ r=4 \sin (4 \theta) $$

Eliminate the parameter \(t\) to rewrite the parametric equation as a Cartesian equation. $$ \left\\{\begin{array}{l} x(t)=4 \log (t) \\ y(t)=3+2 t \end{array}\right. $$
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Probability and Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Decision Maths

Read Explanation

Pure Maths

Read Explanation

Logic and Functions

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.