/*! 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 59 Convert to a rectangular equatio... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 8: Problem 59

Convert to a rectangular equation. $$r=5$$

Short Answer

Expert verified
The rectangular equation is \( x^2 + y^2 = 25 \).

Step by step solution

01

Identify the Polar Equation

The given polar equation is: \[ r = 5 \] This specifies that the radius is always 5 units.
02

Recall the Relationship Between Polar and Rectangular Coordinates

Recall the relationships between polar and rectangular coordinates: \[ x = r \, \cos(\theta) \] \[ y = r \, \sin(\theta) \] \[ r^2 = x^2 + y^2 \]
03

Substitute the Given Polar Equation into the Rectangular Coordinate Formula

Substitute \( r = 5 \) into the equation \( r^2 = x^2 + y^2 \): \[ (5)^2 = x^2 + y^2 \] Simplify to get: \[ 25 = x^2 + y^2 \]
04

Write the Final Rectangular Equation

The rectangular equation corresponding to the given polar equation is: \[ x^2 + y^2 = 25 \]

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.

Polar Coordinates
Polar coordinates offer a different way to describe the position of a point in the plane. Instead of using rectangular coordinates \(x, y\), we use a radius \(r\) and an angle \(\theta\). This is particularly useful for modeling circular and spiral shapes.

In this system:
  • The radius \(r\) measures the distance from the origin to the point.
  • The angle \(\theta\) represents the direction from the origin.
For example, in the given exercise, the equation \(r = 5\) tells us that the distance from the origin to the point is always 5 units, regardless of the angle \(\theta\). This describes a circle centered at the origin with a radius of 5.
Coordinate Transformation
To transform from polar coordinates to rectangular coordinates, we can use specific transformation formulas. These help convert the information from one coordinate system into another, making the equations easier to work with in different scenarios.

Key transformations include:
  • \(x = r \, \cos(\theta)\)
  • \(y = r \, \sin(\theta)\)
  • \(r^2 = x^2 + y^2\)
In our problem, we need to convert \(r = 5\) into a rectangular equation. We use the relationship \(r^2 = x^2 + y^2\). Substituting \(r = 5\) gives us \(25 = x^2 + y^2\), which is the equation of a circle in rectangular coordinates.
Equation Conversion
Equation conversion involves changing the form of an equation to make it fit into a different coordinate system. This can simplify solving problems or provide more insight into their geometry. In this exercise, we were given a polar equation \(r = 5\) and needed to convert it into a rectangular equation.

Steps to convert:
  • Start with the polar equation: \(r = 5\)
  • Use the transformation: \(r^2 = x^2 + y^2\)
  • Substitute \(r = 5\): \( (5)^2 = x^2 + y^2 \)
  • Simplify: \(25 = x^2 + y^2\)
The result is \(x^2 + y^2 = 25\), which is a circle centered at the origin with a radius of 5 in rectangular coordinates. This makes it easier to understand and visualize the problem!

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

Convert to radian measure. $$-225^{\circ}$$

Determine the direction angle \(\theta\) of the vector, to the nearest degree. $$\mathbf{t}=\langle 5,6\rangle$$

Raise the number to the given power and write standard notation for the answer. $$\left[2\left(\cos 10^{\circ}+i \sin 10^{\circ}\right)\right]^{9}$$

Graph the equation by plotting points. Then check your work using a graphing calculator. $$r=1-\cos \theta$$

Prove the following area formulas for a general triangle \(A B C\) with area represented by \(K\) $$\begin{array}{l}K=\frac{a^{2} \sin B \sin C}{2 \sin A}, \quad K=\frac{c^{2} \sin A \sin B}{2 \sin C} \\\K=\frac{b^{2} \sin C \sin A}{2 \sin B}\end{array}$$
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Logic and Functions

Read Explanation

Geometry

Read Explanation

Applied Mathematics

Read Explanation

Calculus

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.