/*! 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 65 Convert the polar equation to re... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 10: Problem 65

Convert the polar equation to rectangular form. $$r=4 \sin \theta$$

Short Answer

Expert verified
The rectangular form of the polar equation \(r = 4\sin(\theta)\) is \(yx^2 = 4x^2 - 4(x^2 + y^2)\), after substituting and simplifying.

Step by step solution

01

Review the conversion relation between coordinate systems

The connection between polar and rectangular coordinates is given by \(x = r\cos(\theta)\) and \(y = r\sin(\theta)\).
02

Substitute the given polar equation into rectangular form

The given polar equation is \(r = 4\sin(\theta)\). We can substitute \(r\) into the expression for \(y\) as follows: \(y = r\sin(\theta) = 4\sin^2(\theta)\).
03

Reduce the equation

To obtain the most common representation of this equation, use the Pythagorean identity \(1-\cos^2(\theta)=\sin^2(\theta)\). This will give us: \(y = 4(1-\cos^2(\theta))\). Next, substitute \(x = r\cos(\theta)\) into our equation to replace \(\cos(\theta)\). Solving for r, we find that \(r = \frac{x}{\cos(\theta)}\). Substituting this gives: \(y = 4(1-(r/x)^2)\). This needs to be further reduced in order to give x and y in terms of r.
04

Solve the final equation in terms of x and y

To get y in terms of x, multiply both sides by x^2: \(yx^2 = 4x^2 - 4r^2\). Our equation: \(r^2 = x^2 + y^2\), will replace \(r^2\), so finally we get \(yx^2 = 4x^2 - 4(x^2 + y^2)\).

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

Check for symmetry with respect to both axes and to the origin. Then determine whether the function is even, odd, or neither. $$(x-2)^{2}=y+4$$

Find the exact value of the trigonometric expression when \(u\) and \(v\) are in Quadrant IV and \(\sin u=-\frac{3}{5}\) and \(\cos v=1 / \sqrt{2}\). $$\cos (u-v)$$

Check for symmetry with respect to both axes and to the origin. Then determine whether the function is even, odd, or neither. $$f(x)=\frac{4 x^{2}}{x^{2}+1}$$

Convert the rectangular equation to polar form. Assume \(a<0\) $$y=-\sqrt{3} x$$

Convert the rectangular equation to polar form. Assume \(a<0\) $$x^{2}+y^{2}-2 a y=0$$
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Decision Maths

Read Explanation

Statistics

Read Explanation

Applied Mathematics

Read Explanation

Geometry

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.