/*! 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 36 Convert the following equations ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 11: Problem 36

Convert the following equations to Cartesian coordinates. Describe the resulting curve. $$r=\frac{1}{2 \cos \theta+3 \sin \theta}$$

Short Answer

Expert verified
Question: Convert the polar equation \(r = \frac{1}{2\cos\theta + 3\sin\theta}\) to a Cartesian equation and describe the resulting curve. Answer: The Cartesian equation for the given polar equation is \(y^2(x^2 + y^2) = 4x^2 + 12xy + 9y^2\). The curve represented by this equation has a complex interaction between the x and y variables, as it contains terms involving \(x^2y^2\), \(x^2\), and \(xy\). We cannot directly identify it as a specific type of curve like a circle, ellipse, or parabola.

Step by step solution

01

Find an expression for "r" in the given polar equation

Given the polar equation, $$r=\frac{1}{2 \cos \theta+3 \sin \theta}$$. We will use this equation as is for further conversion.
02

Replace r, x, and y using conversion formulas

We use the conversion formulas: \(x = r\cos\theta\) and \(y = r\sin\theta\). We can rewrite them as \(r = \frac{x}{\cos\theta}\) and \(\frac{y}{\sin\theta} = r\). Now, substitute the Cartesian expressions for the polar equation: $$\frac{y}{\sin \theta}=\frac{1}{2 \cos \theta+3 \sin \theta}$$.
03

Eliminate 胃

To eliminate 胃, we will use trigonometric identities. We know that \(\sin^2\theta + \cos^2\theta = 1\). Multiply both sides of the equation we got in Step 2 by \(2\cos\theta + 3\sin\theta\): $$y(\sin^2\theta+\cos^2\theta) = 2\cos\theta + 3\sin\theta$$. Now, we will express \(\sin\theta\) and \(\cos\theta\) in terms of \(x\) and \(y\). We know that $$\cos\theta = \frac{x}{\sqrt{x^2 + y^2}}$$ and $$\sin\theta = \frac{y}{\sqrt{x^2 + y^2}}$$. Substitute these formulas into the above expression: $$y = 2\left(\frac{x}{\sqrt{x^2+y^2}}\right) + 3\left(\frac{y}{\sqrt{x^2+y^2}}\right)$$
04

Simplify the expression

Multiply both sides of the equation by \(\sqrt{x^2 + y^2}\) to simplify further: $$y\sqrt{x^2 + y^2} = 2x + 3y$$ Now, we can square both sides of the equation to eliminate the square root: $$(y\sqrt{x^2 + y^2})^2 = (2x + 3y)^2$$ $$y^2(x^2 + y^2) = 4x^2 + 12xy + 9y^2$$
05

Observe the resulting curve

Now that we have our Cartesian equation: $$y^2(x^2 + y^2) = 4x^2 + 12xy + 9y^2$$ Upon simplification, we get 3 terms containing \(x^2y^2\), \(x^2\), and \(xy\). Since there is no simple form we can reduce it to, we can't directly identify it as a specific type of curve like circle, ellipse, or parabola. However, we can observe that it is a curve that possesses both \(x\) and \(y\) terms, which suggests a complex interaction between the two variables.

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

Graph the following conic sections, labeling the vertices, foci, directrices, and asymptotes (if they exist). Use a graphing utility to check your work. $$r=\frac{6}{3+2 \sin \theta}$$

Find an equation of the following parabolas, assuming the vertex is at the origin. A parabola that opens downward with directrix \(y=6\)

Explain and carry out a method for graphing the curve \(x=1+\cos ^{2} y-\sin ^{2} y\) using parametric equations and a graphing utility.

Consider the curve \(r=f(\theta)=\cos \left(a^{\theta}\right)-1.5\) where \(a=(1+12 \pi)^{1 / 2 \pi} \approx 1.78933\) (see figure). a. Show that \(f(0)=f(2 \pi)\) and find the point on the curve that corresponds to \(\theta=0\) and \(\theta=2 \pi\) b. Is the same curve produced over the intervals \([-\pi, \pi]\) and \([0,2 \pi] ?\) c. Let \(f(\theta)=\cos \left(a^{\theta}\right)-b,\) where \(a=(1+2 k \pi)^{1 / 2 \pi}, k\) is an integer, and \(b\) is a real number. Show that \(f(0)=f(2 \pi)\) and that the curve closes on itself. d. Plot the curve with various values of \(k\). How many fingers can you produce?

Find an equation of the following curves, assuming the center is at the origin. Sketch a graph labeling the vertices, foci, asymptotes, and directrices. Use a graphing utility to check your work. A hyperbola with vertices (卤1,0) and eccentricity 3
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Logic and Functions

Read Explanation

Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Probability and Statistics

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.