Skip to main content
Ch. 11 - Parametric Equations and Polar Coordinates
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 11 - Parametric Equations and Polar CoordinatesProblem 11.3.48
Chapter 11, Problem 11.3.48

Polar to Cartesian Equations


Replace the polar equations in Exercises 27–52 with equivalent Cartesian equations. Then describe or identify the graph.


r = 3 cos θ

Verified step by step guidance
1
Recall the relationships between polar and Cartesian coordinates: \(x = r \cos \theta\) and \(y = r \sin \theta\), and also \(r^2 = x^2 + y^2\).
Start with the given polar equation: \(r = 3 \cos \theta\).
Multiply both sides of the equation by \(r\) to eliminate the \(\cos \theta\) term: \(r \cdot r = 3r \cos \theta\), which gives \(r^2 = 3r \cos \theta\).
Substitute the Cartesian equivalents: replace \(r^2\) with \(x^2 + y^2\) and \(r \cos \theta\) with \(x\), resulting in the equation \(x^2 + y^2 = 3x\).
Rewrite the equation in standard Cartesian form by moving all terms to one side: \(x^2 - 3x + y^2 = 0\). Then, complete the square for the \(x\) terms to identify the graph type.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
3m
Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Polar and Cartesian Coordinate Systems

Polar coordinates represent points using a radius and an angle (r, θ), while Cartesian coordinates use (x, y) positions on a plane. Understanding how these systems relate is essential for converting equations between them.
Recommended video:
05:32
Intro to Polar Coordinates

Conversion Formulas Between Polar and Cartesian Coordinates

The key formulas are x = r cos θ and y = r sin θ, with r = √(x² + y²) and θ = arctan(y/x). These allow rewriting polar equations in terms of x and y, enabling analysis in the Cartesian plane.
Recommended video:
05:32
Intro to Polar Coordinates

Graphing and Identifying Curves from Equations

After converting to Cartesian form, recognizing the type of curve (circle, line, etc.) involves analyzing the equation's structure. For example, r = 3 cos θ corresponds to a circle in Cartesian coordinates.
Recommended video:
06:38
Determining Concavity from the Graph of f
Related Practice
Textbook Question

Centroids


Find the coordinates of the centroid of the curve x = cos t, y = t + sin t, 0 ≤ t ≤ π.

1
views
Textbook Question

Finding Cartesian from Parametric Equations


In Exercises 19–24, match the parametric equations with the parametric curves labeled A through F.


x = cos t, y = sin 3t


1
views
Textbook Question

Tangent Lines to Parametrized Curves


In Exercises 1−14, find an equation for the line tangent to the curve at the point defined by the given value of t. Also, find the value of d²y/dx² at this point.


x = sec² t − 1, y = tan t, t = −π/4

1
views
Textbook Question

Parabolas


Exercises 9-16 give equations of parabolas. Find each parabola's focus and directrix. Then sketch the parabola. Include the focus and directrix in your sketch.


x² = 6y

Textbook Question

Shifting Conic Sections


You may wish to review Section 1.2 before solving Exercises 39-56.


Exercises 53-56 give equations for hyperbolas and tell how many units up or down and to the right or left each hyperbola is to be shifted. Find an equation for the new hyperbola, and find the new center, foci, vertices, and asymptotes.


x²/4 − y²/5 = 1, right 2, up 2

Textbook Question

Hyperbolas


Exercises 27-34 give equations for hyperbolas. Put each equation in standard form and find the hyperbola's asymptotes. Then sketch the hyperbola. Include the asymptotes and foci in your sketch.


8x² − 2y² = 16