Question

Identify and graph each polar equation. $$r=2-2 \cos \theta$$

   Identify and graph each polar equation.
$$r=2-2 \cos \theta$$

Precalculus

Michael Sullivan 9th Edition

Chapter 9, Problem 40

Instant Answer

Solved by Expert Foster Wisusik

07/05/2020

Step 1

The given equation is in the form of a cardioid, which is a heart-shaped curve. The general form of a cardioid is \(r=a+b\cos(\theta)\) or \(r=a+b\sin(\theta)\). In this case, \(a=2\) and \(b=-2\), and the function is cosine, so it is a horizontal cardioid. Show more…

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Identify and graph each polar equation. $$r=2-2 \cos \theta$$

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Key Concepts

Polar Coordinate System

The polar coordinate system represents points in the plane using a radius and an angle, which is particularly useful for expressing curves with rotational or circular symmetry. It provides an alternative to rectangular (Cartesian) coordinates, where a point is determined by how far it is from the origin and the angle measured from the positive x-axis.

Polar Equations

Polar equations define relationships between the radius r and the angle ?. Rather than expressing a curve in terms of x and y coordinates, these equations often simplify the representation of curves that are naturally circular or spiral, facilitating easier analysis and graphing.

Graphing Polar Equations

Graphing polar equations involves plotting points for various angles ? and corresponding radii r, identifying symmetries (such as symmetry about the polar axis, line ? = ?/2, or the pole), and then connecting the points to form the complete curve. This process often requires converting polar coordinates to Cartesian form for verification or analysis.

Cardioid

A cardioid is a specific type of curve that typically appears in polar equations of the form r = a ± a cos(?) or r = a ± a sin(?). It has a heart-like shape with a cusp at one end, and understanding its features is crucial for graphing and analyzing the behavior of curves that take on this form.

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