Question

Sketch the curve in polar coordinates. $$ r=3 $$

   Sketch the curve in polar coordinates.
$$
r=3
$$

Calculus Early Transcendentals

Calculus Early Transcendentals

Howard Anton, Irl… 9th Edition

Chapter 10, Problem 23

Instant Answer

Solved by Expert Carson Merrill

12/11/2020

Step 1

In polar coordinates, $r$ represents the distance from the origin to a point in the plane. So, the equation $r=3$ means that all points are at a distance of 3 units from the origin. Show more…

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Sketch the curve in polar coordinates. $$ r=3 $$

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Key Concepts

Polar Coordinate System

The polar coordinate system represents points in the plane using a distance from a fixed reference point (the pole) and an angle from a fixed direction (the polar axis). This method of representation is often more suitable for curves that exhibit circular or rotational symmetry, as it simplifies the equation and the visualization of such curves.

Graphing Circles in Polar Coordinates

An equation of the form r = constant describes a circle in polar coordinates where every point on the circle is at the same distance from the origin. This direct relationship makes it easier to sketch the curve, as it represents a circle centered at the origin with the given constant as its radius.

Conversion to Cartesian Coordinates

To better understand the geometric nature of polar equations, they can be converted to Cartesian coordinates using the relationships x = r*cos(?) and y = r*sin(?). For example, converting r = 3 leads to the equation x² + y² = 9, which confirms that the graph is a circle with radius 3 centered at the origin.

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